Turing instability in reaction-diffusion systems with nonlinear diffusion
Zemskov, E. P., E-mail: zemskov@ccas.ru [Russian Academy of Sciences, Dorodnicyn Computing Center (Russian Federation)
2013-10-15
The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.
Fluctuation in nonextensive reaction-diffusion systems
Wu Junlin; Chen Huaijun [Department of Physics, Shaanxi Normal University, Xian 710062 (China)
2007-05-15
The density fluctuation in a nonextensive reaction-diffusion system is investigated, where the nonequilibrium stationary-state distribution is described by the generalized Maxwell-Boltzmann distribution in the framework of Tsallis statistics (or nonextensive statistics). By using the density operator technique, the nonextensive pressure effect is introduced into the master equation and thus the generalized master equation is derived for the system. As an example, we take the{sup 3}He reaction-diffusion model inside stars to analyse the nonextensive effect on the density fluctuation and we find that the nonextensive parameter q different from one plays a very important role in determining the characteristics of the fluctuation waves.
Distribution in flowing reaction-diffusion systems
Kamimura, Atsushi
2009-12-28
A power-law distribution is found in the density profile of reacting systems A+B→C+D and 2A→2C under a flow in two and three dimensions. Different densities of reactants A and B are fixed at both ends. For the reaction A+B, the concentration of reactants asymptotically decay in space as x-1/2 and x-3/4 in two dimensions and three dimensions, respectively. For 2A, it decays as log (x) /x in two dimensions. The decay of A+B is explained considering the effect of segregation of reactants in the isotropic case. The decay for 2A is explained by the marginal behavior of two-dimensional diffusion. A logarithmic divergence of the diffusion constant with system size is found in two dimensions. © 2009 The American Physical Society.
Galactic disks as reaction-diffusion systems
Smolin, L
1996-01-01
A model of a galactic disk is presented which extends the homogeneous one zone models by incorporating propagation of material and energy in the disk. For reasonable values of the parameters the homogeneous steady state is unstable to the development of inhomogeneities, leading to the development of spatial and temporal structure. At the linearized level a prediction for the length and time scales of the patterns is found. These instabilities arise for the same reason that pattern formation is seen in non-equilibrium chemical and biological systems, which is that the positive and negative feedback effects which govern the rates of the critical processes act over different distance scales, as in Turing's reaction-diffusion models. This shows that patterns would form in the disk even in the absence of gravitational effects, density waves, rotation, shear and external perturbations. These nonlinear effects may thus explain the spiral structure seen in the star forming regions of isolated flocculent galaxies.
NONLINEAR SINGULARLY PERTURBED PREDATOR-PREY REACTION DIFFUSION SYSTEMS
MoJiaqi; TangRongrong
2004-01-01
A class of nonlinear predator-prey reaction diffusion systems for singularly perturbedproblems are considered. Under suitable conditions, by using theory of differential inequalitiesthe existence and asymptotic behavior of solution for initial boundary value problems arestudied.
Energy diffusion controlled reaction rate in dissipative Hamiltonian systems
Deng Mao-Lin; Zhu Wei-Qiu
2007-01-01
In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first-passage time (MFPT) of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kramers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.
Target Patterns in Reaction-Diffusion Systems,
1981-01-01
new variable xd, the diffusion matrix in (1.1) is just DM and the velocities Vph and v,, in (4.20) and (4.21) are multiplied by C-I/2, Finally except...the (ksr)-’ / factors in (4.4), (4.19), (5.1), and (5.2) are absent in one dimension. However, the analysis for three dimensions hinges on whether (4.2
Internal Stabilization of a Mutualistic Reaction Diffusion System
Wang Yuan DONG
2007-01-01
We study the internal stabilization of steady-state solutions to a 2-species mutualistic reaction diffusion system via finite-dimensional feedback controllers. Our main idea is to use differ- ent internal controllers to stabilize different steady-state solutions. The controllers are provided by considering LQ problems associated with the lineaxized systems at steady-state solutions.
Parametric spatiotemporal oscillation in reaction-diffusion systems.
Ghosh, Shyamolina; Ray, Deb Shankar
2016-03-01
We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions.
A Weak Comparison Principle for Reaction-Diffusion Systems
José Valero
2012-01-01
Full Text Available We prove a weak comparison principle for a reaction-diffusion system without uniqueness of solutions. We apply the abstract results to the Lotka-Volterra system with diffusion, a generalized logistic equation, and to a model of fractional-order chemical autocatalysis with decay. Moreover, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions L∞ is proved for at least one solution of the problem.
A weak comparison principle for reaction-diffusion systems
Valero, José
2012-01-01
In this paper we prove a weak comparison principle for a reaction-diffusion system without uniqueness of solutions. We apply the abstract results to the Lotka-Volterra system with diffusion, a generalized logistic equation and to a model of fractional-order chemical autocatalysis with decay. Morever, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions $L^{\\infty}$ is proved for at least one solution of the problem.
Controllability of Degenerating Reaction-Diffusion System in Electrocardiology
Bendahmane, Mostafa
2011-01-01
This paper is devoted to analyze the null controllability of a nonlinear reaction-diffusion system approximating a parabolic-elliptic system modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by a single control force acting on a subdomain is shown. The proof needs a precisely estimate with respect to the degenerating parameter and it is done combining Carleman estimates and energy inequalities.
Resonant Phase Patterns in a Reaction-Diffusion System
Lin, Anna L.; Bertram, Matthias; Martinez, Karl; Swinney, Harry L.; Ardelea, Alexandre; Carey, Graham F.
2000-05-01
Resonance regions similar to the Arnol'd tongues found in single oscillator frequency locking are observed in experiments using a spatially extended periodically forced Belousov-Zhabotinsky system. We identify six distinct 2:1 subharmonic resonant patterns and describe them in terms of the position-dependent phase and magnitude of the oscillations. Some experimentally observed features are also found in numerical studies of a forced Brusselator reaction-diffusion model. (c) 2000 The American Physical Society.
Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion.
Zemskov, Evgeny P; Tsyganov, Mikhail A; Horsthemke, Werner
2017-01-01
We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.
Global dynamics of a reaction-diffusion system
Yuncheng You
2011-02-01
Full Text Available In this work the existence of a global attractor for the semiflow of weak solutions of a two-cell Brusselator system is proved. The method of grouping estimation is exploited to deal with the challenge in proving the absorbing property and the asymptotic compactness of this type of coupled reaction-diffusion systems with cubic autocatalytic nonlinearity and linear coupling. It is proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite. Moreover, the existence of an exponential attractor for this solution semiflow is shown.
A discrete model to study reaction-diffusion-mechanics systems.
Weise, Louis D; Nash, Martyn P; Panfilov, Alexander V
2011-01-01
This article introduces a discrete reaction-diffusion-mechanics (dRDM) model to study the effects of deformation on reaction-diffusion (RD) processes. The dRDM framework employs a FitzHugh-Nagumo type RD model coupled to a mass-lattice model, that undergoes finite deformations. The dRDM model describes a material whose elastic properties are described by a generalized Hooke's law for finite deformations (Seth material). Numerically, the dRDM approach combines a finite difference approach for the RD equations with a Verlet integration scheme for the equations of the mass-lattice system. Using this framework results were reproduced on self-organized pacemaking activity that have been previously found with a continuous RD mechanics model. Mechanisms that determine the period of pacemakers and its dependency on the medium size are identified. Finally it is shown how the drift direction of pacemakers in RDM systems is related to the spatial distribution of deformation and curvature effects.
A discrete model to study reaction-diffusion-mechanics systems.
Louis D Weise
Full Text Available This article introduces a discrete reaction-diffusion-mechanics (dRDM model to study the effects of deformation on reaction-diffusion (RD processes. The dRDM framework employs a FitzHugh-Nagumo type RD model coupled to a mass-lattice model, that undergoes finite deformations. The dRDM model describes a material whose elastic properties are described by a generalized Hooke's law for finite deformations (Seth material. Numerically, the dRDM approach combines a finite difference approach for the RD equations with a Verlet integration scheme for the equations of the mass-lattice system. Using this framework results were reproduced on self-organized pacemaking activity that have been previously found with a continuous RD mechanics model. Mechanisms that determine the period of pacemakers and its dependency on the medium size are identified. Finally it is shown how the drift direction of pacemakers in RDM systems is related to the spatial distribution of deformation and curvature effects.
Distributed order reaction-diffusion systems associated with Caputo derivatives
Saxena, R. K.; Mathai, A. M.; Haubold, H. J.
2014-08-01
This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. ["The fundamental solution of the space-time fractional diffusion equation," Fractional Calculus Appl. Anal. 4, 153-202 (2001); Mainardi et al. "Fox H-functions in fractional diffusion," J. Comput. Appl. Math. 178, 321-331 (2005)] for the fundamental solution of the space-time fractional equation, including Haubold et al. ["Solutions of reaction-diffusion equations in terms of the H-function," Bull. Astron. Soc. India 35, 681-689 (2007)] and Saxena et al. ["Fractional reaction-diffusion equations," Astrophys. Space Sci. 305, 289-296 (2006a)] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion, and time-fractional diffusion, see Schneider and Wyss ["Fractional diffusion and wave equations," J. Math. Phys. 30, 134-144 (1989)]. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables
Theory and application of stability for stochastic reaction diffusion systems
LUO Qi; DENG FeiQi; MAO XueRong; BAO JunDong; ZHANG YuTian
2008-01-01
So far, the Lyapunov direct method is still the moat effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding Ito formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the Ito stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probablity, end exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results ob-tained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.
Distributed order reaction-diffusion systems associated with Caputo derivatives
Saxena, R K; Haubold, H J
2011-01-01
This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. [23,24], for the fundamental solution of the space-time fractional equation, including Haubold et al. [13] and Saxena et al. [38] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion...
Cross-diffusional effect in a telegraph reaction diffusion Lotka-Volterra two competitive system
Abdusalam, H.A E-mail: hosny@operamail.com; Fahmy, E.S
2003-10-01
It is known now that, telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion in several branches of sciences. Telegraph reaction diffusion Lotka-Volterra two competitive system is considered. We observed that this system can give rise to diffusive instability only in the presence of cross-diffusion. Local and global stability analysis in the cross-diffusional effect are studied by considering suitable Lyapunov functional.
Self-similar fast-reaction limits for reaction-diffusion systems on unbounded domains
Crooks, E. C. M.; Hilhorst, D.
2016-08-01
We present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit k → ∞ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed k, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.
Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces.
Dhillon, Daljit Singh J; Milinkovitch, Michel C; Zwicker, Matthias
2017-04-01
In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction-diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace-Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.
Salem Abdelmalek
2014-11-01
Full Text Available In this article we construct the invariant regions for m-component reaction-diffusion systems with a tridiagonal symmetric Toeplitz matrix of diffusion coefficients and with nonhomogeneous boundary conditions. We establish the existence of global solutions, and use Lyapunov functional methods. The nonlinear reaction term is assumed to be of polynomial growth.
Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix
Hugo Leiva
2009-01-01
Full Text Available We prove the interior approximate controllability for the following 2×2 reaction-diffusion system with cross-diffusion matrix ut=aΔu−β(−Δ1/2u+bΔv+1ωf1(t,x in (0,τ×Ω, vt=cΔu−dΔv−β(−Δ1/2v+1ωf2(t,x in (0,τ×Ω, u=v=0, on (0,T×∂Ω, u(0,x=u0(x, v(0,x=v0(x, x∈Ω, where Ω is a bounded domain in ℝN (N≥1, u0,v0∈L2(Ω, the 2×2 diffusion matrix D=[abcd] has semisimple and positive eigenvalues 0<ρ1≤ρ2, β is an arbitrary constant, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, and the distributed controls f1,f2∈L2([0,τ];L2(Ω. Specifically, we prove the following statement: if λ11/2ρ1+β>0 (where λ1 is the first eigenvalue of −Δ, then for all τ>0 and all open nonempty subset ω of Ω the system is approximately controllable on [0,τ].
Accelerated stochastic and hybrid methods for spatial simulations of reaction-diffusion systems
Rossinelli, D; Bayati, B; Koumoutsakos, P.
2008-01-01
Spatial distributions characterize the evolution of reaction-diffusion models of several physical, chemical, and biological systems. We present two novel algorithms for the efficient simulation of these models: Spatial т-Leaping (Sт -Leaping), employing a unified acceleration of the stochastic simulation of reaction and diffusion, and Hybrid т-Leaping (Hт-Leaping), combining a deterministic diffusion approximation with a т-Leaping acceleration of the stochastic reactions. The algorithms are v...
Estimation and prediction of convection-diffusion-reaction systems from point measurement
Vries, D.
2008-01-01
Different procedures with respect to estimation and prediction of systems characterized by convection, diffusion and reactions on the basis of point measurement data, have been studied. Two applications of these convection-diffusion-reaction (CDR) systems have been used as a case study of the propos
Spiral and Antispiral Waves in Reaction-Diffusion Systems
LIUYu-Fang; WUYan-Ning; XUHou-Ju; SUNJin-Feng
2004-01-01
Spiral waves are ubiquitous phenomena in nonlinear chemical, physical, and biological systems. But antispiral waves are infrequent to date. The transition between spiral and antispiral waves has been rarely explored. We have analyzed the extended Brusselator model and the extended Oregonator model by linear stability analysis. We have demonstrated that it is possible and plausible to realize the transition between them by control of diffusion coefficient of inactivator from theoretical analysis and numerical simulations.
Traveling wave solutions for reaction-diffusion systems
Pedersen, Michael; Lin, Zhigui; Tian, Canrong
2010-01-01
This paper is concerned with traveling waves of reaction–diffusion systems. The definition of coupled quasi-upper and quasi-lower solutions is introduced for systems with mixed quasimonotone functions, and the definition of ordered quasi-upper and quasi-lower solutions is also given for systems...... with quasimonotone nondecreasing functions. By the monotone iteration method, it is shown that if the system has a pair of coupled quasi-upper and quasi-lower solutions, then there exists at least a traveling wave solution. Moreover, if the system has a pair of ordered quasi-upper and quasi-lower solutions......, then there exists at least a traveling wavefront. As an application we consider the delayed system of a mutualistic model....
An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems.
Chiu, Chichia; Yu, Jui-Ling
2007-04-01
Reaction-diffusion-chemotaxis systems have proven to be fairly accurate mathematical models for many pattern formation problems in chemistry and biology. These systems are important for computer simulations of patterns, parameter estimations as well as analysis of the biological systems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. In this paper, a general reaction-diffusion-chemotaxis system is considered for specific numerical issues of pattern simulations. We propose a fully explicit discretization combined with a variable optimal time step strategy for solving the reaction-diffusion-chemotaxis system. Theorems about stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results on a model problem are given for comparison with other numerical methods. Simulations on two real biological experiments will also be shown.
Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems
Wang, Haiyan
2010-01-01
Much has been studied on the spreading speed and traveling wave solutions for cooperative reaction-diffusion systems. In this paper, we shall establish the spreading speed for a large class of non-cooperative reaction-diffusion systems and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. As an application, our results are applied to a partially cooperative system describing interactions between ungulates and grass.
Scaling of morphogenetic patterns in reaction-diffusion systems.
Rasolonjanahary, Manan'Iarivo; Vasiev, Bakhtier
2016-09-07
Development of multicellular organisms is commonly associated with the response of individual cells to concentrations of chemical substances called morphogens. Concentration fields of morphogens form a basis for biological patterning and ensure its properties including ability to scale with the size of the organism. While mechanisms underlying the formation of morphogen gradients are reasonably well understood, little is known about processes responsible for their scaling. Here, we perform a formal analysis of scaling for chemical patterns forming in continuous systems. We introduce a quantity representing the sensitivity of systems to changes in their size and use it to analyse scaling properties of patterns forming in a few different systems. Particularly, we consider how scaling properties of morphogen gradients forming in diffusion-decay systems depend on boundary conditions and how the scaling can be improved by passive modulation of morphogens or active transport in the system. We also analyse scaling of morphogenetic signal caused by two opposing gradients and consider scaling properties of patterns forming in activator-inhibitor systems. We conclude with a few possible mechanisms which allow scaling of morphogenetic patterns.
Nonlinear predator-prey singularly perturbed Robin Problems for reaction diffusion systems
莫嘉琪; 韩祥临
2003-01-01
The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.
莫嘉琪
2003-01-01
The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.
Jia-qi Mo; Wan-tao Lin
2006-01-01
In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of the solution for the problem are studied.
Fedotov, Sergei
1998-10-01
An asymptotic method is presented for the analysis of the traveling waves in the one-dimensional reaction-diffusion system with the diffusion with a finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics. The analysis makes use of the path-integral approach, scaling procedure, and the singular perturbation techniques involving the large deviations theory for the Poisson random walk. The exact formula for the position and speed of reaction front is derived. It is found that the reaction front dynamics is formally associated with the relativistic Hamiltonian/Lagrangian mechanics.
Cherniha, Roman
2017-01-01
This book presents several fundamental results in solving nonlinear reaction-diffusion equations and systems using symmetry-based methods. Reaction-diffusion systems are fundamental modeling tools for mathematical biology with applications to ecology, population dynamics, pattern formation, morphogenesis, enzymatic reactions and chemotaxis. The book discusses the properties of nonlinear reaction-diffusion systems, which are relevant for biological applications, from the symmetry point of view, providing rigorous definitions and constructive algorithms to search for conditional symmetry (a nontrivial generalization of the well-known Lie symmetry) of nonlinear reaction-diffusion systems. In order to present applications to population dynamics, it focuses mainly on two- and three-component diffusive Lotka-Volterra systems. While it is primarily a valuable guide for researchers working with reaction-diffusion systems and those developing the theoretical aspects of conditional symmetry conception,...
Reaction-Diffusion Systems: Front Propagation and Spatial Structures
Cencini, Massimo; Lopez, Cristobal; Vergni, Davide
After the pioneering works of Kolmogorov, Petrovskii and Piskunov [1] and Fisher [2] in 1937 on the nonlinear diffusion equation and its traveling wave solutions, scientists from many different disciplines have been captivated by questions about structure, formation and dynamics of patterns in reactive media. Combustion, spreading of epidemics, diffusive transport of chemicals in cells and population dynamics are just a few examples bearing witness of the influence of those works in different areas of modern science.
Stability Analysis of a Reaction-Diffusion System Modeling Atherogenesis
Ibragimov, Akif
2010-01-01
This paper presents a linear, asymptotic stability analysis for a reaction-diffusionconvection system modeling atherogenesis, the initiation of atherosclerosis, as an inflammatory instability. Motivated by the disease paradigm articulated by Ross, atherogenesis is viewed as an inflammatory spiral with a positive feedback loop involving key cellular and chemical species interacting and reacting within the intimal layer of muscular arteries. The inflammatory spiral is initiated as an instability from a healthy state which is defined to be an equilibrium state devoid of certain key inflammatory markers. Disease initiation is studied through a linear, asymptotic stability analysis of a healthy equilibrium state. Various theorems are proved, giving conditions on system parameters guaranteeing stability of the health state, and a general framework is developed for constructing perturbations from a healthy state that exhibit blow-up, which are interpreted as corresponding to disease initiation. The analysis reveals key features that arterial geometry, antioxidant levels, and the source of inflammatory components (through coupled third-kind boundary conditions or through body sources) play in disease initiation. © 2010 Society for Industrial and Applied Mathematics.
Accelerated stochastic and hybrid methods for spatial simulations of reaction diffusion systems
Rossinelli, Diego; Bayati, Basil; Koumoutsakos, Petros
2008-01-01
Spatial distributions characterize the evolution of reaction-diffusion models of several physical, chemical, and biological systems. We present two novel algorithms for the efficient simulation of these models: Spatial τ-Leaping ( Sτ-Leaping), employing a unified acceleration of the stochastic simulation of reaction and diffusion, and Hybrid τ-Leaping ( Hτ-Leaping), combining a deterministic diffusion approximation with a τ-Leaping acceleration of the stochastic reactions. The algorithms are validated by solving Fisher's equation and used to explore the role of the number of particles in pattern formation. The results indicate that the present algorithms have a nearly constant time complexity with respect to the number of events (reaction and diffusion), unlike the exact stochastic simulation algorithm which scales linearly.
Instability and pattern formation in reaction-diffusion systems: a higher order analysis.
Riaz, Syed Shahed; Sharma, Rahul; Bhattacharyya, S P; Ray, D S
2007-08-14
We analyze the condition for instability and pattern formation in reaction-diffusion systems beyond the usual linear regime. The approach is based on taking into account perturbations of higher orders. Our analysis reveals that nonlinearity present in the system can be instrumental in determining the stability of a system, even to the extent of destabilizing one in a linearly stable parameter regime. The analysis is also successful to account for the observed effect of additive noise in modifying the instability threshold of a system. The analytical study is corroborated by numerical simulation in a standard reaction-diffusion system.
Concentration fluctuations in non-isothermal reaction-diffusion systems. II. The nonlinear case
Bedeaux, D.; Ortiz de Zárate, J.M.; Pagonabarraga, I.; Sengers, J.V.; Kjelstrup, S.
2011-01-01
In this paper, we consider a simple reaction-diffusion system, namely, a binary fluid mixture with an association-dissociation reaction between two species. We study fluctuations at hydrodynamic spatiotemporal scales when this mixture is driven out of equilibrium by the presence of a temperature gra
Wang, Jing [Iowa State Univ., Ames, IA (United States)
2013-01-11
We analyze the spatiotemporal behavior of species concentrations in a diffusion-mediated conversion reaction which occurs at catalytic sites within linear pores of nanometer diameter. A strict single-file (no passing) constraint occurs in the diffusion within such narrow pores. Both transient and steady-state behavior is precisely characterized by kinetic Monte Carlo simulations of a spatially discrete lattice–gas model for this reaction–diffusion process considering various distributions of catalytic sites. Exact hierarchical master equations can also be developed for this model. Their analysis, after application of mean-field type truncation approximations, produces discrete reaction–diffusion type equations (mf-RDE). For slowly varying concentrations, we further develop coarse-grained continuum hydrodynamic reaction–diffusion equations (h-RDE) incorporating a precise treatment of single-file diffusion (SFD) in this multispecies system. Noting the shortcomings of mf-RDE and h-RDE, we then develop a generalized hydrodynamic (GH) formulation of appropriate gh-RDE which incorporates an unconventional description of chemical diffusion in mixed-component quasi-single-file systems based on a refined picture of tracer diffusion for finite-length pores. The gh-RDE elucidate the non-exponential decay of the steady-state reactant concentration into the pore and the non-mean-field scaling of the reactant penetration depth. Then an extended model of a catalytic conversion reaction within a functionalized nanoporous material is developed to assess the effect of varying the reaction product – pore interior interaction from attractive to repulsive. The analysis is performed utilizing the generalized hydrodynamic formulation of the reaction-diffusion equations which can reliably capture the complex interplay between reaction and restricted transport for both irreversible and reversible reactions.
Sample Duplication Method for Monte Carlo Simulation of Large Reaction-Diffusion System
张红东; 陆建明; 杨玉良
1994-01-01
The sample duplication method for the Monte Carlo simulation of large reaction-diffusion system is proposed in this paper. It is proved that the sample duplication method will effectively raise the efficiency and statistical precision of the simulation without changing the kinetic behaviour of the reaction-diffusion system and the critical condition for the bifurcation of the steady-states. The method has been applied to the simulation of spatial and time dissipative structure of Brusselator under the Dirichlet boundary condition. The results presented in this paper definitely show that the sample duplication method provides a very efficient way to sol-’e the master equation of large reaction-diffusion system. For the case of two-dimensional system, it is found that the computation time is reduced at least by a factor of two orders of magnitude compared to the algorithm reported in literature.
Dynamical Behavior of Core 3 He Nuclear Reaction-Diffusion Systems and Sun's Gravitational Field
DU Jiulin; SHEN Hong
2005-01-01
The coupling of the sun's gravitational field with processes of diffusion and convection exerts a significant influence on the dynamical behavior of the core 3He nuclear reaction-diffusion system. Stability analyses of the system are made in this paper by using the theory of nonequilibrium dynamics. It is showed that, in the nuclear reaction regions extending from the center to about 0.38 times of the radius of the sun, the gravitational field enables the core 3He nuclear reaction-diffusion system to become unstable and, after the instability, new states to appear in the system have characteristic of time oscillation. This may change the production rates of both 7Be and 8B neutrinos.
Turing bifurcation in a reaction-diffusion system with density-dependent dispersal
Kumar, Niraj; Horsthemke, Werner
2010-05-01
Motivated by the recent finding [N. Kumar, G.M. Viswanathan, V.M. Kenkre, Physica A 388 (2009) 3687] that the dynamics of particles undergoing density-dependent nonlinear diffusion shows sub-diffusive behaviour, we study the Turing bifurcation in a two-variable system with this kind of dispersal. We perform a linear stability analysis of the uniform steady state to find the conditions for the Turing bifurcation and compare it with the standard Turing condition in a reaction-diffusion system, where dispersal is described by simple Fickian diffusion. While activator-inhibitor kinetics are a necessary condition for the Turing instability as in standard two-variable systems, the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator. We apply these results to two model systems, the Brusselator and the Gierer-Meinhardt model.
Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains
Madzvamuse, Anotida
2009-08-29
By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth. © Springer-Verlag 2009.
The entropy dissipation method for spatially inhomogeneous reaction-diffusion-type systems
Di Francesco, M.
2008-12-08
We study the long-time asymptotics of reaction-diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion-convection equations, and the main goal of this paper is to study its generalization to systems of partial differential equations that contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of a reaction-diffusion-convection system arising in solid-state physics as a paradigm for general nonlinear systems. © 2008 The Royal Society.
Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach
Kim, Changho; Nonaka, Andy; Bell, John B.; Garcia, Alejandro L.; Donev, Aleksandar
2017-03-01
We develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuating hydrodynamics (FHD). For hydrodynamic systems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poisson fluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamic fluctuations to the formation of a two
Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems
Keldermann, R. H.; Nash, M. P.; Panfilov, A. V.
2009-06-01
In many practically important cases, wave propagation described by the reaction-diffusion equation initiates deformation of the medium. Mathematically, such processes are described by coupled reaction-diffusion-mechanics (RDM) systems. RDM systems were recently used to study the effects of deformation on wave propagation in cardiac tissue, so called mechano-electrical feedback (MEF). In this article, we review the results of some of these studies, in particular those relating to the effects of deformation on pacemaker activity and spiral wave dynamics in the heart. We also provide brief descriptions of the numerical methods used, and the underlying cardiac physiology.
Relation between the complex Ginzburg-Landau equation and reaction-diffusion System
Shao Xin; Ren Yi; Ouyang Qi
2006-01-01
The complex Ginzburg-Landau equation(CGLE)has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation,and is not valid when a RD system is away from the onset.To test this,we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding CGLE.Numerical simulations confirm that the CGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.
von Kameke, A.; Huhn, F.; Muñuzuri, A. P.; Pérez-Muñuzuri, V.
2013-02-01
In the absence of advection, reaction-diffusion systems are able to organize into spatiotemporal patterns, in particular spiral and target waves. Whenever advection is present that can be parametrized in terms of effective or turbulent diffusion D*, these patterns should be attainable on a much greater, boosted length scale. However, so far, experimental evidence of these boosted patterns in a turbulent flow was lacking. Here, we report the first experimental observation of boosted target and spiral patterns in an excitable chemical reaction in a quasi-two-dimensional turbulent flow. The wave patterns observed are ˜50 times larger than in the case of molecular diffusion only. We vary the turbulent diffusion coefficient D* of the flow and find that the fundamental Fisher-Kolmogorov-Petrovsky-Piskunov equation, vf∝D*, for the asymptotic speed of a reactive wave remains valid. However, not all measures of the boosted wave scale with D* as expected from molecular diffusion, since the wave fronts turn out to be highly filamentous.
Travelling Wave Solutions in Delayed Reaction Diffusion Systems with Partial Monotonicity
Jian-hua Huang; Xing-fu Zou
2006-01-01
This paper deals with the existence of travelling wave fronts of delayed reaction diffusion systems with partial quasi-monotonicity. We propose a concept of "desirable pair of upper-lower solutions", through which a subset can be constructed. We then apply the Schauder's fixed point theorem to some appropriate operator in this subset to obtain the existence of the travelling wave fronts.
Singh, Jagdev; Rashidi, M. M.; Kumar, Devendra; Swroop, Ram
2016-12-01
In this paper, we study a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions with time fractional Caputo derivative. The present article involves a more generalized effective approach, proposed for the Brusselator system say q-homotopy analysis transform method (q-HATM), providing the family of series solutions with nonlocal generalized effects. The convergence of the q-HATM series solution is adjusted and controlled by auxiliary parameter ℏ and asymptotic parameter n. The numerical results are demonstrated graphically. The outcomes of the study show that the q-HATM is computationally very effective and accurate to analyze nonlinear fractional differential equations.
Dierckx, Hans; Bernus, Olivier; Verschelde, Henri
2011-09-02
The dependency of wave velocity in reaction-diffusion (RD) systems on the local front curvature determines not only the stability of wave propagation, but also the fundamental properties of other spatial configurations such as vortices. This Letter gives the first derivation of a covariant eikonal-curvature relation applicable to general RD systems with spatially varying anisotropic diffusion properties, such as cardiac tissue. The theoretical prediction that waves which seem planar can nevertheless possess a nonvanishing geometrical curvature induced by local anisotropy is confirmed by numerical simulations, which reveal deviations up to 20% from the nominal plane wave speed.
Partitioned coupling of advection-diffusion-reaction systems and Brinkman flows
Lenarda, Pietro; Paggi, Marco; Ruiz Baier, Ricardo
2017-09-01
We present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection-diffusion-reaction systems and incompressible, viscous flow. The space discretisation of the governing equations is based on mixed finite element methods defined on unstructured meshes, whereas the time integration hinges on an operator splitting strategy that exploits the differences in scales between the reaction, advection, and diffusion processes, considering the global system as a number of sequentially linked sets of partial differential, and algebraic equations. The flow solver presents the advantage that all unknowns in the system (here vorticity, velocity, and pressure) can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective. The robustness of the proposed method is illustrated with a series of numerical tests in 2D and 3D, relevant in the modelling of bacterial bioconvection and Boussinesq systems.
Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System
Qianiqian Zheng
2017-01-01
Full Text Available Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS, Stochastic Reaction-Diffusion System (SRDS is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.
Dichotomous-noise-induced pattern formation in a reaction-diffusion system
Das, Debojyoti; Ray, Deb Shankar
2013-06-01
We consider a generic reaction-diffusion system in which one of the parameters is subjected to dichotomous noise by controlling the flow of one of the reacting species in a continuous-flow-stirred-tank reactor (CSTR) -membrane reactor. The linear stability analysis in an extended phase space is carried out by invoking Furutzu-Novikov procedure for exponentially correlated multiplicative noise to derive the instability condition in the plane of the noise parameters (correlation time and strength of the noise). We demonstrate that depending on the correlation time an optimal strength of noise governs the self-organization. Our theoretical analysis is corroborated by numerical simulations on pattern formation in a chlorine-dioxide-iodine-malonic acid reaction-diffusion system.
Hybrid stochastic simulation of reaction-diffusion systems with slow and fast dynamics
Strehl, Robert; Ilie, Silvana, E-mail: silvana@ryerson.ca [Department of Mathematics, Ryerson University, Toronto, Ontario M5B 2K3 (Canada)
2015-12-21
In this paper, we present a novel hybrid method to simulate discrete stochastic reaction-diffusion models arising in biochemical signaling pathways. We study moderately stiff systems, for which we can partition each reaction or diffusion channel into either a slow or fast subset, based on its propensity. Numerical approaches missing this distinction are often limited with respect to computational run time or approximation quality. We design an approximate scheme that remedies these pitfalls by using a new blending strategy of the well-established inhomogeneous stochastic simulation algorithm and the tau-leaping simulation method. The advantages of our hybrid simulation algorithm are demonstrated on three benchmarking systems, with special focus on approximation accuracy and efficiency.
ASYMPTOTIC SOLUTION OF ACTIVATOR INHIBITOR SYSTEMS FOR NONLINEAR REACTION DIFFUSION EQUATIONS
Jiaqi MO; Wantao LIN
2008-01-01
A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.
Generalized monotone method and numerical approach for coupled reaction diffusion systems
Sowmya, M.; Vatsala, Aghalaya S.
2017-01-01
Study of coupled reaction diffusion systems are very useful in various branches of science and engineering. In this paper, we provide a methodology to construct the solution for the coupled reaction diffusion systems, with initial and boundary conditions, where the forcing function is the sum of an increasing and decreasing function. It is known that the generalized monotone method coupled with coupled lower and upper solutions yield monotone sequences which converges uniformly and monotonically to coupled minimal and maximal solutions. In addition, the interval of existence is guaranteed by the lower and upper solutions, which are relatively easy to compute. Using the lower and upper solutions as the initial approximation, we develop a method to compute the sequence of coupled lower and upper solutions on the interval or on the desired interval of existence. Further, if the uniqueness conditions are satisfied, the coupled minimal and maximal solutions converge to the unique solution of the reaction diffusion systems. We will provide some numerical results as an application of our numerical methodology.
Wave reflection in a reaction-diffusion system: breathing patterns and attenuation of the echo.
Tsyganov, M A; Ivanitsky, G R; Zemskov, E P
2014-05-01
Formation and interaction of the one-dimensional excitation waves in a reaction-diffusion system with the piecewise linear reaction functions of the Tonnelier-Gerstner type are studied. We show that there exists a parameter region where the established regime of wave propagation depends on initial conditions. Wave phenomena with a complex behavior are found: (i) the reflection of waves at a growing distance (the remote reflection) upon their collision with each other or with no-flux boundaries and (ii) the periodic transformation of waves with the jumping from one regime of wave propagation to another (the periodic trigger wave).
An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems
Padgett, Jill M. A.; Ilie, Silvana, E-mail: silvana@ryerson.ca [Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3 (Canada)
2016-03-15
Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.
An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems
Padgett, Jill M. A.; Ilie, Silvana
2016-03-01
Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.
Szalai, István; Cuiñas, Daniel; Takács, Nándor; Horváth, Judit; De Kepper, Patrick
2012-08-06
In his seminal 1952 paper, Alan Turing predicted that diffusion could spontaneously drive an initially uniform solution of reacting chemicals to develop stable spatially periodic concentration patterns. It took nearly 40 years before the first two unquestionable experimental demonstrations of such reaction-diffusion patterns could be made in isothermal single phase reaction systems. The number of these examples stagnated for nearly 20 years. We recently proposed a design method that made their number increase to six in less than 3 years. In this report, we formally justify our original semi-empirical method and support the approach with numerical simulations based on a simple but realistic kinetic model. To retain a number of basic properties of real spatial reactors but keep calculations to a minimal complexity, we introduce a new way to collapse the confined spatial direction of these reactors. Contrary to similar reduced descriptions, we take into account the effect of the geometric size in the confinement direction and the influence of the differences in the diffusion coefficient on exchange rates of species with their feed environment. We experimentally support the method by the observation of stationary patterns in red-ox reactions not based on oxihalogen chemistry. Emphasis is also brought on how one of these new systems can process different initial conditions and memorize them in the form of localized patterns of different geometries.
Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK.
Wang, Kaier; Steyn-Ross, Moira L; Steyn-Ross, D Alistair; Wilson, Marcus T; Sleigh, Jamie W; Shiraishi, Yoichi
2014-04-11
Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system's set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This "code-based" approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution. As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations. The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming
Judd, S L; Judd, Stephen L.; Silber, Mary
1998-01-01
This paper investigates the competition between both simple (e.g. stripes, hexagons) and ``superlattice'' (super squares, super hexagons) Turing patterns in two-component reaction-diffusion systems. ``Superlattice'' patterns are formed from eight or twelve Fourier modes, and feature structure at two different length scales. Using perturbation theory, we derive simple analytical expressions for the bifurcation equation coefficients on both rhombic and hexagonal lattices. These expressions show that, no matter how complicated the reaction kinectics, the nonlinear reaction terms reduce to just four effective terms within the bifurcation equation coefficients. Moreover, at the hexagonal degeneracy -- when the quadratic term in the hexagonal bifurcation equation disappears -- the number of effective system parameters drops to two, allowing a complete characterization of the possible bifurcation results at this degeneracy. The general results are then applied to specific model equations, to investigate the stabilit...
Breathing spiral waves in the chlorine dioxide-iodine-malonic acid reaction-diffusion system
Berenstein, Igal; Muñuzuri, Alberto P.; Yang, Lingfa; Dolnik, Milos; Zhabotinsky, Anatol M.; Epstein, Irving R.
2008-08-01
Breathing spiral waves are observed in the oscillatory chlorine dioxide-iodine-malonic acid reaction-diffusion system. The breathing develops within established patterns of multiple spiral waves after the concentration of polyvinyl alcohol in the feeding chamber of a continuously fed, unstirred reactor is increased. The breathing period is determined by the period of bulk oscillations in the feeding chamber. Similar behavior is obtained in the Lengyel-Epstein model of this system, where small amplitude parametric forcing of spiral waves near the spiral wave frequency leads to the formation of breathing spiral waves in which the period of breathing is equal to the period of forcing.
Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion
Fatima, Tasnim; Ptashnyk, Mariya
2011-01-01
We use the periodic unfolding technique to derive corrector estimates for a reaction-diffusion system describing concrete corrosion penetration in the sewer pipes. The system, defined in a periodically-perforated domain, is semi-linear, partially dissipative, and coupled via a non-linear ordinary differential equation posed on the solid-water interface at the pore level. After discussing the solvability of the pore scale model, we apply the periodic unfolding techniques (adapted to treat the presence of perforations) not only to get upscaled model equations, but also to prepare a proper framework for getting a convergence rate (corrector estimates) of the averaging procedure.
A Reaction-diffusion System with Nonlinear Absorption Terms and Boundary Flux
2008-01-01
This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.
Velikanov, M V; Velikanov, Mikhail V.; Kapral, Raymond
1998-01-01
Spatially-distributed, nonequilibrium chemical systems described by a Markov chain model are considered. The evolution of such systems arises from a combination of local birth-death reactive events and random walks executed by the particles on a lattice. The parameter \\gamma, the ratio of characteristic time scales of reaction and diffusion, is used to gauge the relative contributions of these two processes to the overall dynamics. For the case of relatively fast diffusion, i.e. \\gamma 0 these memory terms vanish and the mass-action law is recovered; b) the memory kernel is found to assume a simple exponential form. A comparison with numerical results from lattice gas automaton simulations is also carried out.
Global existence and asymptotic stability of equilibria to reaction-diffusion systems
Wang, Rong-Nian; Tang, Zhong Wei
2009-06-01
In this paper, we study weakly coupled reaction-diffusion systems in unbounded domains of {\\bb R}^2 or {\\bb R}^3 , where the reaction terms are sums of quasimonotone nondecreasing and nonincreasing functions. Such systems are more complicated than those in many previous publications and little is known about them. A comparison principle and global existence, and boundedness theorems for solutions to these systems are established. Sufficient conditions on the nonlinearities, ensuring the positively Ljapunov stability of the zero solution with respect to H2-perturbations, are also obtained. As samples of applications, these results are applied to an autocatalytic chemical model and a concrete problem, whose nonlinearities are nonquasimonotone. Our results are novel. In particular, we present a solution to an open problem posed by Escher and Yin (2005 J. Nonlinear Anal. Theory Methods Appl. 60 1065-84).
Yamada, H; Ito, M
1998-01-01
The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on the basis of spatio-temporal pattern formation by local contraction-oscillators. This biological system can be regarded as a reaction-diffusion system which has spatial interaction by active flow of protoplasmic sol in the cell. Paying attention to the physiological evidence that the flow is determined by contraction pattern in the plasmodium, a reaction-diffusion system having self-determined flow arises. Such a coupling of reaction-diffusion-advection is a characteristic of the biological system, and is expected to relate with control mechanism of amoeboid behaviours. Hence, we have studied effects of the self-determined flow on pattern formation of simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial solution, the envelope dynamics follows the complex Ginzburg-Landau type equation just after bifurcation occurs at finite wave number. The flow term affects the nonlinear term of the equation through the critic...
Ducrot, Arnaud; Giletti, Thomas
2014-09-01
In this work we study the asymptotic behaviour of the Kermack-McKendrick reaction-diffusion system in a periodic environment with non-diffusive susceptible population. This problem was proposed by Kallen et al. as a model for the spatial spread for epidemics, where it can be reasonable to assume that the susceptible population is motionless. For arbitrary dimensional space we prove that large classes of solutions of such a system have an asymptotic spreading speed in large time, and that the infected population has some pulse-like asymptotic shape. The analysis of the one-dimensional problem is more developed, as we are able to uncover a much more accurate description of the profile of solutions. Indeed, we will see that, for some initially compactly supported infected population, the profile of the solution converges to some pulsating travelling wave with minimal speed, that is to some entire solution moving at a constant positive speed and whose profile's shape is periodic in time.
Critical behavior in reaction-diffusion systems exhibiting absorbing phase transition
Ódor, G
2003-01-01
Phase transitions of reaction-diffusion systems with site occupation restriction and with particle creation that requires n>1 parents and where explicit diffusion of single particles (A) exists are reviewed. Arguments based on mean-field approximation and simulations are given which support novel kind of non-equilibrium criticality. These are in contradiction with the implications of a suggested phenomenological, multiplicative noise Langevin equation approach and with some of recent numerical analysis. Simulation results for the one and two dimensional binary spreading 2A -> 4A, 4A -> 2A model display a new type of mean-field criticality characterized by alpha=1/3 and beta=1/2 critical exponents suggested in cond-mat/0210615.
Complex patterns in reaction-diffusion systems a tale of two front instabilities
Hagberg, A; Aric Hagberg; Ehud Meron
1994-01-01
Two front instabilities in a reaction-diffusion system are shown to lead to the formation of complex patterns. The first is an instability to transverse modulations that drives the formation of labyrinthine patterns. The second is a Nonequilibrium Ising-Bloch (NIB) bifurcation that renders a stationary planar front unstable and gives rise to a pair of counterpropagating fronts. Near the NIB bifurcation the relation of the front velocity to curvature is highly nonlinear and transitions between counterpropagating fronts become feasible. Nonuniformly curved fronts may undergo local front transitions that nucleate spiral-vortex pairs. These nucleation events provide the ingredient needed to initiate spot splitting and spiral turbulence. Similar spatio-temporal processes have been observed recently in the ferrocyanide-iodate-sulfite reaction.
Scenarios of domain pattern formation in a reaction-diffusion system
Muratov, C B
1996-01-01
We performed an extensive numerical study of a two-dimensional reaction-diffusion system of the activator-inhibitor type in which domain patterns can form. We showed that both multidomain and labyrinthine patterns may form spontaneously as a result of Turing instability. In the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus. Depending on the parameters and the excitation level of the system stripes, spots, wriggled stripes, or labyrinthine patterns form. The labyrinthine patterns may be both connected and disconnected. In the the stable homogeneous system with the slow inhibitor one can excite self-replicating spots, breathing patterns, autowaves and turbulence. The parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101 (1996). The dynamics of the patterns observed i...
A Lagrangian particle method for reaction-diffusion systems on deforming surfaces.
Bergdorf, Michael; Sbalzarini, Ivo F; Koumoutsakos, Petros
2010-11-01
Reaction-diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction-diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction-diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction-diffusion equations on complex and deforming geometries.
Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment
Fort, Joaquim [Departament de Fisica, Universitat de Girona, Girona (Spain)]. E-mail: joaquim.fort@udg.es; Mendez, Vicenc [Facultat de Ciencies de la Salut, Universitat Internacional de Catalunya, Ssant Cugat del Valles (Spain)]. E-mail: vmendez@csc.unica.edu
2002-06-01
We review the recent theoretical progress in the formulation and solution of the front speed problem for time-delayed reaction-diffusion systems. Most of the review is focused on hyperbolic equations. They have been widely used in recent years, because they allow for analytical solutions and yield a realistic description of some relevant phenomena. The theoretical methods are applied to a range of applications, including population dynamics, forest fire models, bistable systems and combustion wavefronts. We also present a detailed account of successful predictions of the models, as assessed by comparison to experimental data for some biophysical systems, without making use of any free parameters. Areas where the work reviewed may contribute to future progress are discussed. (author)
KPP reaction-diffusion system with a nonlinear loss inside a cylinder
Giletti, Thomas
2010-09-01
We consider in this paper a reaction-diffusion system in the presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a nonlinear space-dependent loss term inside the domain. In particular, we will point out the existence of a minimal speed, above which any real value is an admissible speed. We will also give some spreading results for initial conditions decaying exponentially at infinity.
Fu, Jin; Wu, Sheng; Li, Hong; Petzold, Linda R.
2014-10-01
The inhomogeneous stochastic simulation algorithm (ISSA) is a fundamental method for spatial stochastic simulation. However, when diffusion events occur more frequently than reaction events, simulating the diffusion events by ISSA is quite costly. To reduce this cost, we propose to use the time dependent propensity function in each step. In this way we can avoid simulating individual diffusion events, and use the time interval between two adjacent reaction events as the simulation stepsize. We demonstrate that the new algorithm can achieve orders of magnitude efficiency gains over widely-used exact algorithms, scales well with increasing grid resolution, and maintains a high level of accuracy.
Tulzer, Gerhard; Heitzinger, Clemens
2016-04-22
In this work, we develop a 2D algorithm for stochastic reaction-diffusion systems describing the binding and unbinding of target molecules at the surfaces of affinity-based sensors. In particular, we simulate the detection of DNA oligomers using silicon-nanowire field-effect biosensors. Since these devices are uniform along the nanowire, two dimensions are sufficient to capture the kinetic effects features. The model combines a stochastic ordinary differential equation for the binding and unbinding of target molecules as well as a diffusion equation for their transport in the liquid. A Brownian-motion based algorithm simulates the diffusion process, which is linked to a stochastic-simulation algorithm for association at and dissociation from the surface. The simulation data show that the shape of the cross section of the sensor yields areas with significantly different target-molecule coverage. Different initial conditions are investigated as well in order to aid rational sensor design. A comparison of the association/hybridization behavior for different receptor densities allows optimization of the functionalization setup depending on the target-molecule density.
Brownian-motion based simulation of stochastic reaction-diffusion systems for affinity based sensors
Tulzer, Gerhard; Heitzinger, Clemens
2016-04-01
In this work, we develop a 2D algorithm for stochastic reaction-diffusion systems describing the binding and unbinding of target molecules at the surfaces of affinity-based sensors. In particular, we simulate the detection of DNA oligomers using silicon-nanowire field-effect biosensors. Since these devices are uniform along the nanowire, two dimensions are sufficient to capture the kinetic effects features. The model combines a stochastic ordinary differential equation for the binding and unbinding of target molecules as well as a diffusion equation for their transport in the liquid. A Brownian-motion based algorithm simulates the diffusion process, which is linked to a stochastic-simulation algorithm for association at and dissociation from the surface. The simulation data show that the shape of the cross section of the sensor yields areas with significantly different target-molecule coverage. Different initial conditions are investigated as well in order to aid rational sensor design. A comparison of the association/hybridization behavior for different receptor densities allows optimization of the functionalization setup depending on the target-molecule density.
Spatiotemporal patterns in reaction-diffusion system and in a vibrated granular bed
Swinney, H.L.; Lee, K.J.; McCormick, W.D. [Univ. of Texas, Austin, TX (United States)
1995-12-31
Experiments on a quasi-two-dimensional reaction-diffusion system reveal transitions from a uniform state to stationary hexagonal, striped, and rhombic spatial patterns. For other reactor conditions lamellae and self-replicating spot patterns are observed. These patterns form in continuously fed thin gel reactors that can be maintained indefinitely in well-defined nonequilibrium states. Reaction-diffusion models with two chemical species yield patterns similar to those observed in the experiments. Pattern formation is also being examined in vertically oscillated thin granular layers (typically 3-30 particle diameters deep). For small acceleration amplitudes, a granular layer is flat, but above a well-defined critical acceleration amplitude, spatial patterns spontaneously form. Disordered time-dependent granular patterns are observed as well as regular patterns of squares, stripes, and hexagons. A one-dimensional model consisting of a completely inelastic ball colliding with a sinusoidally oscillating platform provides a semi-quantitative description of most of the observed bifurcations between the different spatiotemporal regimes.
Nicola Ernesto M
2010-11-01
Full Text Available Abstract Background A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze. Results Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC, can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations, but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations. Conclusions This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward
Existence of global solutions to reaction-diffusion systems via a Lyapunov functional
Said Kouachi
2001-10-01
Full Text Available The purpose of this paper is to construct polynomial functionals (according to solutions of the coupled reaction-diffusion equations which give $L^{p}$-bounds for solutions. When the reaction terms are sufficiently regular, using the well known regularizing effect, we deduce the existence of global solutions. These functionals are obtained independently of work done by Malham and Xin [11].
Vorticity field, helicity integral and persistence of entanglement in reaction-diffusion systems
Trueba, J L; Arrayas, M [Area de Electromagnetismo, Universidad Rey Juan Carlos, Camino del Molino s/n, 28943 Fuenlabrada, Madrid (Spain)
2009-07-17
We show that a global description of the stability of entangled structures in reaction-diffusion systems can be made by means of a helicity integral. A vorticity vector field is defined for these systems, as in electromagnetism or fluid dynamics. We have found under which conditions the helicity is conserved or lost through the boundaries of the medium, so the entanglement of structures observed is preserved or disappears during time evolution. We illustrate the theory with an example of knotted entanglement in a FitzHugh-Nagumo model. For this model, we introduce new non-trivial initial conditions using the Hopf fibration and follow the time evolution of the entanglement. (fast track communication)
An observer for an occluded reaction-diffusion system with spatially varying parameters
Kramer, Sean; Bollt, Erik M.
2017-03-01
Spatially dependent parameters of a two-component chaotic reaction-diffusion partial differential equation (PDE) model describing ocean ecology are observed by sampling a single species. We estimate the model parameters and the other species in the system by autosynchronization, where quantities of interest are evolved according to misfit between model and observations, to only partially observed data. Our motivating example comes from oceanic ecology as viewed by remote sensing data, but where noisy occluded data are realized in the form of cloud cover. We demonstrate a method to learn a large-scale coupled synchronizing system that represents the spatio-temporal dynamics and apply a network approach to analyze manifold stability.
Turing pattern formation in the chlorine dioxide-iodine- malonic acid reaction-diffusion system
Setayeshgar, Sima
The formation of localized structures in the chlorine dioxide-idodine-malonic acid (CDIMA) reaction-diffusion system is investigated numerically using a realistic model of this system. We analyze the one-dimensional patterns formed along the gradients imposed by boundary feeds, and study their linear stability to symmetry- breaking perturbations (the Turing instability) in the plane transverse to these gradients. We establish that an often-invoked simple local linear analysis which neglects longitudinal diffusion is inappropriate for predicting the linear stability of these patterns. Using a fully nonuniform analysis, we investigate the structure of the patterns formed along the gradients and their stability to transverse Turing pattern formation as a function of the values of two control parameters: the malonic acid feed concentration and the size of the reactor in the dimension along the gradients. The results from this investigation are compared with existing experimental results. We also verify that the two-variable reduction of the chemical model employed in the linear stability analysis is justified. Finally, we present numerical solution of the CDIMA system in two dimensions which is in qualitative agreement with experiments. This result also confirms our linear stability analysis, while demonstrating the feasibility of numerical exploration of realistic chemical models.
Pattern formation in reaction-diffusion systems: From spiral waves to turbulence
Davidsen, Joern
2009-05-01
Almost all systems we encounter in nature possess some sort of form or structure. In many cases, the structures arise from an initially unstructured state without the action of an agent that predetermines the pattern. Such self-organized structures emerge from cooperative interactions among the constituents of the system and often exhibit properties that are distinct from those of their constituent elements or molecules. For example, chemical waves in reaction-diffusion systems are at the core of a huge variety of physical, chemical, and biological processes. In (quasi) two-dimensional situations, spiral wave patterns are especially prevalent and determine the characteristics of processes such as surface catalytic oxidation reactions, contraction of the heart muscle, and various signaling mechanisms in biological systems. In this talk, I will review and discuss recent theoretical and experimental results regarding the dynamics, properties and stability of spiral waves and their three-dimensional analog (scroll waves). Special emphasis will be given to synchronization defect lines which generically arise in complex-oscillatory media, and the phenomenon of defect-mediated turbulence or filament turbulence where the dynamics of a pattern is dominated by the rapid motion, nucleation, and annihilation of spirals or scroll waves, respectively. The latter is of direct relevance in the context of ventricular fibrillation - a turbulent electrical wave activity that destroys the coherent contraction of the ventricular muscle and its main pumping function leading to sudden cardiac death.
Lueptow, Richard M.; Schlick, Conor P.; Umbanhowar, Paul B.; Ottino, Julio M.
2013-11-01
We investigate chaotic advection and diffusion in competitive autocatalytic reactions. To study this subject, we use a computationally efficient method for solving advection-reaction-diffusion equations for periodic flows using a mapping method with operator splitting. In competitive autocatalytic reactions, there are two species, B and C, which both react autocatalytically with species A (A +B -->2B and A +C -->2C). If there is initially a small amount of spatially localized B and C and a large amount of A, all three species will be advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that the small scale interactions associated with the chaotic velocity field, specifically the local finite-time Lyapunov exponents (FTLEs), can accurately predict the final average concentrations of B and C after the reaction is complete. The species, B or C, that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If species B and C start in regions having similar FTLEs, their average concentrations at the end of the reaction will also be similar. Funded by NSF Grant CMMI-1000469.
Drift and breakup of spiral waves in reaction-diffusion-mechanics systems.
Panfilov, A V; Keldermann, R H; Nash, M P
2007-05-08
Rotating spiral waves organize excitation in various biological, physical, and chemical systems. They underpin a variety of important phenomena, such as cardiac arrhythmias, morphogenesis processes, and spatial patterns in chemical reactions. Important insights into spiral wave dynamics have been obtained from theoretical studies of the reaction-diffusion (RD) partial differential equations. However, most of these studies have ignored the fact that spiral wave rotation is often accompanied by substantial deformations of the medium. Here, we show that joint consideration of the RD equations with the equations of continuum mechanics for tissue deformations (RD-mechanics systems), yield important effects on spiral wave dynamics. We show that deformation can induce the breakup of spiral waves into complex spatiotemporal patterns. We also show that mechanics leads to spiral wave drift throughout the medium approaching dynamical attractors, which are determined by the parameters of the model and the size of the medium. We study mechanisms of these effects and discuss their applicability to the theory of cardiac arrhythmias. Overall, we demonstrate the importance of RD-mechanics systems for mathematics applied to life sciences.
Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system.
Muhammad Abbas
Full Text Available In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann method. The accuracy of the proposed scheme is demonstrated by applying it on a test problem. The performance of this scheme is shown by computing L∞ and L2 error norms for different time levels. The numerical results are found to be in good agreement with known exact solutions.
Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system.
Abbas, Muhammad; Majid, Ahmad Abd; Md Ismail, Ahmad Izani; Rashid, Abdur
2014-01-01
In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann method. The accuracy of the proposed scheme is demonstrated by applying it on a test problem. The performance of this scheme is shown by computing L∞ and L2 error norms for different time levels. The numerical results are found to be in good agreement with known exact solutions.
STEPS: modeling and simulating complex reaction-diffusion systems with Python
Stefan Wils
2009-06-01
Full Text Available We describe how the use of the Python language improved the user interface of the program STEPS. STEPS is a simulation platform for modeling and stochastic simulation of coupled reaction-diffusion systems with complex 3-dimensional boundary conditions. Setting up such models is a complicated process that consists of many phases. Initial versions of STEPS relied on a static input format that did not cleanly separate these phases, limiting modelers in how they could control the simulation and becoming increasingly complex as new features and new simulation algorithms were added. We solved all of these problems by tightly integrating STEPS with Python, using SWIG to expose our existing simulation code.
Minimal wave speed for a class of non-cooperative diffusion-reaction system
Zhang, Tianran; Wang, Wendi; Wang, Kaifa
2016-02-01
In this paper, we consider a class of non-cooperative diffusion-reaction systems, which include prey-predator models and disease-transmission models. The concept of weak traveling wave solutions is proposed. The necessary and sufficient conditions for the existence of such solutions are obtained by the Schauder's fixed-point theorem and persistence theory. The introduction of persistence theory is very technical and crucial. The LaSalle's invariance principle is applied to show that traveling wave solutions connect two equilibria. The nonexistence of traveling wave solutions is proved by introducing a negative one-sided Laplace transform. The results are applied to a prey-predator model and a disease-transmission model with specific interaction functions. We find that the profile of traveling wave solutions may depend on different eigenvalues according to the corresponding condition, which is a new phenomenon.
Budroni, M. A.; De Wit, A.
2016-06-01
When two solutions containing separate reactants A and B of an oscillating reaction are put in contact in a gel, localized spatiotemporal patterns can develop around the contact zone thanks to the interplay of reaction and diffusion processes. Using the Brusselator model, we explore analytically the deployment in space and time of the bifurcation diagram of such an A +B → oscillator system. We provide a parametric classification of possible instabilities as a function of the ratio of the initial reactant concentrations and of the reaction intermediate species diffusion coefficients. Related one-dimensional reaction-diffusion dynamics are studied numerically. We find that the system can spatially localize waves and Turing patterns as well as induce more complex dynamics such as zigzag spatiotemporal waves when Hopf and Turing modes interact.
Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems
Woolley, Thomas E.
2012-05-22
Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases. © 2012 American Physical Society.
Yu, Pengfei; Wang, Hailong; Chen, Jianyong; Shen, Shengping
2017-07-01
In this study, the conservation laws οf dissipative mechanical-diffusion-electrochemical reaction system are systematically obtained based on Noether's theorem. According to linear, irreversible thermodynamics, dissipative phenomena can be described by an irreversible force and an irreversible flow. Additionally, the Lagrange function, L and the generalized Hamilton least-action principle are proposed to be used to obtain the conservation integrals. A group of these integrals, including the J-, M-, and L-integrals, can be then obtained using the classical Noether approach for dissipative processes. The relation between the J-integral and the energy release rate is illustrated. The path-independence of the J-integral is then proven. The J-integral, derived based on Noether's theorem, is a line integral, contrary to the propositions of existing published works that describe it both as a line and an area integral. Herein, we prove that the outcomes are identical, and identify the physical meaning of the area integral, a concept that was not explained previously. To show that the J-integral can dominate the distribution of the corresponding field quantities, an example of a partial, stress-diffusion coupling process is disscussed.
Bursting regimes in a reaction-diffusion system with action potential-dependent equilibrium.
Stephen R Meier
Full Text Available The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed.
Convective wave front locking for a reaction-diffusion system in a conical flow reactor
Kuptsov, P.V.; Kuznetsov, S.P.; Knudsen, Carsten
2002-01-01
We consider reaction-diffusion instabilities in a flow reactor whose cross-section slowly expands with increasing longitudinal coordinate (cone shaped reactor). Due to deceleration of the flow in this reactor, the instability is convective near the inlet to the reactor and absolute at the downstr...
Weakly nonlinear dynamics in reaction-diffusion systems with Levy flights
Nec, Y; Nepomnyashchy, A A [Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000 (Israel); Golovin, A A [Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208 (United States)], E-mail: flyby@techunix.technion.ac.il
2008-12-15
Reaction-diffusion equations with a fractional Laplacian are reduced near a long wave Hopf bifurcation. The obtained amplitude equation is shown to be the complex Ginzburg-Landau equation with a fractional Laplacian. Some of the properties of the normal complex Ginzburg-Landau equation are generalized for the fractional analogue. In particular, an analogue of the Kuramoto-Sivashinsky equation is derived.
Parallel Solutions for Voxel-Based Simulations of Reaction-Diffusion Systems
Daniele D’Agostino
2014-01-01
Full Text Available There is an increasing awareness of the pivotal role of noise in biochemical processes and of the effect of molecular crowding on the dynamics of biochemical systems. This necessity has given rise to a strong need for suitable and sophisticated algorithms for the simulation of biological phenomena taking into account both spatial effects and noise. However, the high computational effort characterizing simulation approaches, coupled with the necessity to simulate the models several times to achieve statistically relevant information on the model behaviours, makes such kind of algorithms very time-consuming for studying real systems. So far, different parallelization approaches have been deployed to reduce the computational time required to simulate the temporal dynamics of biochemical systems using stochastic algorithms. In this work we discuss these aspects for the spatial TAU-leaping in crowded compartments (STAUCC simulator, a voxel-based method for the stochastic simulation of reaction-diffusion processes which relies on the Sτ-DPP algorithm. In particular we present how the characteristics of the algorithm can be exploited for an effective parallelization on the present heterogeneous HPC architectures.
Parallel solutions for voxel-based simulations of reaction-diffusion systems.
D'Agostino, Daniele; Pasquale, Giulia; Clematis, Andrea; Maj, Carlo; Mosca, Ettore; Milanesi, Luciano; Merelli, Ivan
2014-01-01
There is an increasing awareness of the pivotal role of noise in biochemical processes and of the effect of molecular crowding on the dynamics of biochemical systems. This necessity has given rise to a strong need for suitable and sophisticated algorithms for the simulation of biological phenomena taking into account both spatial effects and noise. However, the high computational effort characterizing simulation approaches, coupled with the necessity to simulate the models several times to achieve statistically relevant information on the model behaviours, makes such kind of algorithms very time-consuming for studying real systems. So far, different parallelization approaches have been deployed to reduce the computational time required to simulate the temporal dynamics of biochemical systems using stochastic algorithms. In this work we discuss these aspects for the spatial TAU-leaping in crowded compartments (STAUCC) simulator, a voxel-based method for the stochastic simulation of reaction-diffusion processes which relies on the Sτ-DPP algorithm. In particular we present how the characteristics of the algorithm can be exploited for an effective parallelization on the present heterogeneous HPC architectures.
Deriving reaction-diffusion models in ecology from interacting particle systems.
Cantrell, R S; Cosner, C
2004-02-01
We use a scaling procedure based on averaging Poisson distributed random variables to derive population level models from local models of interactions between individuals. The procedure is suggested by using the idea of hydrodynamic limits to derive reaction-diffusion models for population interactions from interacting particle systems. The scaling procedure is formal in the sense that we do not address the issue of proving that it converges; instead we focus on methods for computing the results of the scaling or deriving properties of rescaled systems. To that end we treat the scaling procedure as a transform, in analogy with the Laplace or Fourier transform, and derive operational formulas to aid in the computation of rescaled systems or the derivation of their properties. Since the limiting procedure is adapted from work by Durrett and Levin, we refer to the transform as the Durrett-Levin transform. We examine the effects of rescaling in various standard models, including Lotka-Volterra models, Holling type predator-prey models, and ratio-dependent models. The effects of scaling are mostly quantitative in models with smooth interaction terms, but ratio-dependent models are profoundly affected by the scaling. The scaling transforms ratio-dependent terms that are singular at the origin into smooth terms. Removing the singularity at the origin eliminates some of the unique dynamics that can arise in ratio-dependent models.
Parallel Solutions for Voxel-Based Simulations of Reaction-Diffusion Systems
D'Agostino, Daniele; Pasquale, Giulia; Clematis, Andrea; Maj, Carlo; Mosca, Ettore; Milanesi, Luciano; Merelli, Ivan
2014-01-01
There is an increasing awareness of the pivotal role of noise in biochemical processes and of the effect of molecular crowding on the dynamics of biochemical systems. This necessity has given rise to a strong need for suitable and sophisticated algorithms for the simulation of biological phenomena taking into account both spatial effects and noise. However, the high computational effort characterizing simulation approaches, coupled with the necessity to simulate the models several times to achieve statistically relevant information on the model behaviours, makes such kind of algorithms very time-consuming for studying real systems. So far, different parallelization approaches have been deployed to reduce the computational time required to simulate the temporal dynamics of biochemical systems using stochastic algorithms. In this work we discuss these aspects for the spatial TAU-leaping in crowded compartments (STAUCC) simulator, a voxel-based method for the stochastic simulation of reaction-diffusion processes which relies on the Sτ-DPP algorithm. In particular we present how the characteristics of the algorithm can be exploited for an effective parallelization on the present heterogeneous HPC architectures. PMID:25045716
李新政; 白占国; 李燕; 贺亚峰; 赵昆
2015-01-01
The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have infl uences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will con-vert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.
Sigaut, Lorena; Villarruel, Cecilia; Ponce, María Laura; Ponce Dawson, Silvina
2017-06-01
Many cell signaling pathways involve the diffusion of messengers that bind and unbind to and from intracellular components. Quantifying their net transport rate under different conditions then requires having separate estimates of their free diffusion coefficient and binding or unbinding rates. In this paper, we show how performing sets of fluorescence correlation spectroscopy (FCS) experiments under different conditions, it is possible to quantify free diffusion coefficients and on and off rates of reaction-diffusion systems. We develop the theory and present a practical implementation for the case of the universal second messenger, calcium (Ca2 +) and single-wavelength dyes that increase their fluorescence upon Ca2 + binding. We validate the approach with experiments performed in aqueous solutions containing Ca2 + and Fluo4 dextran (both in its high and low affinity versions). Performing FCS experiments with tetramethylrhodamine-dextran in Xenopus laevis oocytes, we infer the corresponding free diffusion coefficients in the cytosol of these cells. Our approach can be extended to other physiologically relevant reaction-diffusion systems to quantify biophysical parameters that determine the dynamics of various variables of interest.
Diffusion and reaction in crowded environments
EcheverIa, Carlos [Laboratorio de Fisica Aplicada y Computacional, Departamento de Matematica y Fisica, Universidad Nacional Experimental del Tachira, San Cristobal 5001 (Venezuela); Tucci, Kay [Centro de Fisica Fundamental, Universidad de Los Andes, Merida 5101 (Venezuela); Kapral, Raymond [Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON, M5S 3H6 (Canada)
2007-02-14
The effects of molecular crowding on small molecule diffusion and chemical reaction rate coefficients are investigated. The systems considered comprise a random distribution of stationary spherical obstacles occupying a volume fraction {phi} of the system and a large number of small molecules whose dynamics are followed. Chemical reactions are studied in such crowded systems where, in addition to the obstacles, a large reactive sphere C is present that catalyses the reaction A+C {yields} B+C. Using a mesoscopic description of the dynamics employing multiparticle collisions among the small molecules, the {phi} dependence of the diffusion and reaction rate coefficients is computed. Both the diffusion and reaction rate coefficients decrease with increase of the obstacle volume fraction as expected but variations of these quantities with {phi} are not predicted by simple models of the dynamics.
Finite Travelling Waves for a Semilinear Degenerate Reaction-Diffusion System
Shu WANG; Cheng Fu WANG; Dang LUO
2001-01-01
In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion systemis studied, where 0 ＜αi ＜ 1, mij ≥ 0 and ∑nj=1mij ＞ 0, i,j = 1,.. N. Necessary and sufficientconditions on existence and large time behaviours of FTWs of (I) are obtained by using the matrixtheory, Schauder's fixed point theorem, and upper and lower solutions method.
Ram K. Saxena
2015-04-01
Full Text Available This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampé de Fériet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given.
M.H. Tiwana
2017-04-01
Full Text Available This work investigates the fractional non linear reaction diffusion (FNRD system of Lotka-Volterra type. The system of equations together with the boundary conditions are solved by Homotopy perturbation transform method (HPTM. The series solutions are obtained for the two cases (homogeneous and non-homogeneous of FNRD system. The effect of fractional parameter on the mass concentration of two species are shown and discussed with the help of 3D graphs.
Reaction-diffusion systems in natural sciences and new technology transfer
Keller, André A.
2012-12-01
Diffusion mechanisms in natural sciences and innovation management involve partial differential equations (PDEs). This is due to their spatio-temporal dimensions. Functional semi-discretized PDEs (with lattice spatial structures or time delays) may be even more adapted to real world problems. In the modeling process, PDEs can also formalize behaviors, such as the logistic growth of populations with migration, and the adopters’ dynamics of new products in innovation models. In biology, these events are related to variations in the environment, population densities and overcrowding, migration and spreading of humans, animals, plants and other cells and organisms. In chemical reactions, molecules of different species interact locally and diffuse. In the management of new technologies, the diffusion processes of innovations in the marketplace (e.g., the mobile phone) are a major subject. These innovation diffusion models refer mainly to epidemic models. This contribution introduces that modeling process by using PDEs and reviews the essential features of the dynamics and control in biological, chemical and new technology transfer. This paper is essentially user-oriented with basic nonlinear evolution equations, delay PDEs, several analytical and numerical methods for solving, different solutions, and with the use of mathematical packages, notebooks and codes. The computations are carried out by using the software Wolfram Mathematica®7, and C++ codes.
Cluster geometry and survival probability in systems driven by reaction-diffusion dynamics
Windus, Alastair; Jensen, Henrik J [The Institute for Mathematical Sciences, 53 Prince' s Gate, South Kensington, London SW7 2PG (United Kingdom)], E-mail: h.jensen@imperial.ac.uk
2008-11-15
We consider a reaction-diffusion model incorporating the reactions A{yields}{phi}, A{yields}2A and 2A{yields}3A. Depending on the relative rates for sexual and asexual reproduction of the quantity A, the model exhibits either a continuous or first-order absorbing phase transition to an extinct state. A tricritical point separates the two phase lines. While we comment on this critical behaviour, the main focus of the paper is on the geometry of the population clusters that form. We observe the different cluster structures that arise at criticality for the three different types of critical behaviour and show that there exists a linear relationship for the survival probability against initial cluster size at the tricritical point only.
Cluster geometry and survival probability in systems driven by reaction diffusion dynamics
Windus, Alastair; Jensen, Henrik J.
2008-11-01
We consider a reaction-diffusion model incorporating the reactions A→phi, A→2A and 2A→3A. Depending on the relative rates for sexual and asexual reproduction of the quantity A, the model exhibits either a continuous or first-order absorbing phase transition to an extinct state. A tricritical point separates the two phase lines. While we comment on this critical behaviour, the main focus of the paper is on the geometry of the population clusters that form. We observe the different cluster structures that arise at criticality for the three different types of critical behaviour and show that there exists a linear relationship for the survival probability against initial cluster size at the tricritical point only.
Resonance Reaction in Diffusion-Influenced Bimolecular Reactions
Kolb, Jakob J; Dzubiella, Joachim
2016-01-01
We investigate the influence of a stochastically fluctuating step-barrier potential on bimolecular reaction rates by exact analytical theory and stochastic simulations. We demonstrate that the system exhibits a new resonant reaction behavior with rate enhancement if an appropriately defined fluctuation decay length is of the order of the system size. Importantly, we find that in the proximity of resonance the standard reciprocal additivity law for diffusion and surface reaction rates is violated due to the dynamical coupling of multiple kinetic processes. Together, these findings may have important repercussions on the correct interpretation of various kinetic reaction problems in complex systems, as, e.g., in biomolecular association or catalysis.
Reaction rates for reaction-diffusion kinetics on unstructured meshes
Hellander, Stefan; Petzold, Linda
2017-02-01
The reaction-diffusion master equation is a stochastic model often utilized in the study of biochemical reaction networks in living cells. It is applied when the spatial distribution of molecules is important to the dynamics of the system. A viable approach to resolve the complex geometry of cells accurately is to discretize space with an unstructured mesh. Diffusion is modeled as discrete jumps between nodes on the mesh, and the diffusion jump rates can be obtained through a discretization of the diffusion equation on the mesh. Reactions can occur when molecules occupy the same voxel. In this paper, we develop a method for computing accurate reaction rates between molecules occupying the same voxel in an unstructured mesh. For large voxels, these rates are known to be well approximated by the reaction rates derived by Collins and Kimball, but as the mesh is refined, no analytical expression for the rates exists. We reduce the problem of computing accurate reaction rates to a pure preprocessing step, depending only on the mesh and not on the model parameters, and we devise an efficient numerical scheme to estimate them to high accuracy. We show in several numerical examples that as we refine the mesh, the results obtained with the reaction-diffusion master equation approach those of a more fine-grained Smoluchowski particle-tracking model.
Simultaneous and non-simultaneous blow-up and uniform blow-up profiles for reaction-diffusion system
Zhengqiu Ling
2012-11-01
Full Text Available This article concerns the blow-up solutions of a reaction-diffusion system with nonlocal sources, subject to the homogeneous Dirichlet boundary conditions. The criteria used to identify simultaneous and non-simultaneous blow-up of solutions by using the parameters p and q in the model are proposed. Also, the uniform blow-up profiles in the interior domain are established.
Cherniha, Roman
2010-01-01
New definitions of $Q$-conditional symmetry for systems of PDEs are presented, which generalize the standard notation of non-classical (conditional) symmetry. It is shown that different types of $Q$-conditional symmetry of a system generate a hierarchy of conditional symmetry operators. A class of two-component nonlinear RD systems is examined to demonstrate applicability of the definitions proposed and it is shown when different definitions of $Q$-conditional symmetry lead to the same operators.
Reaction diffusion in the NiCrAl and CoCrAl systems
Levine, S. R.
1978-01-01
The paper assesses the effect of overlay coating and substrate composition on the kinetics of coating depletion by interdiffusion. This is accomplished by examining the constitution, kinetics and activation energies for a series of diffusion couples primarily of the NiCrAl/Ni-10Cr or CoCrAl/Ni-10Cr type annealed at temperatures in the range 1000-1205 C for times up to 500 hr. A general procedure is developed for analyzing diffusion in multicomponent multiphase systems. It is shown that by introducing the concept of beta-source strength, which can be determined from appropriate phase diagrams, the Wagner solution for consumption of a second phase in a semiinfinite couple is successfully applied to the analysis of MCrAl couples. Thus, correlation of beta-recession rate constants with couple composition, total and diffusional activation energies, and interdiffusion coefficients are determined.
Bifurcation and pattern formation in a coupled higher autocatalator reaction diffusion system
无
2007-01-01
Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst,are discussed.The stability of the unique homogeneous steady state is obtained by the linearized theory.A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given.Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory.Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.
Hamiltonian perspective on compartmental reaction-diffusion networks
Seslija, Marko; van der Schaft, Arjan; Scherpen, Jacquelien M. A.
2014-01-01
Inspired by the recent developments in modeling and analysis of reaction networks, we provide a geometric formulation of the reversible reaction networks under the influence of diffusion. Using the graph knowledge of the underlying reaction network, the obtained reaction diffusion system is a distri
Reaction-Diffusion Automata Phenomenology, Localisations, Computation
Adamatzky, Andrew
2013-01-01
Reaction-diffusion and excitable media are amongst most intriguing substrates. Despite apparent simplicity of the physical processes involved the media exhibit a wide range of amazing patterns: from target and spiral waves to travelling localisations and stationary breathing patterns. These media are at the heart of most natural processes, including morphogenesis of living beings, geological formations, nervous and muscular activity, and socio-economic developments. This book explores a minimalist paradigm of studying reaction-diffusion and excitable media using locally-connected networks of finite-state machines: cellular automata and automata on proximity graphs. Cellular automata are marvellous objects per se because they show us how to generate and manage complexity using very simple rules of dynamical transitions. When combined with the reaction-diffusion paradigm the cellular automata become an essential user-friendly tool for modelling natural systems and designing future and emergent computing arch...
Walker, Christoph
2010-01-01
The paper focuses on positive solutions to a coupled system of parabolic equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. By using global bifurcation techniques, we describe the structure of the set of positive solutions with respect to two parameters measuring the intensities of the fertility of the species. In particular, we establish co-existence steady-states, i.e. solutions which are nonnegative and nontrivial in both components.
Dynamical Analysis of a Delayed Reaction-Diffusion Predator-Prey System
Yanuo Zhu
2012-01-01
Full Text Available This work deals with the analysis of a delayed diffusive predator-prey system under Neumann boundary conditions. The dynamics are investigated in terms of the stability of the nonnegative equilibria and the existence of Hopf bifurcation by analyzing the characteristic equations. The direction of Hopf bifurcation and the stability of bifurcating periodic solution are also discussed by employing the normal form theory and the center manifold reduction. Furthermore, we prove that the positive equilibrium is asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than the critical value.
Diffusion and Surface Reaction in Heterogeneous Catalysis
Baiker, A.; Richarz, W.
1978-01-01
Ethylene hydrogenation on a platinum catalyst, electrolytically applied to a tube wall, is a good system for the study of the interactions between diffusion and surface reaction in heterogeneous catalysis. Theoretical background, apparatus, procedure, and student performance of this experiment are discussed. (BB)
Reaction phases and diffusion paths in SiC/metal systems
Naka, M.; Fukai, T. [Osaka Univ., Osaka (Japan); Schuster, J.C. [Vienna Univ., Vienna (Austria)
2004-07-01
The interface structures between SiC and metal are reviewed at SiC/metal systems. Metal groups are divided to carbide forming metals and non-carbide forming metals. Carbide forming metals form metal carbide granular or zone at metal side, and metal silicide zone at SiC side. The further diffusion of Si and C from SiC causes the formation of T ternary phase depending metal. Non-carbide forming metals form silicide zone containing graphite or the layered structure of metal silicide and metal silicide containing graphite. The diffusion path between SiC and metal are formed along tie-lines connecting SiC and metal on the corresponding ternary Si-C-M system. The reactivity of metals is dominated by the forming ability of carbide or silicide. Te reactivity tendency of elements are discussed on the periodical table of elements, and Ti among elements shows the highest reactivity among carbide forming metals. For non-carbide forming metals the reactivity sequence of metals is Fe>Ni>Co. (orig.)
Pattern formation in a two-component reaction-diffusion system with delayed processes on a network
Petit, Julien; Asllani, Malbor; Fanelli, Duccio; Lauwens, Ben; Carletti, Timoteo
2016-11-01
Reaction-diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator-inhibitor variant without delay. Numerical results gained from the Mimura-Murray model support the theoretical approach.
Basavaraja, C.; Park, Do Young; Yamaguchi, Tomohiko; Huh, Do Sung
2007-06-01
Periodic waves were studied via the process of wet stamping, in a heterogeneous media soaked in different solutions of the Belousov-Zhabotinsky (BZ) reaction system containing ferroin and malonic acid. The experiments were performed by making an initial contact of soaked polyacrylamide (PAA) gel with agarose stamp for a short period of time, and then placing the gel onto the glass plate after exposing both of its surfaces. This led to the development of propagating waves on the surface of the gel. The results show that patterns have a dependence on the concentration of individual species, which are represented in the form of bifurcation diagrams. The behavioral trend of this system is also compared with the current perspectives of the reaction diffusion system.
Tezuka, Ken-ichi; Wada, Yoshitaka; Takahashi, Akiyuki; Kikuchi, Masanori
2005-01-01
Bone is a complex system with functions including those of adaptation and repair. To understand how bone cells can create a structure adapted to the mechanical environment, we propose a simple bone remodeling model based on a reaction-diffusion system influenced by mechanical stress. Two-dimensional bone models were created and subjected to mechanical loads. The conventional finite element method (FEM) was used to calculate stress distribution. A stress-reactive reaction-diffusion model was constructed and used to simulate bone remodeling under mechanical loads. When an external mechanical stress was applied, stimulated bone formation and subsequent activation of bone resorption produced an efficient adaptation of the internal shape of the model bone to a given stress, and demonstrated major structures of trabecular bone seen in the human femoral neck. The degree of adaptation could be controlled by modulating the diffusion constants of hypothetical local factors. We also tried to demonstrate the deformation of bone structure during osteoporosis by the modulation of a parameter affecting the balance between formation and resorption. This simple model gives us an insight into how bone cells can create an architecture adapted to environmental stress, and will serve as a useful tool to understand both physiological and pathological states of bone based on structural information.
Na An
2016-01-01
Full Text Available We present a new numerical method for solving nonlinear reaction-diffusion systems with cross-diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two-dimensional system is discretized by the local discontinuous Galerkin (LDG method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (Δt=O(hmin2, the implicit integration factor (IIF method is employed for the temporal discretization so that the time step can be relaxed as Δt=O(hmin. Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.
Ghosh, Pushpita; Sen, Shrabani; Riaz, Syed Shahed; Ray, Deb Shankar
2009-05-01
The photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system has been an experimental paradigm for the study of Turing pattern over the last several years. When subjected to illumination of varied intensity by visible light the patterns undergo changes from spots to stripes, vice versa, and their mixture. We carry out a nonlinear analysis of the underlying model in terms of a Galerkin scheme with finite number of modes to explore the nature of the stability and existence of various modes responsible for the type and crossover of the light-induced patterns.
Felicia Shirly Peace
2014-01-01
Full Text Available A mathematical model of the dynamics of the self-ignition of a reaction-diffusion system is studied in this paper. An approximate analytical method (modified Adomian decomposition method is used to solve nonlinear differential equations under steady-state condition. Analytical expressions for concentrations of the gas reactant and the temperature have been derived for Lewis number (Le and parameters β, γ, and ϕ2. Furthermore, in this work, the numerical simulation of the problem is also reported using MATLAB program. An agreement between analytical and numerical results is noted.
New mechanism of spiral wave initiation in a reaction-diffusion-mechanics system.
Louis D Weise
Full Text Available Spiral wave initiation in the heart muscle is a mechanism for the onset of dangerous cardiac arrhythmias. A standard protocol for spiral wave initiation is the application of a stimulus in the refractory tail of a propagating excitation wave, a region that we call the "classical vulnerable zone." Previous studies of vulnerability to spiral wave initiation did not take the influence of deformation into account, which has been shown to have a substantial effect on the excitation process of cardiomyocytes via the mechano-electrical feedback phenomenon. In this work we study the effect of deformation on the vulnerability of excitable media in a discrete reaction-diffusion-mechanics (dRDM model. The dRDM model combines FitzHugh-Nagumo type equations for cardiac excitation with a discrete mechanical description of a finite-elastic isotropic material (Seth material to model cardiac excitation-contraction coupling and stretch activated depolarizing current. We show that deformation alters the "classical," and forms a new vulnerable zone at longer coupling intervals. This mechanically caused vulnerable zone results in a new mechanism of spiral wave initiation, where unidirectional conduction block and rotation directions of the consequently initiated spiral waves are opposite compared to the mechanism of spiral wave initiation due to the "classical vulnerable zone." We show that this new mechanism of spiral wave initiation can naturally occur in situations that involve wave fronts with curvature, and discuss its relation to supernormal excitability of cardiac tissue. The concept of mechanically induced vulnerability may lead to a better understanding about the onset of dangerous heart arrhythmias via mechano-electrical feedback.
Fatima, Tasnim; Aiki, Toyohiko
2012-01-01
This paper treats the solvability of a semilinear reaction-diffusion system, which incorporates transport (diffusion) and reaction effects emerging from two separated spatial scales: $x$ - macro and $y$ - micro. The system's origin connects to the modeling of concrete corrosion in sewer concrete pipes. It consists of three partial differential equations which are mass-balances of concentrations, as well as, one ordinary differential equation tracking the damage-by-corrosion. The system is semilinear, partially dissipative, and coupled via the solid-water interface at the microstructure (pore) level. The structure of the model equations is obtained in \\cite{tasnim1} by upscaling of the physical and chemical processes taking place within the microstructure of the concrete. Herein we ensure the positivity and $L^\\infty-$bounds on concentrations, and then prove the global-in-time existence and uniqueness of a suitable class of positive and bounded solutions that are stable with respect to the two-scale data and m...
Interface reaction and a special form of grain boundary diffusion in the Cr-W system
Broeder, F.J.A.
1972-01-01
The two phase diffusion between chromium and tungsten, forming b.c.c. solid solutions, has been followed right from the beginning by means of a special X-ray diffraction technique, light microscopy and electron microprobe analysis. It appears that in a first stage only tungsten diffuses into the chr
Barreira, Rebeca Iglesias; García, Belén Bardán; López, Mónica Granero; Legazpi, Iria Rodríguez; Díaz, Hortensia Álvarez; Penín, Isaura Rodríguez
2012-10-01
To report a paradoxical reaction of Raynaud phenomenon following the repeated administration of iloprost in a patient with diffuse cutaneous systemic sclerosis with vascular involvement. In January 2006, a 40-year-old male was diagnosed with diffuse cutaneous systemic sclerosis with pulmonary, esophageal, cutaneous, and vascular involvement (Raynaud phenomenon, with digital ulcers on his hands). In December 2008, treatment with iloprost was started due to worsening disease. Nine cycles of iloprost were administered at a rate of 0.5-1 ng/kg/min (6 hours per day, for 5 days every 6-8 weeks); the patient tolerated this treatment well. However, on the fourth day of cycles 10 and 11, the patient developed paradoxical Raynaud phenomenon in the hand with perfusion when the infusion was increased to 1 ng/kg/min, requiring treatment to be stopped. Treatment was continued during cycles 12 and 13 at 0.5 ng/kg/min; the patient tolerated the treatment well, although paradoxical Raynaud phenomenon occurred when the rate of infusion was increased. Raynaud phenomenon is extremely common in patients with scleroderma, and often is severe. Iloprost has vasodilating, antiplatelet, cytoprotective, and immunomodulating properties, and has been found to be an efficacious alternative to nifedipine for the treatment of Raynaud phenomenon in patients with scleroderma. The Naranjo probability scale indicated that iloprost was the probable cause of the paradoxical Raynaud phenomenon in this patient. This case demonstrates a probable relationship between the rate of infusion of iloprost and the paradoxical reaction of Raynaud phenomenon.
Reaction rates for a generalized reaction-diffusion master equation.
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
Reaction rates for a generalized reaction-diffusion master equation
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
Fernandes, Ryan I
2012-01-01
An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only $O({\\cal N})$ operations where ${\\cal N}$ is the number of unknowns. Moreover,it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties.
Reaction rates for mesoscopic reaction-diffusion kinetics.
Hellander, Stefan; Hellander, Andreas; Petzold, Linda
2015-02-01
The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic scale-dependent reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a Robin boundary condition at the reaction radius of two molecules. We also establish fundamental limits on the range of mesh resolutions for which this approach yields accurate results and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics. We show that for mesh sizes below the fundamental lower limit, results are less accurate. Thus, the lower limit determines the mesh size for which we obtain the most accurate results.
Reaction rates for mesoscopic reaction-diffusion kinetics
Hellander, Stefan; Hellander, Andreas; Petzold, Linda
2015-02-01
The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic scale-dependent reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a Robin boundary condition at the reaction radius of two molecules. We also establish fundamental limits on the range of mesh resolutions for which this approach yields accurate results and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics. We show that for mesh sizes below the fundamental lower limit, results are less accurate. Thus, the lower limit determines the mesh size for which we obtain the most accurate results.
Dellar, Paul
2016-11-01
We present discrete kinetic and lattice Boltzmann formulations for reaction cross-diffusion systems, as commonly used to model microbiological chemotaxis and macroscopic predator-prey interactions, and their hyperbolic extensions with fluid-like persistence terms. For example, the canonical Patlak-Keller-Segal model for chemotaxis involves a flux of cells up the gradient of a chemical secreted by the cells, in addition to the usual down-gradient diffusive fluxes. Existing lattice Boltzmann approaches for such systems use finite difference approximations to compute the flux of cells due to the chemical gradient. The resulting coupling between, and necessary synchronisation of the evolution of, adjacent grid points greatly complicates boundary conditions, and efficient implementation on graphical processing units (GPUs). We present a kinetic formulation using cross-collisions between bases of moments for the two sets of distribution functions to couple the fluxes of the two species, from which we construct lattice Boltzmann algorithms using second-order Strang splitting. We demonstrate an efficient GPU implementation, and verify second-order spatial convergence towards spectral solutions for benchmark problems such as the finite-time blow-up in the Patlak-Keller-Segal model.
Solid state reaction studies in Fe{sub 3}O{sub 4}–TiO{sub 2} system by diffusion couple method
Ren, Zhongshan [State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing 100083 (China); School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing 100083 (China); Hu, Xiaojun, E-mail: huxiaojun@ustb.edu.cn [State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing 100083 (China); School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing 100083 (China); Xue, Xiangxin [School of Materials and Metallurgy, Northeastern University, Shenyang 110006 (China); Chou, Kuochih [State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing 100083 (China); School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing 100083 (China)
2013-12-15
Highlights: •The solid state reactions of Fe2O3-TiO2 system was studied by the diffusion couple method. •Different products were formed by diffusion, and the FeTiO3 was more stable phase. •The inter-diffusion coefficients and diffusion activation energy were estimated. -- Abstract: The solid state reactions in Fe{sub 3}O{sub 4}–TiO{sub 2} system has been studied by diffusion couple experiments at 1323–1473 K, in which the oxygen partial pressure was controlled by the CO–CO{sub 2} gas mixture. The XRD analysis was used to confirm the phases of the inter-compound, and the concentration profiles were determined by electron probe microanalysis (EPMA). Based on the concentration profile of Ti, the inter-diffusion coefficients in Fe{sub 3}O{sub 4} phase, which were both temperature and concentration of Ti ions dependent, were calculated by the modified Boltzmann–Matano method. According to the relation between the thickness of diffusion layer and temperature, the diffusion coefficient of the Fe{sub 3}O{sub 4}–TiO{sub 2} system was obtained. According to the Arrhenius equation, the estimated diffusion activation energy was about 282.1 ± 18.8 kJ mol{sup −1}.
Giletti, Thomas
2010-01-01
We consider in this paper a reaction-diffusion system under a KPP hypothesis in a cylindrical domain in the presence of a shear flow. Such systems arise in predator-prey models as well as in combustion models with heat losses. Similarly to the single equation case, the existence of a minimal speed c* and of traveling front solutions for every speed c > c* has been shown both in the cases of heat losses distributed inside the domain or on the boundary. Here, we deal with the accordance between the two models by choosing heat losses inside the domain which tend to a Dirac mass located on the boundary. First, using the characterizations of the corresponding minimal speeds, we will see that they converge to the minimal speed of the limiting problem. Then, we will take interest in the convergence of the traveling front solutions of our reaction-di?usion systems. We will show the convergence under some assumptions on those solutions, which in particular can be satis?ed in dimension 2.
Speed ot travelling waves in reaction-diffusion equations
Benguria, R D; Méndez, V
2002-01-01
Reaction diffusion equations arise in several problems of population dynamics, flame propagation and others. In one dimensional cases the systems may evolve into travelling fronts. Here we concentrate on a reaction diffusion equation which arises as a simple model for chemotaxis and present results for the speed of the travelling fronts. (Author)
Speed ot travelling waves in reaction-diffusion equations
Benguria, R.D.; Depassier, M.C. [Facultad de Fisica, Pontificia Universidad Catolica de Chile, Avda. Vicuna Mackenna 4860, Santiago (Chile); Mendez, V. [Facultat de Ciencies de la Salut, Universidad Internacional de Catalunya, Gomera s/n 08190 Sant Cugat del Valles, Barcelona (Spain)
2002-07-01
Reaction diffusion equations arise in several problems of population dynamics, flame propagation and others. In one dimensional cases the systems may evolve into travelling fronts. Here we concentrate on a reaction diffusion equation which arises as a simple model for chemotaxis and present results for the speed of the travelling fronts. (Author)
Fluorescence Correlation Spectroscopy and Nonlinear Stochastic Reaction-Diffusion
Del Razo, Mauricio J; Qian, Hong; Lin, Guang
2014-01-01
The currently existing theory of fluorescence correlation spectroscopy(FCS) is based on the linear fluctuation theory originally developed by Einstein, Onsager, Lax, and others as a phenomenological approach to equilibrium fluctuations in bulk solutions. For mesoscopic reaction-diffusion systems with nonlinear chemical reactions among a small number of molecules, a situation often encountered in single-cell biochemistry, it is expected that FCS time correlation functions of a reaction-diffusion system can deviate from the classic results of Elson and Magde. We first discuss this nonlinear effect for reaction systems without diffusion. For nonlinear stochastic reaction-diffusion systems here are no closed solutions; therefore, stochastic Monte-Carlo simulations are carried out. We show that the deviation is small for a simple bimolecular reaction; the most significant deviations occur when the number of molecules is small and of the same order. Our results show that current linear FCS theory could be adequate ...
Pattern Formation in the Turing-Hopf Codimension-2 Phase Space in a Reaction-Diffusion System
YUAN Xu-Jin; SHAO Xin; LIAO Hui-Min; OUYANG Qi
2009-01-01
We systematically investigate the behaviour of pattern formation in a reaction-diffusion system when the system is located near the Turing-Hopf codimension-2 point in phase space. The chloride-iodide-malonic acid (CIMA) reaction is used in this study. A phase diagram is obtained using the concentration of polyvinyl alcohol (PVA) and malonic acid (MA) as control parameters. It is found that the Turing-Hopf mixed state appears only in a small vicinity near the codimension-2 point, and has the form of hexagonal pattern overlapped with anti-target wave; the boundary line separating the Thring state and the wave state is independent of the concentration of MA, only relies on the concentration of PVA. The corresponding numerical simulation using the Lengyel-Epstein (LE) model gives a similar phase diagram as the experiment; it reproduces most patterns observed in the experiment. However, the mixed state we obtain in simulation only appears in the anti-wave tip area, implying that the 3-D effect in the experiments may change the pattern forming behaviour in the codimension-2 regime.
Saxena, R K; Haubold, H J
2011-01-01
The object of this paper is to present a computable solution of a fractional partial differential equation associated with a Riemann-Liouville derivative of fractional order as the time-derivative and Riesz-Feller fractional derivative as the space derivative. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computable form in terms of the H-function. It provides an elegant extension of the results given earlier by Debnath, Chen et al., Haubold et al., Mainardi et al., Saxena et al., and Pagnini et al. The results obtained are presented in the form of four theorems. Some results associated with fractional Schroeodinger equation and fractional diffusion-wave equation are also derived as special cases of the findings.
Uniqueness of solution of a quasi-linear Reaction-diffusion system
无
2007-01-01
In this paper, we consider nonnegative classical solutions of a Quasi-linear reactiondiffusion system with nonlinear boundary conditions. We prove the uniqueness of a nonnegative classical solution to this problem.
Wave Instability and Spatiotemporal Chaos in Reaction-Diffusion System with Oscillatory Dynamics
XIE Fa-Gen; YANG Jun-Zhong; LI Hong-Gang
2006-01-01
We investigate the Turing-like wave instability of the uniform oscillator in oscillatory mediums using theoretical and numerical methods. A propagating wave pattern originated at the corner of the system emerges when the uniform oscillator becomes unstable via Turing-like wave instability. Bifurcations from periodically propagated wave patterns to quasi-periodically propagated wave patterns, then to spatiotemporal chaos occur, as the system size increases from the instability threshold of the uniform oscillator.
Asymptotic analysis of Complex Automata models for reaction-diffusion systems
Caiazzo, A.; Falcone, J.-L.; Chopard, B.; Hoekstra, A.G.
2009-01-01
Complex Automata (CxA) have been recently introduced as a paradigm to simulate multiscale multiscience systems as a collection of generalized Cellular Automata on different scales. The approach yields numerical and computational challenges and can become a powerful tool for the simulation of
Finite time blow-up for a reaction-diffusion system in bounded domain
Bai, Xueli
2014-02-01
This paper mainly considers the coupled parabolic system in a bounded domain: u t = Δ u + u α v p , v t = Δ v + u q v β in Ω × (0, T) with null Dirichlet boundary value condition which had been discussed by Wang in (Z Angew Math Phys 51:160-167, 2000). The aim of this paper is to solve the open problem mentioned in the Remark of Wang (Z Angew Math Phys 51:160-167, 2000).
Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion
Bidaut-Véron, Marie-Françoise; Yarur, Cecilia
2010-01-01
In this article we study the positive solutions of the parabolic semilinear system of competitive type \\[\\{\\begin{array} [c]{c}% u_{t}-\\Delta u+v^{p}=0, v_{t}-\\Delta v+u^{q}=0, \\end{array}.\\] in $\\Omega\\times(0,T)$, where $\\Omega$ is a domain of $\\mathbb{R}^{N},$ and $p,q>0,$ $pq\
1981-08-01
AD-AI03 864 WISCONSIN UN! V-ADISON MATHEMATICS RESEARCH CENTER F/G 7/1 BIFURCATION OF SINGULAR SOLUTIONS IN REVERSIBLE SYSTEMS AND APP--ETC (U) AUG...al M RENAROT OAAG29-Al C 0041 UNCLASSIFIED MRC-TSR-2261 NL so Ih sonhlnsoo *MRC Technical Summary Report #2261 BIFURCATION OF SINGULAR SOLUTIONS IN i...and views expressed in this descriptive summary lies with MRC, and not with the author of this report. BIFURCATION OF SINGULAR SOLUTIONS IN REVERSIBLE
Kelbert, M. Ya.; Suhov, Yu. M.
1995-02-01
A general model of a branching random walk in R 1 is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable 220_2005_Article_BF02101538_TeX2GIFE1.gif X^0 = mathop {sup max}limits_{ n ≥q 0 1 ≤q k ≤q N_n } X_{n,k} to be finite. Here, X n, k is the position of the k th particle in the n th generation, N n is the number of particles in the n th generation (regardless of their type). It turns out that the distribution of X 0 gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.
Fluorescence Correlation Spectroscopy and Nonlinear Stochastic Reaction-Diffusion
Del Razo, Mauricio; Pan, Wenxiao; Qian, Hong; Lin, Guang
2014-05-30
The currently existing theory of fluorescence correlation spectroscopy (FCS) is based on the linear fluctuation theory originally developed by Einstein, Onsager, Lax, and others as a phenomenological approach to equilibrium fluctuations in bulk solutions. For mesoscopic reaction-diffusion systems with nonlinear chemical reactions among a small number of molecules, a situation often encountered in single-cell biochemistry, it is expected that FCS time correlation functions of a reaction-diffusion system can deviate from the classic results of Elson and Magde [Biopolymers (1974) 13:1-27]. We first discuss this nonlinear effect for reaction systems without diffusion. For nonlinear stochastic reaction-diffusion systems there are no closed solutions; therefore, stochastic Monte-Carlo simulations are carried out. We show that the deviation is small for a simple bimolecular reaction; the most significant deviations occur when the number of molecules is small and of the same order. Extending Delbrück-Gillespie’s theory for stochastic nonlinear reactions with rapidly stirring to reaction-diffusion systems provides a mesoscopic model for chemical and biochemical reactions at nanometric and mesoscopic level such as a single biological cell.
Connectionist and diffusion models of reaction time.
Ratcliff, R; Van Zandt, T; McKoon, G
1999-04-01
Two connectionist frameworks, GRAIN (J. L. McClelland, 1993) and brain-state-in-a-box (J. A. Anderson, 1991), and R. Ratcliff's (1978) diffusion model were evaluated using data from a signal detection task. Dependent variables included response probabilities, reaction times for correct and error responses, and shapes of reaction-time distributions. The diffusion model accounted for all aspects of the data, including error reaction times that had previously been a problem for all response-time models. The connectionist models accounted for many aspects of the data adequately, but each failed to a greater or lesser degree in important ways except for one model that was similar to the diffusion model. The findings advance the development of the diffusion model and show that the long tradition of reaction-time research and theory is a fertile domain for development and testing of connectionist assumptions about how decisions are generated over time.
Lenk, Claudia; Einax, Mario; Maass, Philipp
2013-04-01
Spatiotemporal excitation patterns in the FitzHugh-Nagumo model are studied, which result from the disturbance of a primary pacemaker by a secondary pacemaker. The primary and secondary pacemakers generate regular waves with frequencies f(pace) and f(pert), respectively. The pacemakers are spatially separated, but waves emanating from them encounter each other via a small bridge. This leads to three different types I-III of irregular excitation patterns in disjunct domains of the f(pace)-f(pert) plane. Types I and II are caused by detachments of waves coming from the two pacemakers at corners of the bridge. Type III irregularities are confined to a boundary region of the system and originate from a partial penetration of the primary waves into a space, where circular wave fronts from the secondary pacemaker prevail. For this type, local frequencies can significantly exceed f(pace) and f(pert). The degree of irregularity found for the three different types is quantified by the entropy of the local frequency distribution and an order parameter for phase coherence.
Diffusion-reaction compromise the polymorphs of precipitated calcium carbonate
Han Wang; Wenlai Huang; Yongsheng Han
2013-01-01
Diffusion is seldom considered by chemists and materialists in the preparation of materials while it plays an important role in the field of chemical engineering.If we look at crystallization at the atomic level,crystal growth in a solution starts from the diffusion of ions to the growing surface followed by the incorporation of ions into its lattice.Diffusion can be a rate determining step for the growth of crystals.In this paper,we take the crystallization of calcium carbonate as an example to illustrate the microscopic processes of diffusion and reaction and their compromising influence on the morphology of the crystals produced.The diffusion effect is studied in a specially designed three-cell reactor.Experiments show that a decrease of diffusion leads to retardation of supersaturation and the formation of a continuous concentration gradient in the reaction cell,thus promoting the formation of cubic calcite particles.The reaction rate is regulated by temperature.Increase of reaction rate favors the formation of needle-like aragonite particles.When diffusion and reaction play joint roles in the reaction system,their compromise dominates the formation of products,leading to a mixture of cubic and needle-like particles with a controllable ratio.Since diffusion and reaction are universal factors in the preparation of materials,the finding of this paper could be helpful in the controlled synthesis of other materials.
Karniadakis, George Em [Brown University
2014-03-11
The main objective of this project is to develop new computational tools for uncertainty quantifica- tion (UQ) of systems governed by stochastic partial differential equations (SPDEs) with applications to advection-diffusion-reaction systems. We pursue two complementary approaches: (1) generalized polynomial chaos and its extensions and (2) a new theory on deriving PDF equations for systems subject to color noise. The focus of the current work is on high-dimensional systems involving tens or hundreds of uncertain parameters.
Langevin Equations for Reaction-Diffusion Processes
Benitez, Federico; Duclut, Charlie; Chaté, Hugues; Delamotte, Bertrand; Dornic, Ivan; Muñoz, Miguel A.
2016-09-01
For reaction-diffusion processes with at most bimolecular reactants, we derive well-behaved, numerically tractable, exact Langevin equations that govern a stochastic variable related to the response field in field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encountered in field-theoretical approaches and paves the way for systematic numerical and theoretical analyses of reaction-diffusion problems.
Johnson, Margaret E.; Hummer, Gerhard
2014-07-01
We present a new algorithm for simulating reaction-diffusion equations at single-particle resolution. Our algorithm is designed to be both accurate and simple to implement, and to be applicable to large and heterogeneous systems, including those arising in systems biology applications. We combine the use of the exact Green's function for a pair of reacting particles with the approximate free-diffusion propagator for position updates to particles. Trajectory reweighting in our free-propagator reweighting (FPR) method recovers the exact association rates for a pair of interacting particles at all times. FPR simulations of many-body systems accurately reproduce the theoretically known dynamic behavior for a variety of different reaction types. FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone. FPR applications include the modeling of pathways and networks of protein-driven processes where reaction rates can vary widely and thousands of proteins may participate in the formation of large assemblies. With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events. Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.
Physarum machines: encapsulating reaction-diffusion to compute spanning tree
Adamatzky, Andrew
2007-12-01
The Physarum machine is a biological computing device, which employs plasmodium of Physarum polycephalum as an unconventional computing substrate. A reaction-diffusion computer is a chemical computing device that computes by propagating diffusive or excitation wave fronts. Reaction-diffusion computers, despite being computationally universal machines, are unable to construct certain classes of proximity graphs without the assistance of an external computing device. I demonstrate that the problem can be solved if the reaction-diffusion system is enclosed in a membrane with few ‘growth points’, sites guiding the pattern propagation. Experimental approximation of spanning trees by P. polycephalum slime mold demonstrates the feasibility of the approach. Findings provided advance theory of reaction-diffusion computation by enriching it with ideas of slime mold computation.
Laser Spot Detection Based on Reaction Diffusion
Vázquez-Otero, Alejandro; Khikhlukha, Danila; Solano-Altamirano, J. M.; Dormido, Raquel; Duro, Natividad
2016-01-01
Center-location of a laser spot is a problem of interest when the laser is used for processing and performing measurements. Measurement quality depends on correctly determining the location of the laser spot. Hence, improving and proposing algorithms for the correct location of the spots are fundamental issues in laser-based measurements. In this paper we introduce a Reaction Diffusion (RD) system as the main computational framework for robustly finding laser spot centers. The method presented is compared with a conventional approach for locating laser spots, and the experimental results indicate that RD-based computation generates reliable and precise solutions. These results confirm the flexibility of the new computational paradigm based on RD systems for addressing problems that can be reduced to a set of geometric operations. PMID:26938537
Laser Spot Detection Based on Reaction Diffusion
Alejandro Vázquez-Otero
2016-03-01
Full Text Available Center-location of a laser spot is a problem of interest when the laser is used for processing and performing measurements. Measurement quality depends on correctly determining the location of the laser spot. Hence, improving and proposing algorithms for the correct location of the spots are fundamental issues in laser-based measurements. In this paper we introduce a Reaction Diffusion (RD system as the main computational framework for robustly finding laser spot centers. The method presented is compared with a conventional approach for locating laser spots, and the experimental results indicate that RD-based computation generates reliable and precise solutions. These results confirm the flexibility of the new computational paradigm based on RD systems for addressing problems that can be reduced to a set of geometric operations.
Theory for Diffusion-Limited Oscillating Chemical Reactions
Bussemaker, H J
1997-01-01
A kinetic description of lattice-gas automaton models for reaction-diffusion systems is presented. It provides corrections to the mean-field rate equations in the diffusion-limited regime. When applied to the two-species Maginu model, the theory gives an excellent quantitative prediction of the effect of slow diffusion on the periodic oscillations of the average concentrations in a spatially homogeneous state.
Victor Kardashov
2002-01-01
Full Text Available This paper has considered a novel approach to structural recognition and control of nonlinear reaction-diffusion systems (systems with density dependent diffusion. The main consistence of the approach is interactive variation of the nonlinear diffusion and sources structural parameters that allows to implement a qualitative control and recognition of transitional system conditions (transients. The method of inverse solutions construction allows formulating the new analytic conditions of compactness and periodicity of the transients that is also available for nonintegrated systems. On the other hand, using of energy conservations laws, allows transfer to nonlinear dynamics models that gives the possiblity to apply the modern deterministic chaos theory (particularly the Feigenboum's universal constants and scenario of chaotic transitions.
Multiresolution stochastic simulations of reaction-diffusion processes.
Bayati, B; Chatelain, P.; Koumoutsakos, P.
2008-01-01
Stochastic simulations of reaction-diffusion processes are used extensively for the modeling of complex systems in areas ranging from biology and social sciences to ecosystems and materials processing. These processes often exhibit disparate scales that render their simulation prohibitive even for massive computational resources. The problem is resolved by introducing a novel stochastic multiresolution method that enables the efficient simulation of reaction-diffusion processes as modeled by ...
Gaskins, Delora K.; Pruc, Emily E.; Epstein, Irving R.; Dolnik, Milos
2016-07-01
Turing patterns in the chlorine dioxide-iodine-malonic acid reaction were modified through additions of sodium halide salt solutions. The range of wavelengths obtained is several times larger than in the previously reported literature. Pattern wavelength was observed to significantly increase with sodium bromide or sodium chloride. A transition to a uniform state was found at high halide concentrations. The observed experimental results are qualitatively well reproduced in numerical simulations with the Lengyel-Epstein model with an additional chemically realistic kinetic term to account for the added halide and an adjustment of the activator diffusion rate to allow for interhalogen formation.
Modeling of Reaction Processes Controlled by Diffusion
Revelli, J
2003-01-01
Stochastic modeling is quite powerful in science and technology.The technics derived from this process have been used with great success in laser theory, biological systems and chemical reactions.Besides, they provide a theoretical framework for the analysis of experimental results on the field of particle's diffusion in ordered and disordered materials.In this work we analyze transport processes in one-dimensional fluctuating media, which are media that change their state in time.This fact induces changes in the movements of the particles giving rise to different phenomena and dynamics that will be described and analyzed in this work.We present some random walk models to describe these fluctuating media.These models include state transitions governed by different dynamical processes.We also analyze the trapping problem in a lattice by means of a simple model which predicts a resonance-like phenomenon.Also we study effective diffusion processes over surfaces due to random walks in the bulk.We consider differe...
Approximating parameters in nonlinear reaction diffusion equations
Robert R. Ferdinand
2001-07-01
Full Text Available We present a model describing population dynamics in an environment. The model is a nonlinear, nonlocal, reaction diffusion equation with Neumann boundary conditions. An inverse method, involving minimization of a least-squares cost functional, is developed to identify unknown model parameters. Finally, numerical results are presented which display estimates of these parameters using computationally generated data.
Exact solutions for logistic reaction-diffusion equations in biology
Broadbridge, P.; Bradshaw-Hajek, B. H.
2016-08-01
Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.
Reaction-Diffusion in the NEURON Simulator
Robert A. McDougal
2013-11-01
Full Text Available In order to support research on the role of cell biological principles (genomics, proteomics, signaling cascades and reaction dynamics on the dynamics of neuronal response in health and disease, NEURON has developed a Reaction-Diffusion (rxd module in Python which provides specification and simulation for these dynamics, coupled with the electrophysiological dynamics of the cell membrane. Arithmetic operations on species and parameters are overloaded, allowing arbitrary reaction formulas to be specified using Python syntax. These expressions are then transparently compiled into bytecode that uses NumPy for fast vectorized calculations. At each time step, rxd combines NEURON's integrators with SciPy’s sparse linear algebra library.
Reaction-diffusion in the NEURON simulator.
McDougal, Robert A; Hines, Michael L; Lytton, William W
2013-01-01
In order to support research on the role of cell biological principles (genomics, proteomics, signaling cascades and reaction dynamics) on the dynamics of neuronal response in health and disease, NEURON's Reaction-Diffusion (rxd) module in Python provides specification and simulation for these dynamics, coupled with the electrophysiological dynamics of the cell membrane. Arithmetic operations on species and parameters are overloaded, allowing arbitrary reaction formulas to be specified using Python syntax. These expressions are then transparently compiled into bytecode that uses NumPy for fast vectorized calculations. At each time step, rxd combines NEURON's integrators with SciPy's sparse linear algebra library.
Simple computation of reaction-diffusion processes on point clouds.
Macdonald, Colin B; Merriman, Barry; Ruuth, Steven J
2013-06-04
The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulk-surface coupling are provided for a variety of complex point cloud surfaces.
Voter Model Perturbations and Reaction Diffusion Equations
Cox, J Theodore; Perkins, Edwin
2011-01-01
We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \\ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the ...
Reaction-diffusion basis of retroviral infectivity
Sadiq, S. Kashif
2016-11-01
Retrovirus particle (virion) infectivity requires diffusion and clustering of multiple transmembrane envelope proteins (Env3) on the virion exterior, yet is triggered by protease-dependent degradation of a partially occluding, membrane-bound Gag polyprotein lattice on the virion interior. The physical mechanism underlying such coupling is unclear and only indirectly accessible via experiment. Modelling stands to provide insight but the required spatio-temporal range far exceeds current accessibility by all-atom or even coarse-grained molecular dynamics simulations. Nor do such approaches account for chemical reactions, while conversely, reaction kinetics approaches handle neither diffusion nor clustering. Here, a recently developed multiscale approach is considered that applies an ultra-coarse-graining scheme to treat entire proteins at near-single particle resolution, but which also couples chemical reactions with diffusion and interactions. A model is developed of Env3 molecules embedded in a truncated Gag lattice composed of membrane-bound matrix proteins linked to capsid subunits, with freely diffusing protease molecules. Simulations suggest that in the presence of Gag but in the absence of lateral lattice-forming interactions, Env3 diffuses comparably to Gag-absent Env3. Initial immobility of Env3 is conferred through lateral caging by matrix trimers vertically coupled to the underlying hexameric capsid layer. Gag cleavage by protease vertically decouples the matrix and capsid layers, induces both matrix and Env3 diffusion, and permits Env3 clustering. Spreading across the entire membrane surface reduces crowding, in turn, enhancing the effect and promoting infectivity. This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.
A Note on a Nonlocal Nonlinear Reaction-Diffusion Model
Walker, Christoph
2011-01-01
We give an application of the Crandall-Rabinowitz theorem on local bifurcation to a system of nonlinear parabolic equations with nonlocal reaction and cross-diffusion terms as well as nonlocal initial conditions. The system arises as steady-state equations of two interacting age-structured populations.
Reaction-diffusion pulses: a combustion model
Campos, Daniel [Grup de FIsica EstadIstica, Dept. de FIsica, Universitat Autonoma de Barcelona, E-08193 Bellaterrra (Spain); Llebot, Josep Enric [Grup de FIsica EstadIstica, Dept. de FIsica, Universitat Autonoma de Barcelona, E-08193 Bellaterrra (Spain); Fort, Joaquim [Dept. de FIsica, Univ. de Girona, Campus de Montilivi, 17071 Girona, Catalonia (Spain)
2004-07-02
We focus on a reaction-diffusion approach proposed recently for experiments on combustion processes, where the heat released by combustion follows first-order reaction kinetics. This case allows us to perform an exhaustive analytical study. Specifically, we obtain the exact expressions for the speed of the thermal pulses, their maximum temperature and the condition of self-sustenance. Finally, we propose two generalizations of the model, namely, the case of several reactants burning together, and that of time-delayed heat conduction. We find an excellent agreement between our analytical results and simulations.
无
2007-01-01
Taking the Lindemann model as a sample system in which there exist chemical reactions, diffusion and heat conduction, we found the theoretical framework of linear stability analysis for a unidimensional nonhomogeneous two-variable system with one end subject to Dirichlet conditions, while the other end no-flux conditions. Furthermore, the conditions for the emergence of temperature waves are found out by the linear stabiliy analysis and verified by a diagram for successive steps of evolution of spatial profile of temperature during a period that is plotted by numerical simulations on a computer. Without doubt, these results are in favor of the heat balance in chemical reactor designs.
Reaction diffusion equations with boundary degeneracy
Huashui Zhan
2016-03-01
Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.
Zhang, Chunxia; Zhang, Hong; Ouyang, Qi; Hu, Bambi; Gunaratne, Gemunu H.
2003-09-01
The transition from spiral waves to defect-mediated turbulence was studied in a spatial open reactor using Belousov-Zhabotinsky reaction. The experimental results show a new mechanism of the transition from spirals to spatiotemporal chaos, in which the gradient effects in the three-dimensional system are essential. The transition scenario consists of two stages: first, the effects of gradients in the third dimension cause a splitting of the spiral tip and a deletion of certain wave segments, generating new wave sources; second, the waves sent by the new wave sources undergo a backfire instability, and the back waves are laterally unstable. As a result, defects are automatically generated and fill all over the system. The result of numerical simulation using the FitzHugh-Nagumo model essentially agrees with the experimental observation.
Diffusion-limited reactions in crowded environments.
Dorsaz, N; De Michele, C; Piazza, F; De Los Rios, P; Foffi, G
2010-09-17
Diffusion-limited reactions are usually described within the Smoluchowski theory, which neglects interparticle interactions. We propose a simple way to incorporate excluded-volume effects building on simulations of hard sphere in the presence of a sink. For large values of the sink-to-particle size ratio R(s), the measured encounter rate is in good agreement with a simple generalization of the Smoluchowski equation at high densities. Reducing R(s), the encounter rate is substantially depressed and becomes even nonmonotonic for R(s)saturation of the rate, stationary density waves set in close to the sink. A mean-field analysis helps to shed light on the subtle link between such ordering and the slowing down of the encounter dynamics. Finally, we show how an infinitesimal amount of nonreacting impurities can equally slow down dramatically the reaction.
Chemical computing with reaction-diffusion processes.
Gorecki, J; Gizynski, K; Guzowski, J; Gorecka, J N; Garstecki, P; Gruenert, G; Dittrich, P
2015-07-28
Chemical reactions are responsible for information processing in living organisms. It is believed that the basic features of biological computing activity are reflected by a reaction-diffusion medium. We illustrate the ideas of chemical information processing considering the Belousov-Zhabotinsky (BZ) reaction and its photosensitive variant. The computational universality of information processing is demonstrated. For different methods of information coding constructions of the simplest signal processing devices are described. The function performed by a particular device is determined by the geometrical structure of oscillatory (or of excitable) and non-excitable regions of the medium. In a living organism, the brain is created as a self-grown structure of interacting nonlinear elements and reaches its functionality as the result of learning. We discuss whether such a strategy can be adopted for generation of chemical information processing devices. Recent studies have shown that lipid-covered droplets containing solution of reagents of BZ reaction can be transported by a flowing oil. Therefore, structures of droplets can be spontaneously formed at specific non-equilibrium conditions, for example forced by flows in a microfluidic reactor. We describe how to introduce information to a droplet structure, track the information flow inside it and optimize medium evolution to achieve the maximum reliability. Applications of droplet structures for classification tasks are discussed.
Cohabitation reaction-diffusion model for virus focal infections
Amor, Daniel R.; Fort, Joaquim
2014-12-01
The propagation of virus infection fronts has been typically modeled using a set of classical (noncohabitation) reaction-diffusion equations for interacting species. However, for some single-species systems it has been recently shown that noncohabitation reaction-diffusion equations may lead to unrealistic descriptions. We argue that previous virus infection models also have this limitation, because they assume that a virion can simultaneously reproduce inside a cell and diffuse away from it. For this reason, we build a several-species cohabitation model that does not have this limitation. Furthermore, we perform a sensitivity analysis for the most relevant parameters of the model, and we compare the predicted infection speed with observed data for two different strains of the T7 virus.
Reaction diffusion equation with spatio-temporal delay
Zhao, Zhihong; Rong, Erhua
2014-07-01
We investigate reaction-diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction-diffusion equation with spatio-temporal delay. Applying this theory to Lotka-Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem's steady-state solution.
Multiresolution stochastic simulations of reaction-diffusion processes.
Bayati, Basil; Chatelain, Philippe; Koumoutsakos, Petros
2008-10-21
Stochastic simulations of reaction-diffusion processes are used extensively for the modeling of complex systems in areas ranging from biology and social sciences to ecosystems and materials processing. These processes often exhibit disparate scales that render their simulation prohibitive even for massive computational resources. The problem is resolved by introducing a novel stochastic multiresolution method that enables the efficient simulation of reaction-diffusion processes as modeled by many-particle systems. The proposed method quantifies and efficiently handles the associated stiffness in simulating the system dynamics and its computational efficiency and accuracy are demonstrated in simulations of a model problem described by the Fisher-Kolmogorov equation. The method is general and can be applied to other many-particle models of physical processes.
Exact solutions of certain nonlinear chemotaxis diffusion reaction equations
MISHRA AJAY; KAUSHAL R S; PRASAD AWADHESH
2016-05-01
Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in theconstructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.
Reaction and diffusion in turbulent combustion
Pope, S.B. [Mechanical and Aerospace Engineering, Ithaca, NY (United States)
1993-12-01
The motivation for this project is the need to obtain a better quantitative understanding of the technologically-important phenomenon of turbulent combustion. In nearly all applications in which fuel is burned-for example, fossil-fuel power plants, furnaces, gas-turbines and internal-combustion engines-the combustion takes place in a turbulent flow. Designers continually demand more quantitative information about this phenomenon-in the form of turbulent combustion models-so that they can design equipment with increased efficiency and decreased environmental impact. For some time the PI has been developing a class of turbulent combustion models known as PDF methods. These methods have the important virtue that both convection and reaction can be treated without turbulence-modelling assumptions. However, a mixing model is required to account for the effects of molecular diffusion. Currently, the available mixing models are known to have some significant defects. The major motivation of the project is to seek a better understanding of molecular diffusion in turbulent reactive flows, and hence to develop a better mixing model.
Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme
Li, Shanbing; Wu, Jianhua; Dong, Yaying
2015-09-01
In this paper, we consider a reaction-diffusion model with Degn-Harrison reaction scheme. Some fundamental analytic properties of nonconstant positive solutions are first investigated. We next study the stability of constant steady-state solution to both ODE and PDE models. Our result also indicates that if either the size of the reactor or the effective diffusion rate is large enough, then the system does not admit nonconstant positive solutions. Finally, we establish the global structure of steady-state bifurcations from simple eigenvalues by bifurcation theory and the local structure of the steady-state bifurcations from double eigenvalues by the techniques of space decomposition and implicit function theorem.
Stochastic reaction-diffusion kinetics in the microscopic limit
Fange, David; Berg, Otto G.; Sjöberg, Paul; Elf, Johan
2010-01-01
Quantitative analysis of biochemical networks often requires consideration of both spatial and stochastic aspects of chemical processes. Despite significant progress in the field, it is still computationally prohibitive to simulate systems involving many reactants or complex geometries using a microscopic framework that includes the finest length and time scales of diffusion-limited molecular interactions. For this reason, spatially or temporally discretized simulations schemes are commonly used when modeling intracellular reaction networks. The challenge in defining such coarse-grained models is to calculate the correct probabilities of reaction given the microscopic parameters and the uncertainty in the molecular positions introduced by the spatial or temporal discretization. In this paper we have solved this problem for the spatially discretized Reaction-Diffusion Master Equation; this enables a seamless and physically consistent transition from the microscopic to the macroscopic frameworks of reaction-diffusion kinetics. We exemplify the use of the methods by showing that a phosphorylation-dephosphorylation motif, commonly observed in eukaryotic signaling pathways, is predicted to display fluctuations that depend on the geometry of the system. PMID:21041672
A Lattice Boltzmann Model for Oscillating Reaction-Diffusion
Rodríguez-Romo, Suemi; Ibañez-Orozco, Oscar; Sosa-Herrera, Antonio
2016-07-01
A computational algorithm based on the lattice Boltzmann method (LBM) is proposed to model reaction-diffusion systems. In this paper, we focus on how nonlinear chemical oscillators like Belousov-Zhabotinsky (BZ) and the chlorite-iodide-malonic acid (CIMA) reactions can be modeled by LBM and provide with new insight into the nature and applications of oscillating reactions. We use Gaussian pulse initial concentrations of sulfuric acid in different places of a bidimensional reactor and nondiffusive boundary walls. We clearly show how these systems evolve to a chaotic attractor and produce specific pattern images that are portrayed in the reactions trajectory to the corresponding chaotic attractor and can be used in robotic control.
ALTERNATING DIRECTION FINITE ELEMENT METHOD FOR SOME REACTION DIFFUSION MODELS
江成顺; 刘蕴贤; 沈永明
2004-01-01
This paper is concerned with some nonlinear reaction - diffusion models. To solve this kind of models, the modified Laplace finite element scheme and the alternating direction finite element scheme are established for the system of patrical differential equations. Besides, the finite difference method is utilized for the ordinary differential equation in the models. Moreover, by the theory and technique of prior estimates for the differential equations, the convergence analyses and the optimal L2- norm error estimates are demonstrated.
An Application of Equivalence Transformations to Reaction Diffusion Equations
Mariano Torrisi
2015-10-01
Full Text Available In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper.
Basan, Markus; Lenz, Martin; Joanny, Jean-François; Risler, Thomas
2015-01-01
Contact inhibition is the process by which cells switch from a motile growing state to a passive and stabilized state upon touching their neighbors. When two cells touch, an adhesion link is created between them by means of transmembrane E-cadherin proteins. Simultaneously, their actin filaments stop polymerizing in the direction perpendicular to the membrane and reorganize to create an apical belt that colocalizes with the adhesion links. Here, we propose a detailed quantitative model of the role of the cytoplasmic $\\beta$-catenin and $\\alpha$-catenin proteins in this process, treated as a reaction-diffusion system. Upon cell-cell contact, the concentration in $\\alpha$-catenin dimers increases, inhibiting actin branching and thereby reducing cellular motility and expansion pressure. This model provides a mechanism for contact inhibition that could explain previously unrelated experimental findings on the role played by E-cadherin, $\\beta$-catenin and $\\alpha$-catenin in the cellular phenotype and in tumorige...
Villar-Cociña, E.
2005-06-01
Full Text Available A kinetic-diffusive model proposed by the authors in previous papers to describe pozzolanic reaction kinetics in sugar cane straw-clay ash (SCSCA/calcium hydroxide (CH systems is validated in this study. Two different methods (direct and indirect for determining pozzolanic activity were applied and their effect on pozzolanic reaction rate kinetic constants evaluated. Determined by fitting a model to the data, these constants are used to quantitatively characterize pozzolanic activity. The values of the kinetic constants calculated with the model were similar for the two methods. Classic kinetic models, such as the Jander, modified Jander and Zhuravlev models, were also applied to the system studied and the results were compared to the figures calculated with the model proposed. The kinetic-diffusive approach proposed was found to be valid regardless of the method for determining pozzolanic activity used, and to be the most suitable model for describing pozzolanic reaction kinetics in the SCSCA/lime system.
Se valida la aplicación de un modelo cinético-difusivo propuesto por los autores en trabajos anteriores para describir la cinética de reacción puzolánica en sistemas ceniza de paja de caña-arcilla (CPCAñúdróxido de calcio (CH. Se aplican 2 métodos diferentes de actividad puzolánica (directo e indirecto y se valora el efecto que pudieran tener los mismos sobre las constantes cinéticas de velocidad de reacción de la reacción puzolánica. Estas constantes cinéticas son determinadas en el proceso de ajuste del modelo y permiten caracterizar cuantitativamente la actividad puzolánica. Los resultados muestran la similitud de las constantes cinéticas de velocidad de reacción calculadas, aplicando el modelo a los resultados experimentales obtenidos por ambos métodos. Además, fueron aplicados al sistema estudiado modelos cinéticos el chicos como: modelo de Jander, modelo de Jander Modificado y el modelo de Zhuravlev y
Reaction diffusion and solid state chemical kinetics handbook
Dybkov, V I
2010-01-01
This monograph deals with a physico-chemical approach to the problem of the solid-state growth of chemical compound layers and reaction-diffusion in binary heterogeneous systems formed by two solids; as well as a solid with a liquid or a gas. It is explained why the number of compound layers growing at the interface between the original phases is usually much lower than the number of chemical compounds in the phase diagram of a given binary system. For example, of the eight intermetallic compounds which exist in the aluminium-zirconium binary system, only ZrAl3 was found to grow as a separate
Travelling waves in nonlinear diffusion-convection-reaction
Gilding, B.H.; Kersner, R.
2001-01-01
The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the stu
QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION
无
2006-01-01
This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.
Reaction systems with precipitation
Marek Rogalski
2015-04-01
Full Text Available This article proposes expanding Reaction Systems of Ehrenfeucht and Rozenberg by incorporating precipitation reactions into it. This improves the computing power of Reaction Systems by allowing us to implement a stack. This addition enables us to implement a Deterministic Pushdown Automaton.
Marco A. Velasco
2016-10-01
Full Text Available Scaffolds are essential in bone tissue engineering, as they provide support to cells and growth factors necessary to regenerate tissue. In addition, they meet the mechanical function of the bone while it regenerates. Currently, the multiple methods for designing and manufacturing scaffolds are based on regular structures from a unit cell that repeats in a given domain. However, these methods do not resemble the actual structure of the trabecular bone which may work against osseous tissue regeneration. To explore the design of porous structures with similar mechanical properties to native bone, a geometric generation scheme from a reaction-diffusion model and its manufacturing via a material jetting system is proposed. This article presents the methodology used, the geometric characteristics and the modulus of elasticity of the scaffolds designed and manufactured. The method proposed shows its potential to generate structures that allow to control the basic scaffold properties for bone tissue engineering such as the width of the channels and porosity. The mechanical properties of our scaffolds are similar to trabecular tissue present in vertebrae and tibia bones. Tests on the manufactured scaffolds show that it is necessary to consider the orientation of the object relative to the printing system because the channel geometry, mechanical properties and roughness are heavily influenced by the position of the surface analyzed with respect to the printing axis. A possible line for future work may be the establishment of a set of guidelines to consider the effects of manufacturing processes in designing stages.
Kalannagari Viswanath
2015-01-01
Full Text Available Ultrasound imaging is one of the available imaging techniques used for diagnosis of kidney abnormalities, which may be like change in shape and position and swelling of limb; there are also other Kidney abnormalities such as formation of stones, cysts, blockage of urine, congenital anomalies, and cancerous cells. During surgical processes it is vital to recognize the true and precise location of kidney stone. The detection of kidney stones using ultrasound imaging is a highly challenging task as they are of low contrast and contain speckle noise. This challenge is overcome by employing suitable image processing techniques. The ultrasound image is first preprocessed to get rid of speckle noise using the image restoration process. The restored image is smoothened using Gabor filter and the subsequent image is enhanced by histogram equalization. The preprocessed image is achieved with level set segmentation to detect the stone region. Segmentation process is employed twice for getting better results; first to segment kidney portion and then to segment the stone portion, respectively. In this work, the level set segmentation uses two terms, namely, momentum and resilient propagation (Rprop to detect the stone portion. After segmentation, the extracted region of the kidney stone is given to Symlets, Biorthogonal (bio3.7, bio3.9, and bio4.4, and Daubechies lifting scheme wavelet subbands to extract energy levels. These energy levels provide evidence about presence of stone, by comparing them with that of the normal energy levels. They are trained by multilayer perceptron (MLP and back propagation (BP ANN to classify and its type of stone with an accuracy of 98.8%. The prosed work is designed and real time is implemented on both Filed Programmable Gate Array Vertex-2Pro FPGA using Xilinx System Generator (XSG Verilog and Matlab 2012a.
Cumulative signal transmission in nonlinear reaction-diffusion networks.
Diego A Oyarzún
Full Text Available Quantifying signal transmission in biochemical systems is key to uncover the mechanisms that cells use to control their responses to environmental stimuli. In this work we use the time-integral of chemical species as a measure of a network's ability to cumulatively transmit signals encoded in spatiotemporal concentrations. We identify a class of nonlinear reaction-diffusion networks in which the time-integrals of some species can be computed analytically. The derived time-integrals do not require knowledge of the solution of the reaction-diffusion equation, and we provide a simple graphical test to check if a given network belongs to the proposed class. The formulae for the time-integrals reveal how the kinetic parameters shape signal transmission in a network under spatiotemporal stimuli. We use these to show that a canonical complex-formation mechanism behaves as a spatial low-pass filter, the bandwidth of which is inversely proportional to the diffusion length of the ligand.
Parametric pattern selection in a reaction-diffusion model.
Michael Stich
Full Text Available We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.
Adaptive mesh refinement for stochastic reaction-diffusion processes
Bayati, Basil; Chatelain, Philippe; Koumoutsakos, Petros
2011-01-01
We present an algorithm for adaptive mesh refinement applied to mesoscopic stochastic simulations of spatially evolving reaction-diffusion processes. The transition rates for the diffusion process are derived on adaptive, locally refined structured meshes. Convergence of the diffusion process is presented and the fluctuations of the stochastic process are verified. Furthermore, a refinement criterion is proposed for the evolution of the adaptive mesh. The method is validated in simulations of reaction-diffusion processes as described by the Fisher-Kolmogorov and Gray-Scott equations.
Diffusion Reaction of Carbon Monoxide in the Human Lung
Kang, M.-Y.; Guénard, H.; Sapoval, B.
2017-08-01
The capture of CO, a standard lung function test, results from diffusion-reaction processes of CO with hemoglobin inside red blood cells (RBCs). In its current understanding, suggested by Roughton and Forster in 1957, the capture is represented by two independent resistances in series, one for diffusion from the gas to the RBC periphery, the second for internal diffusion reaction. Numerical studies in 3D model structures described here contradict the independence hypothesis. This results from two different theoretical reasons: (i) The RBC peripheries are not equi-concentrations; (ii) diffusion times in series are not additive.
Diffusion Controlled Reactions, Fluctuation Dominated Kinetics, and Living Cell Biochemistry
Konkoli, Zoran
2009-01-01
In recent years considerable portion of the computer science community has focused its attention on understanding living cell biochemistry and efforts to understand such complication reaction environment have spread over wide front, ranging from systems biology approaches, through network analysis (motif identification) towards developing language and simulators for low level biochemical processes. Apart from simulation work, much of the efforts are directed to using mean field equations (equivalent to the equations of classical chemical kinetics) to address various problems (stability, robustness, sensitivity analysis, etc.). Rarely is the use of mean field equations questioned. This review will provide a brief overview of the situations when mean field equations fail and should not be used. These equations can be derived from the theory of diffusion controlled reactions, and emerge when assumption of perfect mixing is used.
Parabolic equations in biology growth, reaction, movement and diffusion
Perthame, Benoît
2015-01-01
This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.
LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS
无
2006-01-01
This work is devoted to the discussion of stochastic reaction diffusion equations and some new theorems on Lagrange stability in mean square of the solution are established via Lyapunov method which is nothing to be done in the past.
Effect of macromolecular crowding on the rate of diffusion-limited enzymatic reaction
Manish Agrawal; S B Santra; Rajat Anand; Rajaram Swaminathan
2008-08-01
The cytoplasm of a living cell is crowded with several macromolecules of different shapes and sizes. Molecular diffusion in such a medium becomes anomalous due to the presence of macromolecules and diffusivity is expected to decrease with increase in macromolecular crowding. Moreover, many cellular processes are dependent on molecular diffusion in the cell cytosol. The enzymatic reaction rate has been shown to be affected by the presence of such macromolecules. A simple numerical model is proposed here based on percolation and diffusion in disordered systems to study the effect of macromolecular crowding on the enzymatic reaction rates. The model qualitatively explains some of the experimental observations.
Mathematical aspects of reacting and diffusing systems
Fife, Paul C
1979-01-01
Modeling and analyzing the dynamics of chemical mixtures by means of differ- tial equations is one of the prime concerns of chemical engineering theorists. These equations often take the form of systems of nonlinear parabolic partial d- ferential equations, or reaction-diffusion equations, when there is diffusion of chemical substances involved. A good overview of this endeavor can be had by re- ing the two volumes by R. Aris (1975), who himself was one of the main contributors to the theory. Enthusiasm for the models developed has been shared by parts of the mathematical community, and these models have, in fact, provided motivation for some beautiful mathematical results. There are analogies between chemical reactors and certain biological systems. One such analogy is rather obvious: a single living organism is a dynamic structure built of molecules and ions, many of which react and diffuse. Other analogies are less obvious; for example, the electric potential of a membrane can diffuse like a chemical, and ...
Restrictive liquid-phase diffusion and reaction in bidispersed catalysts
Lee, S.Y.; Seader, J.D. (Utah Univ., Salt Lake City, UT (United States). Dept. of Chemical Engineering); Tsai, C.H.; Massoth, F.E. (Utah Univ., Salt Lake City, UT (United States). Dept. of Fuels Engineering)
1991-08-01
In this paper, the effect of bidispersed pore-size distribution on liquid-phase diffusion and reaction in NiMo/Al{sub 2}O{sub 3} catalysts is investigated by applying two bidispersed-pore-structure models, the random-pore model and a globular-structure model, to extensive experimental data, which were obtained from sorptive diffusion measurements at ambient conditions and catalytic reaction rate measurements on nitrogen-containing compounds. Transport of the molecules in the catalysts was found to be controlled by micropore diffusion, in accordance with the random-pore model, rather than macropore diffusion as predicted by the globular-structure model. A qualitative criterion for micropore-diffusion control is proposed: relatively small macroporosity and high catalyst pellet density. Since most hydrotreating catalysts have high density, diffusion in these types of catalysts may be controlled by micropore diffusion. Accordingly, it is believed in this case that increasing the size of micropores may be more effective to reduce intraparticle diffusion resistance than incorporating macropores alone.
Reaction-diffusion models of decontamination
Hjorth, Poul G.
A contaminant, which also contains a polymer is in the form of droplets on a solid surface. It is to be removed by the action of a decontaminant, which is applied in aqueous solution. The contaminant is only sparingly soluble in water, so the reaction mechanism is that it slowly dissolves...... in the aqueous solution and then is oxidized by the decontaminant. The polymer is insoluble in water, and so builds up near the interface, where its presence can impede the transport of contaminant. In these circumstances, Dstl wish to have mathematical models that give an understanding of the process, and can...
Guiding brine shrimp through mazes by solving reaction diffusion equations
Singal, Krishma; Fenton, Flavio
Excitable systems driven by reaction diffusion equations have been shown to not only find solutions to mazes but to also to find the shortest path between the beginning and the end of the maze. In this talk we describe how we can use the Fitzhugh-Nagumo model, a generic model for excitable media, to solve a maze by varying the basin of attraction of its two fixed points. We demonstrate how two dimensional mazes are solved numerically using a Java Applet and then accelerated to run in real time by using graphic processors (GPUs). An application of this work is shown by guiding phototactic brine shrimp through a maze solved by the algorithm. Once the path is obtained, an Arduino directs the shrimp through the maze using lights from LEDs placed at the floor of the Maze. This method running in real time could be eventually used for guiding robots and cars through traffic.
Reaction-diffusion problems in the physics of hot plasmas
Wilhelmsson, H
2000-01-01
The physics of hot plasmas is of great importance for describing many phenomena in the universe and is fundamental for the prospect of future fusion energy production on Earth. Nontrivial results of nonlinear electromagnetic effects in plasmas include the self-organization and self-formation in the plasma of structures compact in time and space. These are the consequences of competing processes of nonlinear interactions and can be best described using reaction-diffusion equations. Reaction-Diffusion Problems in the Physics of Hot Plasmas is focused on paradigmatic problems of a reaction-diffusion type met in many branches of science, concerning in particular the nonlinear interaction of electromagnetic fields with plasmas.
Spreading Speed for a Periodic Reaction-diffusion Model with Nonmonotone Birth Function
HUANG Ye-hui; WENG Pei-xuan
2012-01-01
A reaction-diffusion model for a single spccies with age structure and nonlocal reaction for periodic time t is derived.Some results about the model with monotone birth function are firstly introduced,and then by constructing two auxiliary equations and squeezing method,the spreading speed for the system with nonmonotone birth function is obtained.
Reaction-diffusion-branching models of stock price fluctuations
Tang, Lei-Han; Tian, Guang-Shan
Several models of stock trading (Bak et al., Physica A 246 (1997) 430.) are analyzed in analogy with one-dimensional, two-species reaction-diffusion-branching processes. Using heuristic and scaling arguments, we show that the short-time market price variation is subdiffusive with a Hurst exponent H=1/4. Biased diffusion towards the market price and blind-eyed copying lead to crossovers to the empirically observed random-walk behavior ( H=1/2) at long times. The calculated crossover forms and diffusion constants are shown to agree well with simulation data.
Chaotic advection, diffusion, and reactions in open flows
Tel, Tamas [Institute for Theoretical Physics, Eoetvoes University, P.O. Box 32, H-1518 Budapest, (Hungary); Karolyi, Gyoergy [Department of Civil Engineering Mechanics, Technical University of Budapest, Mueegyetem rpk. 3, H-1521 Budapest, (Hungary); Pentek, Aron [Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093-0238 (United States); Scheuring, Istvan [Department of Plant Taxonomy and Ecology, Research Group of Ecology and Theoretical Biology, Eoetvoes University, Ludovika ter 2, H-1083 Budapest, (Hungary); Toroczkai, Zoltan [Department of Physics, University of Maryland, College Park, Maryland 20742-4111 (United States); Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 (United States); Grebogi, Celso [Institute for Plasma Research, University of Maryland, College Park, Maryland 20742 (United States); Kadtke, James [Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093-0238 (United States)
2000-03-01
We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.
Liquid Film Diffusion on Reaction Rate in Submerged Biofilters
Christiansen, Pia; Hollesen, Line; Harremoës, Poul
1995-01-01
Experiments were carried out in order to investigate the influence of liquid film diffusion on reaction rate in a submerged biofilter with denitrification and in order to compare with a theoretical study of the mass transfer coefficient. The experiments were carried out with varied flow, identified...... by the empty bed velocity of inflow and recirculation, respectively 1.3, 2.8, 5.6 and 10.9 m/h. The filter material consisted of 3 mm biostyren spheres. The results indicate that the influence of liquid film diffusion on reaction rate can be ignored....
Layer-adapted meshes for reaction-convection-diffusion problems
Linß, Torsten
2010-01-01
This book on numerical methods for singular perturbation problems - in particular, stationary reaction-convection-diffusion problems exhibiting layer behaviour is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. A classification and a survey of layer-adapted meshes for reaction-convection-diffusion problems are included. This structured and comprehensive account of current ideas in the numerical analysis for various methods on layer-adapted meshes is addressed to researchers in finite element theory and perturbation problems. Finite differences, finite elements and finite volumes are all covered.
魏纯辉; 李俐玫
2011-01-01
作者应用谱理论和规范化Lyapunov-Schmidt约化方法研究了关于基因繁殖的反应扩散系统(0,1)解的定态分歧,在特征值的代数重数为1的情况下,在一定条件下得到了定态方程从二阶非退化奇点处分歧出的正则分歧解.%By using the Spectrum Theory and normalized Lyapunov-Schmidt Reduction method, the steady state bifurcation of the reaction-diffusion system from (0,1)for Gene Propagation model is discussed,bifurcated branches from a second order nondegenerate singular point with algebraic multiplicity one of eigenvalue raised from the given reaction-diffusion equations are investigated. And under some conditions, the regular bifurcated branches are obtained.
The speed of reaction-diffusion wavefronts in nonsteady media
Mendez, Vicenc [Departament de Medicina, Facultat de Ciencies de la Salut, Universitat Internacional de Catalunya. c/Gomera s/n, 08190-Sant Cugat del Valles (Barcelona) (Spain); Fort, Joaquim [Departament de Fisica, Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia (Spain); Pujol, Toni [Departament de Fisica, Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia (Spain)
2003-04-11
The evolution of the speed of wavefronts for reaction-diffusion equations with time-varying parameters is analysed. We make use of singular perturbative analysis to study the temporal evolution of the speed for pushed fronts. The analogy with Hamilton-Jacobi dynamics allows us to consider the problem for pulled fronts, which is described by Kolmogorov-Petrovskii-Piskunov (KPP) reaction kinetics. Both analytical studies are in good agreement with the results of numerical solutions.
The speed of reaction-diffusion wavefronts in nonsteady media
Méndez, V; Pujol, T
2003-01-01
The evolution of the speed of wavefronts for reaction-diffusion equations with time-varying parameters is analysed. We make use of singular perturbative analysis to study the temporal evolution of the speed for pushed fronts. The analogy with Hamilton-Jacobi dynamics allows us to consider the problem for pulled fronts, which is described by Kolmogorov-Petrovskii-Piskunov (KPP) reaction kinetics. Both analytical studies are in good agreement with the results of numerical solutions.
On the perturbation solution of interface-reaction controlled diffusion in solids
Zhi-Wei Cui; Feng Gao; Jian-Min Qu
2012-01-01
Insertion of species A into species B forms a product P through two kinetic processes,namely,(1) the chemical reaction between A and B that occurs at the B-P interface,and (2) the diffusion of species A in product P.These two processes are symbiotic in that the chemical reaction provides the driving force for the diffusion,while the diffusion sustains the chemical reaction by providing sufficient reactant to the reactive interface.In this paper,a mathematical framework is developed for the coupled reactiondiffusion processes.The resulting system of boundary and initial value problem is solved analytically for the case of interface-reaction controlled diffusion,i.e.,the rate of diffusion is much faster than the rate of chemical reaction at the interface so that the final kinetics are limited by the interface chemical reaction.Asymptotic expressions are given for the velocity of the reactive interface and the concentration of diffusing species under two different boundary conditions.
Reaction Diffusion and Chemotaxis for Decentralized Gathering on FPGAs
Bernard Girau
2009-01-01
and rapid simulations of the complex dynamics of this reaction-diffusion model. Then we describe the FPGA implementation of the environment together with the agents, to study the major challenges that must be solved when designing a fast embedded implementation of the decentralized gathering model. We analyze the results according to the different goals of these hardware implementations.
Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations
Alzahrani, Hasnaa H.
2016-07-26
A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.
Diffusion and reaction in microbead agglomerates.
Nunes Kirchner, Carolina; Träuble, Markus; Wittstock, Gunther
2010-04-01
Scanning electrochemical microscopy has been used to analyze the flux of p-aminonophenol (PAP) produced by agglomerates of polymeric microbeads modified with galactosidase as a model system for the bead-based heterogeneous immunoassays. With the use of mixtures of enzyme-modified and bare beads in defined ratio, agglomerates with different saturation levels of the enzyme modification were produced. The PAP flux depends on the intrinsic kinetics of the galactosidase, the local availability of the substrate p-aminophenyl-beta-D-galactopyranoside (PAPG), and the external mass transport conditions in the surrounding of the agglomerate and the internal mass transport within the bead agglomerate. The internal mass transport is influenced by the diffusional shielding of the modified beads by unmodified beads. SECM in combination with optical microscopy was used to determine experimentally the external flux. These data are in quantitative agreement with boundary element simulation considering the SECM microelectrode as an interacting probe and treating the Michaelis-Menten kinetics of the enzyme as nonlinear boundary conditions with two independent concentration variables [PAP] and [PAPG]. The PAPG concentration at the surface of the bead agglomerate was taken as a boundary condition for the analysis of the internal mass transport condition as a function of the enzyme saturation in the bead agglomerate. The results of this analysis are represented as PAP flux per contributing modified bead and the flux from freely suspended galactosidase-modified beads. These numbers are compared to the same number from the SECM experiments. It is shown that depending on the enzyme saturation level a different situation can arise where either beads located at the outer surface of the agglomerate dominate the contribution to the measured external flux or where the contribution of buried beads cannot be neglected for explaining the measured external flux.
De Oliveira Santos, F. [Grand Accelerateur National d' Ions Lourds, UMR 6415, 14 - Caen (France)
2007-07-01
Nuclear reactions can occur at low kinetic energy. Low-energy reactions are characterized by a strong dependence on the structure of the compound nucleus. It turns out that it is possible to study the nuclear structure by measuring these reactions. In this course, three types of reactions are treated: Resonant Elastic Scattering (such as N{sup 14}(p,p)N{sup 14}), Inelastic Scattering (such as N{sup 14}(p,p')N{sup 14*}) and Astrophysical reactions (such as N{sup 14}(p,{gamma})O{sup 15}). (author)
Reaction-diffusion pattern in shoot apical meristem of plants.
Hironori Fujita
Full Text Available A fundamental question in developmental biology is how spatial patterns are self-organized from homogeneous structures. In 1952, Turing proposed the reaction-diffusion model in order to explain this issue. Experimental evidence of reaction-diffusion patterns in living organisms was first provided by the pigmentation pattern on the skin of fishes in 1995. However, whether or not this mechanism plays an essential role in developmental events of living organisms remains elusive. Here we show that a reaction-diffusion model can successfully explain the shoot apical meristem (SAM development of plants. SAM of plants resides in the top of each shoot and consists of a central zone (CZ and a surrounding peripheral zone (PZ. SAM contains stem cells and continuously produces new organs throughout the lifespan. Molecular genetic studies using Arabidopsis thaliana revealed that the formation and maintenance of the SAM are essentially regulated by the feedback interaction between WUSHCEL (WUS and CLAVATA (CLV. We developed a mathematical model of the SAM based on a reaction-diffusion dynamics of the WUS-CLV interaction, incorporating cell division and the spatial restriction of the dynamics. Our model explains the various SAM patterns observed in plants, for example, homeostatic control of SAM size in the wild type, enlarged or fasciated SAM in clv mutants, and initiation of ectopic secondary meristems from an initial flattened SAM in wus mutant. In addition, the model is supported by comparing its prediction with the expression pattern of WUS in the wus mutant. Furthermore, the model can account for many experimental results including reorganization processes caused by the CZ ablation and by incision through the meristem center. We thus conclude that the reaction-diffusion dynamics is probably indispensable for the SAM development of plants.
Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation
Petr Stehlík
2015-01-01
Full Text Available We study reaction-diffusion equations with a general reaction function f on one-dimensional lattices with continuous or discrete time ux′ (or Δtux=k(ux-1-2ux+ux+1+f(ux, x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.
Theory of diffusion-influenced reactions in complex geometries
Galanti, Marta; Fanelli, Duccio; Traytak, Sergey D.; Piazza, Francesco
Chemical reactions involving diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced" (DI). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DI reactions, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DI reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical formulas can be derived easily in most cases.
Reaction-diffusion master equation in the microscopic limit
Hellander, Stefan; Hellander, Andreas; Petzold, Linda
2012-04-01
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.
On the Reaction Diffusion Master Equation in the Microscopic Limit
Hellander, Stefan; Petzold, Linda
2011-01-01
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice Reaction-Diffusion Master Equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. In this paper we give a new, general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model.
Theory of diffusion-influenced reactions in complex geometries
Galanti, Marta; Piazza, Francesco
2015-01-01
Chemical reactions involving diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced" (DI). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DI reactions, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DI reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical form...
Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Cox process representation and inference for stochastic reaction-diffusion processes
Schnoerr, David; Grima, Ramon; Sanguinetti, Guido
2016-05-01
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction-diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reaction-diffusion process from data. Our solution relies on a non-trivial connection between stochastic reaction-diffusion processes and spatio-temporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatio-temporal stochastic systems, and a practical solution to a long-standing problem in computational modelling.
Stochastic flows, reaction-diffusion processes, and morphogenesis
Kozak, J.J.; Hatlee, M.D.; Musho, M.K.; Politowicz, P.A.; Walsh, C.A.
1983-02-01
Recently, an exact procedure has been introduced (C. A. Walsh and J. J. Kozak, Phys. Rev. Lett.. 47: 1500 (1981)) for calculating the expected walk length
MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION
Li Yanling; Ma Yicheng
2005-01-01
The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region.Finally, a few examples of application are given.
Qualitative Analysis on a Reaction-Diffusion Prey Predator Model and the Corresponding Steady-States
Qunyi BIE; Rui PENG
2009-01-01
The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding steady-state problem.The local and global stability of the positive constant steady-state are discussed,and then some results for nonexistence of positive non-constant steady-states are derived.
Does reaction-diffusion support the duality of fragmentation effect?
Roques, Lionel
2009-01-01
There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal -- or similarly harvesting -- term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e. the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields ar...
Untangling knots via reaction-diffusion dynamics of vortex strings
Maucher, Fabian
2016-01-01
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.
Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings
Maucher, Fabian; Sutcliffe, Paul
2016-04-01
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.
Garcia Velarde, M.
1977-07-01
Thermo convective instabilities in horizontal fluid layers are discussed with emphasis on the Rayleigh-Bernard model problem. Steady solutions and time-dependent phenomena (relaxation oscillations and transition to turbulence) are studied within the nonlinear Boussinesq-Oberbeck approximation. Homogeneous steady solutions, limit cycles, and inhomogeneous (ordered) spatial structures are also studied in simple reaction-diffusion systems. Lastly, the non-periodic attractor that appears at large Rayleigh numbers in the truncated Boussinesq-Oberbeck model of Lorenz, is constructed, and a discussion of turbulent behavior is given. (Author) 105 refs.
A Note on the Kinetics of Diffusion-mediated Reactions
Naqvi, K Razi
2014-01-01
The prevalent scheme of a diffusion-mediated bimolecular reaction $A+B\\rightarrow P$ is an adaptation of that proposed by Briggs and Haldane for enzyme action [{\\em Biochem J.\\/}, 19:338--339, 1925]. The purpose of this Note is to explain, {\\em by using an argument involving no mathematics\\/}, why the breakup of the encounter complex cannot be described, except in special circumstances, in terms of a first-order process $\\{AB\\}\\rightarrow A+B$. Briefly, such a description neglects the occurrence of re-encounters, which lie at the heart of Noyes's theory of diffusion-mediated reactions. The relation $k=\\alpha k_{\\mbox{\\scriptsize e}}$ becomes valid only when $\\alpha$ (the reaction probability per encounter) is very much smaller than unity (activation-controlled reactions), or when $\\beta$ (the re-encounter probability) is negligible (as happens in a gas-phase reaction). References to some works (by the author and his collaborators) which propound the correct approach for finding $k$ are also supplied.
Reaction Diffusion Voronoi Diagrams: From Sensors Data to Computing
Alejandro Vázquez-Otero
2015-05-01
Full Text Available In this paper, a new method to solve computational problems using reaction diffusion (RD systems is presented. The novelty relies on the use of a model configuration that tailors its spatiotemporal dynamics to develop Voronoi diagrams (VD as a part of the system’s natural evolution. The proposed framework is deployed in a solution of related robotic problems, where the generalized VD are used to identify topological places in a grid map of the environment that is created from sensor measurements. The ability of the RD-based computation to integrate external information, like a grid map representing the environment in the model computational grid, permits a direct integration of sensor data into the model dynamics. The experimental results indicate that this method exhibits significantly less sensitivity to noisy data than the standard algorithms for determining VD in a grid. In addition, previous drawbacks of the computational algorithms based on RD models, like the generation of volatile solutions by means of excitable waves, are now overcome by final stable states.
Kinetic pathways of diffusion and solid-state reactions in nanostructured thin films
Beke, D. L.; Langer, G. A.; Molnár, G.; Erdélyi, G.; Katona, G. L.; Lakatos, A.; Vad, K.
2013-06-01
Mass transport and solid-state reactions in nanocrystalline thin films are reviewed. It is illustrated that diffusion along different grain boundaries (GBs) can have important effects on the overall intermixing process between two pure films. These processes can be well characterized by a bimodal GB network, with different (fast and slow) diffusivities. First the atoms migrate along fast GBs and accumulate at the film surface. These accumulated atoms form a secondary diffusion source for back diffusion along slow boundaries. Thus the different GBs of the thin films can be gradually filled up with the diffusing atoms and composition depth profiles reflect the result of these processes. Similar processes can be observed in binary systems with intermetallic layers: instead of nucleation and growth of the reaction layer at the initial interface, the reaction takes place in the GBs and the amount of the product phase grows by the motion of its interfaces perpendicular to the GBs. Thus, the entire layer of the pure parent films can be consumed by this GB diffusion-induced solid-state reaction (GBDIREAC), and a fully homogeneous product layer can be obtained.
O'Hara, Kieran
2007-08-01
In the southern Appalachians, the Blue Ridge-Piedmont crystalline thrust sheet was emplaced onto low-grade Late Precambrian and Paleozoic sedimentary rocks in the footwall along a basal detachment consisting of phyllosilicate-rich mylonites (phyllonites). The phyllonites developed first by mechanical breakdown of feldspar followed by chemical breakdown to white mica in the presence of a pore fluid. Diffusion of solute in the pore fluid is the rate limiting step in controlling reaction rate and also the strain rate. Assuming solute diffusion follows the Stokes-Einstein equation, the shear strain rate is given by ⅆγ/ⅆt=2ωkT/5ηrx for shear stress ≥20 MPa, where n is a constant, ω is a geometric factor, k is Boltzmann's constant, T is absolute temperature, η is water viscosity, r is the atomic radius of the diffusing species, and x is the diffusion distance. A bulk diffusion coefficient in the range of ˜10 -10 to 10 -12 m 2/s over distances of 10-100 m results in strain rates of 10 -14 to 10 -13 s -1 in the temperature range 200-400 °C. It is concluded that greenschist grade crystalline thrust sheets develop on pre-existing basement faults that become weak during reaction softening and localize into high strain phyllonite zones in which pore fluid diffusion controls reaction rate and strain rate.
A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model
L. Päivärinta
2011-12-01
Full Text Available In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM. Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT is applied to enlarge the convergence region and rate of solution series given by the HAM.
Chen, Weiliang
2016-01-01
Stochastic, spatial reaction-diffusion simulations have been widely used in systems biology and computational neuroscience. However, the increasing scale and complexity of simulated models and morphologies have exceeded the capacity of any serial implementation. This led to development of parallel solutions that benefit from the boost in performance of modern large-scale supercomputers. In this paper, we describe an MPI-based, parallel Operator-Splitting implementation for stochastic spatial reaction-diffusion simulations with irregular tetrahedral meshes. The performance of our implementation is first examined and analyzed with simulations of a simple model. We then demonstrate its usage in real-world research by simulating the reaction-diffusion components of a published calcium burst model in both Purkinje neuron sub-branch and full dendrite morphologies. Simulation results indicate that our implementation is capable of achieving super-linear speedup for balanced loading simulations with reasonable molecul...
R S Kaushal; Ranjit Kumar; Awadhesh Prasad
2006-08-01
Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.
The Induced Dimension Reduction method applied to convection-diffusion-reaction problems
Astudillo, R.; Van Gijzen, M.B.
2016-01-01
Discretization of (linearized) convection-diffusion-reaction problems yields a large and sparse non symmetric linear system of equations, Ax = b. (1) In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) [10], with other short-recurrences Krylov met
Variational methods applied to problems of diffusion and reaction
Strieder, William
1973-01-01
This monograph is an account of some problems involving diffusion or diffusion with simultaneous reaction that can be illuminated by the use of variational principles. It was written during a period that included sabbatical leaves of one of us (W. S. ) at the University of Minnesota and the other (R. A. ) at the University of Cambridge and we are grateful to the Petroleum Research Fund for helping to support the former and the Guggenheim Foundation for making possible the latter. We would also like to thank Stephen Prager for getting us together in the first place and for showing how interesting and useful these methods can be. We have also benefitted from correspondence with Dr. A. M. Arthurs of the University of York and from the counsel of Dr. B. D. Coleman the general editor of this series. Table of Contents Chapter 1. Introduction and Preliminaries . 1. 1. General Survey 1 1. 2. Phenomenological Descriptions of Diffusion and Reaction 2 1. 3. Correlation Functions for Random Suspensions 4 1. 4. Mean Free ...
Mechanical reaction-diffusion model for bacterial population dynamics
Ngamsaad, Waipot
2015-01-01
The effect of mechanical interaction between cells on the spreading of bacterial population was investigated in one-dimensional space. A nonlinear reaction-diffusion equation has been formulated as a model for this dynamics. In this model, the bacterial cells are treated as the rod-like particles that interact, when contacting each other, through the hard-core repulsion. The repulsion introduces the exclusion process that causes the fast diffusion in bacterial population at high density. The propagation of the bacterial density as the traveling wave front in long time behavior has been analyzed. The analytical result reveals that the front speed is enhanced by the exclusion process---and its value depends on the packing fraction of cell. The numerical solutions of the model have been solved to confirm this prediction.
Turing instability in reaction-diffusion models on complex networks
Ide, Yusuke; Izuhara, Hirofumi; Machida, Takuya
2016-09-01
In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied. Here, we focus on three types of models which generate complex networks, i.e. the Erdős-Rényi, the Watts-Strogatz, and the threshold network models. From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion coefficients completely differ, depending on the network architecture. In addition, we theoretically discuss the stable and unstable regions in the cases of regular enhanced ring lattices which include regular circles, and networks generated by the threshold network model when the number of vertices is large enough.
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Control of DNA replication by anomalous reaction-diffusion kinetics
Bechhoefer, John; Gauthier, Michel
2010-03-01
DNA replication requires two distinct processes: the initiation of pre-licensed replication origins and the propagation of replication forks away from the fired origins. Experiments indicate that these origins are triggered over the whole genome at a rate I(t) (the number of initiations per unreplicated length per time) that increases throughout most of the synthesis (S) phase, before rapidly decreasing to zero at the end of the replication process. We propose a simple model for the control of DNA replication in which the rate of initiation of replication origins is controlled by protein-DNA interactions. Analyzing recent data from Xenopus frog embryos, we find that the initiation rate is reaction limited until nearly the end of replication, when it becomes diffusion limited. Initiation of origins is suppressed when the diffusion-limited search time dominates. To fit the experimental data, we find that the interaction between DNA and the rate-limiting protein must be subdiffusive.
Enhanced Diffusion of Enzymes that Catalyze Exothermic Reactions
Golestanian, Ramin
2015-09-01
Enzymes have been recently found to exhibit enhanced diffusion due to their catalytic activities. A recent experiment [C. Riedel et al., Nature (London) 517, 227 (2015)] has found evidence that suggests this phenomenon might be controlled by the degree of exothermicity of the catalytic reaction involved. Four mechanisms that can lead to this effect, namely, self-thermophoresis, boost in kinetic energy, stochastic swimming, and collective heating are critically discussed, and it is shown that only the last two can be strong enough to account for the observations. The resulting quantitative description is used to examine the biological significance of the effect.
On the solutions of fractional reaction-diffusion equations
Jagdev Singh
2013-05-01
Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative. The results are derived by the application of the Laplace and Fourier transforms in compact and elegant form in terms of Mittag-Leffler function and H-function. The results obtained here are of general nature and include the results investigated earlier by many authors.
ENERGY ESTIMATES FOR DELAY DIFFUSION-REACTION EQUATIONS
J.A.Ferreira; P.M.da Silva
2008-01-01
In this paper we consider nonlinear delay diffusion-reaction equations with initial and Dirichlet boundary conditions.The behaviour and the stability of the solution of such initial boundary value problems(IBVPs)are studied using the energy method.Simple numerical methods are considered for the computation of numerical approximations to the solution of the nonlinear IBVPs.Using the discrete energy method we study the stability and convergence of the numerical approximations.Numerical experiments are carried out to illustrate our theoretical results.
Dulos, E.; Hunding, A.; Boissonade, J.; de Kepper, P.
Since the seminal paper "The chemical basis of morphogenesis" by Alan Turing, the temporal and spatial self-organization phenomena produced in chemically reacting and diffusing systems are often thought as paradigms for biological development. The basic theoretical principles on which the development of stationary concentration patterns (Turing structures) rely on are briefly presented. We review different aspects of our contribution to the experimental observation of reaction-diffusion patterns in iodine-oxychlorine systems. The experimental techniques are emphasized. Phase diagrams gathering different standing and travelling patterns are presented, analyzed and modeled. A special attention is also given to some peculiar pattern growth dynamics (spot division, finger splitting).
Uniform asymptotic approximation of diffusion to a small target: Generalized reaction models
Isaacson, Samuel A.; Mauro, Ava J.; Newby, Jay
2016-10-01
The diffusion of a reactant to a binding target plays a key role in many biological processes. The reaction radius at which the reactant and target may interact is often a small parameter relative to the diameter of the domain in which the reactant diffuses. We develop uniform in time asymptotic expansions in the reaction radius of the full solution to the corresponding diffusion equations for two separate reactant-target interaction mechanisms: the Doi or volume reactivity model and the Smoluchowski-Collins-Kimball partial-absorption surface reactivity model. In the former, the reactant and target react with a fixed probability per unit time when within a specified separation. In the latter, upon reaching a fixed separation, they probabilistically react or the reactant reflects away from the target. Expansions of the solution to each model are constructed by projecting out the contribution of the first eigenvalue and eigenfunction to the solution of the diffusion equation and then developing matched asymptotic expansions in Laplace-transform space. Our approach offers an equivalent, but alternative, method to the pseudopotential approach we previously employed [Isaacson and Newby, Phys. Rev. E 88, 012820 (2013), 10.1103/PhysRevE.88.012820] for the simpler Smoluchowski pure-absorption reaction mechanism. We find that the resulting asymptotic expansions of the diffusion equation solutions are identical with the exception of one parameter: the diffusion-limited reaction rates of the Doi and partial-absorption models. This demonstrates that for biological systems in which the reaction radius is a small parameter, properly calibrated Doi and partial-absorption models may be functionally equivalent.
Theory of diffusion-influenced reactions in complex geometries.
Galanti, Marta; Fanelli, Duccio; Traytak, Sergey D; Piazza, Francesco
2016-06-21
Chemical transformations involving the diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced reactions" (DIR). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DIRs, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory that is able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DIR reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical formulas can be derived easily in most cases.
Application of diffusion-reaction equations to model carious lesion progress
Lewandowska, Katarzyna D.; Kosztołowicz, Tadeusz
2012-04-01
Nonlinear equations that describe the diffusion-reaction process with one static and one mobile substance are used to model a carious lesion process. The system under consideration consists of two initially separated substances A (an acid causing caries) and C (a static enamel mineral) which react chemically according to the formula A+C→0̸(inert). The so-called surface layer, which is formed in this process and in which chemical reactions can be neglected, is also included in this model. Changes in the substance concentrations are calculated approximately using the perturbation method. We show that the experimental data on the enamel mineral concentrations are well described by the analytical solutions of the diffusion-reaction equations.
Describing diffusion, reaction and convection on porous medium
D'Ajello, P C T; Nunes, G L
2013-01-01
In this paper we present a mathematical model for the electrochemical deposition aimed at the production of inverse opals. The real system consists of an arrangement of sub micrometer spheres, through which the species in an electrolytic medium diffuses until they react to the electrode surface and become part thereof. Our model consists in formulating convenient boundary conditions for the transport equation, that somewhat resembles the real system but is nevertheless simple enough to be solved, and then solve it. Similar approach was taken by Nicholson [1, 2], except that, to avoid the difficulties regarding the boundary conditions, he considered none whatsoever, and proposed a modified diffusion coefficient for the porous medium instead. Apropos, our model, with moving boundary condition pertain to the class of problems know as The Stefan problem [3].
Le Novère Nicolas
2010-03-01
Full Text Available Abstract Background Most cellular signal transduction mechanisms depend on a few molecular partners whose roles depend on their position and movement in relation to the input signal. This movement can follow various rules and take place in different compartments. Additionally, the molecules can form transient complexes. Complexation and signal transduction depend on the specific states partners and complexes adopt. Several spatial simulator have been developed to date, but none are able to model reaction-diffusion of realistic multi-state transient complexes. Results Meredys allows for the simulation of multi-component, multi-feature state molecular species in two and three dimensions. Several compartments can be defined with different diffusion and boundary properties. The software employs a Brownian dynamics engine to simulate reaction-diffusion systems at the reactive particle level, based on compartment properties, complex structure, and hydro-dynamic radii. Zeroth-, first-, and second order reactions are supported. The molecular complexes have realistic geometries. Reactive species can contain user-defined feature states which can modify reaction rates and outcome. Models are defined in a versatile NeuroML input file. The simulation volume can be split in subvolumes to speed up run-time. Conclusions Meredys provides a powerful and versatile way to run accurate simulations of molecular and sub-cellular systems, that complement existing multi-agent simulation systems. Meredys is a Free Software and the source code is available at http://meredys.sourceforge.net/.
Reaction-Transport Systems Mesoscopic Foundations, Fronts, and Spatial Instabilities
Horsthemke, Werner; Mendez, Vicenc
2010-01-01
This book is an introduction to the dynamics of reaction-diffusion systems, with a focus on fronts and stationary spatial patterns. Emphasis is on systems that are non-standard in the sense that either the transport is not simply classical diffusion (Brownian motion) or the system is not homogeneous. A important feature is the derivation of the basic phenomenological equations from the mesoscopic system properties. Topics addressed include transport with inertia, described by persistent random walks and hyperbolic reaction-transport equations and transport by anomalous diffusion, in particular subdiffusion, where the mean square displacement grows sublinearly with time. In particular reaction-diffusion systems are studied where the medium is in turn either spatially inhomogeneous, compositionally heterogeneous or spatially discrete. Applications span a vast range of interdisciplinary fields and the systems considered can be as different as human or animal groups migrating under external influences, population...
Multiscale Reaction-Diffusion Algorithms: PDE-Assisted Brownian Dynamics
Franz, Benjamin
2013-06-19
Two algorithms that combine Brownian dynami cs (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface, which partitions the domain, and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that the overlap region is required to accurately compute variances using PBD simulations. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented. © 2013 Society for Industrial and Applied Mathematics.
Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics
Franz, Benjamin; Chapman, S Jonathan; Erban, Radek
2012-01-01
Two algorithms that combine Brownian dynamics (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface which partitions the domain and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that to accurately compute variances using the PBD simulation requires the overlap region. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented.
SDDEs limits solutions to sublinear reaction-diffusion SPDEs
Hassan Allouba
2003-11-01
Full Text Available We start by introducing a new definition of solutions to heat-based SPDEs driven by space-time white noise: SDDEs (stochastic differential-difference equations limits solutions. In contrast to the standard direct definition of SPDEs solutions; this new notion, which builds on and refines our SDDEs approach to SPDEs from earlier work, is entirely based on the approximating SDDEs. It is applicable to, and gives a multiscale view of, a variety of SPDEs. We extend this approach in related work to other heat-based SPDEs (Burgers, Allen-Cahn, and others and to the difficult case of SPDEs with multi-dimensional spacial variable. We focus here on one-spacial-dimensional reaction-diffusion SPDEs; and we prove the existence of a SDDEs limit solution to these equations under less-than-Lipschitz conditions on the drift and the diffusion coefficients, thus extending our earlier SDDEs work to the nonzero drift case. The regularity of this solution is obtained as a by-product of the existence estimates. The uniqueness in law of our SPDEs follows, for a large class of such drifts/diffusions, as a simple extension of our recent Allen-Cahn uniqueness result. We also examine briefly, through order parameters $epsilon_1$ and $epsilon_2$ multiplied by the Laplacian and the noise, the effect of letting $epsilon_1,epsilon_2o 0$ at different speeds. More precisely, it is shown that the ratio $epsilon_2/epsilon_1^{1/4}$ determines the behavior as $epsilon_1,epsilon_2o 0$.
Characterization of Cocycle Attractors for Nonautonomous Reaction-Diffusion Equations
Cardoso, C. A.; Langa, J. A.; Obaya, R.
In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction-diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li-Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee-Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.
A Reaction-Diffusion Model of Cholinergic Retinal Waves
Lansdell, Benjamin; Ford, Kevin; Kutz, J. Nathan
2014-01-01
Prior to receiving visual stimuli, spontaneous, correlated activity in the retina, called retinal waves, drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their inter-wave interval and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In addition to simulation, we are thus able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how parameters such as the spontaneous activation rate and sAHP refractory period contribute to critical wave size variability. PMID:25474327
An Interface Stretching-Diffusion Model for Mixing-Limited Reactions During Convective Mixing
Hidalgo, J. J.; Dentz, M.; Cabeza, Y.; Carrera, J.
2014-12-01
We study the behavior of mixing-limited dissolution reactions under the unstable flow conditions caused by a Rayleigh-Bénard convective instability in a two fluids system. The reactions produce a dissolution pattern that follows the ascending fluids's interface where the largest concentration gradients and maximum mixing are found. Contrary to other chemical systems, the mixing history engraved by the dissolution does not map out the fingering geometry of the unstable flow. The temporal scaling of the mixing Χ and the reaction rate r are explained by a stretching-diffusion model of the interface between the fluids. The model accurately reproduces the three observed regimes: a diffusive regime at which Χ, r ~ t-1/2; a convective regime of at which the interface contracts to the Batchelor scale resulting in a constant Χf and r independent of the Rayleigh number; and an attenuated convection regime in which Χ and r decay faster than diffusion as t-3/2 and t-1, respectevely, because of the decompression of the interface and weakened reactions caused by the accumulation of dissolved fluid below the interface.
Propagating fronts in reaction-transport systems with memory
Yadav, A. [Department of Chemistry, Southern Methodist University, Dallas, TX 75275-0314 (United States)], E-mail: ayadav1@lsu.edu; Fedotov, Sergei [School of Mathematics, University of Manchester, Manchester M60 1DQ (United Kingdom)], E-mail: sergei.fedotov@manchester.ac.uk; Mendez, Vicenc [Grup de Fisica Estadistica, Departament de Fisica, Universitat Autonoma de Barcelona, E-08193 Bellaterra (Spain)], E-mail: vicenc.mendez@uab.es; Horsthemke, Werner [Department of Chemistry, Southern Methodist University, Dallas, TX 75275-0314 (United States)], E-mail: whorsthe@smu.edu
2007-11-26
In reaction-transport systems with non-standard diffusion, the memory of the transport causes a coupling of reactions and transport. We investigate the effect of this coupling for systems with Fisher-type kinetics and obtain a general analytical expression for the front speed. We apply our results to the specific case of subdiffusion.
Hellander, Andreas; Lawson, Michael J.; Drawert, Brian; Petzold, Linda
2014-06-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps were adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the diffusive finite-state projection (DFSP) method, to incorporate temporal adaptivity.
Jacquemet, Vincent
2010-09-01
Microscale electrical propagation in the heart can be modeled by a reaction-diffusion system, describing cell and tissue electrophysiology. Macroscale features of wavefront propagation can be reproduced by an eikonal model, a reduced formulation involving only wavefront shape. In this paper, these two approaches are combined to incorporate global information about reentrant pathways into a reaction-diffusion model. The eikonal-diffusion formulation is generalized to handle reentrant activation patterns and wavefront collisions. Boundary conditions are used to specify pathways of reentry. Finite-element-based numerical methods are presented to solve this nonlinear equation on a coarse triangular mesh. The macroscale eikonal model serves to construct an initial condition for the microscale reaction-diffusion model. Electrical propagation simulated from this initial condition is then compared to the isochrones predicted by the eikonal model. Results in 2-D and thin 3-D test-case geometries demonstrate the ability of this technique to initiate anatomical and functional reentries along prescribed pathways, thus facilitating the development of dedicated models aimed at better understanding clinical case reports.
Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models
Narcisa Apreutesei
2014-05-01
Full Text Available In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.
Ohmori, Shousuke; Yamazaki, Yoshihiro
2016-01-01
Ultradiscrete equations are derived from a set of reaction-diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.
A practical guide to stochastic simulations of reaction-diffusion processes
Erban, Radek; Chapman, Jonathan; Maini, Philip
2007-01-01
A practical introduction to stochastic modelling of reaction-diffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reaction-diffusion methods are presented. The connections between stochastic simul...
A reaction diffusion model of pattern formation in clustering of adatoms on silicon surfaces
Trilochan Bagarti
2012-12-01
Full Text Available We study a reaction diffusion model which describes the formation of patterns on surfaces having defects. Through this model, the primary goal is to study the growth process of Ge on Si surface. We consider a two species reaction diffusion process where the reacting species are assumed to diffuse on the two dimensional surface with first order interconversion reaction occuring at various defect sites which we call reaction centers. Two models of defects, namely a ring defect and a point defect are considered separately. As reaction centers are assumed to be strongly localized in space, the proposed reaction-diffusion model is found to be exactly solvable. We use Green's function method to study the dynamics of reaction diffusion processes. Further we explore this model through Monte Carlo (MC simulations to study the growth processes in the presence of a large number of defects. The first passage time statistics has been studied numerically.
Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds
Desvillettes, Laurent
2008-01-01
In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in L1 to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global L∞ bound via interpolation of a polynomially growing H1 bound with the almost exponential L1 convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.
Giammar, Daniel E; Wang, Fei; Guo, Bin; Surface, J Andrew; Peters, Catherine A; Conradi, Mark S; Hayes, Sophia E
2014-12-16
Reactions of CO2 with magnesium silicate minerals to precipitate magnesium carbonates can result in stable carbon sequestration. This process can be employed in ex situ reactors or during geologic carbon sequestration in magnesium-rich formations. The reaction of aqueous CO2 with the magnesium silicate mineral forsterite was studied in systems with transport controlled by diffusion. The approach integrated bench-scale experiments, an in situ spectroscopic technique, and reactive transport modeling. Experiments were performed using a tube packed with forsterite and open at one end to a CO2-rich solution. The location and amounts of carbonate minerals that formed were determined by postexperiment characterization of the solids. Complementing this ex situ characterization, (13)C NMR spectroscopy tracked the inorganic carbon transport and speciation in situ. The data were compared with the output of reactive transport simulations that accounted for diffusive transport processes, aqueous speciation, and the forsterite dissolution rate. All three approaches found that the onset of magnesium carbonate precipitation was spatially localized about 1 cm from the opening of the forsterite bed. Magnesite was the dominant reaction product. Geochemical gradients that developed in the diffusion-limited zones led to locally supersaturated conditions at specific locations even while the volume-averaged properties of the system remained undersaturated.
Growth and dispersal with inertia: Hyperbolic reaction-transport systems
Méndez, Vicenç; Campos, Daniel; Horsthemke, Werner
2014-10-01
We investigate the behavior of five hyperbolic reaction-diffusion equations most commonly employed to describe systems of interacting organisms or reacting particles where dispersal displays inertia. We first discuss the macroscopic or mesoscopic foundation, or lack thereof, of these reaction-transport equations. This is followed by an analysis of the temporal evolution of spatially uniform states. In particular, we determine the uniform steady states of the reaction-transport systems and their stability properties. We then address the spatiotemporal behavior of pure death processes. We end with a unified treatment of the front speed for hyperbolic reaction-diffusion equations with Kolmogorov-Petrosvskii-Piskunov kinetics. In particular, we obtain an exact expression for the front speed of a general class of reaction correlated random walk systems. Our results establish that three out of the five hyperbolic reaction-transport equations provide physically acceptable models of biological and chemical systems.
The First Integral Method to Study a Class of Reaction-Diffusion Equations
KE Yun-Quan; YU Jun
2005-01-01
In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by using the first integral method.
Studies of the accuracy of time integration methods for reaction-diffusion equations
Ropp, David L.; Shadid, John N.; Ober, Curtis C.
2004-03-01
In this study we present numerical experiments of time integration methods applied to systems of reaction-diffusion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the competing component time scales. Nearly balanced systems can produce a significant coupling of the physical mechanisms and introduce a slow dynamical time scale of interest. These problems provide a challenging test for this evaluation and tend to reveal subtle differences between the various methods. The methods we consider include first- and second-order semi-implicit, fully implicit, and operator-splitting techniques. The test problems include a prototype propagating nonlinear reaction-diffusion wave, a non-equilibrium radiation-diffusion system, a Brusselator chemical dynamics system and a blow-up example. In this evaluation we demonstrate a "split personality" for the operator-splitting methods that we consider. While operator-splitting methods often obtain very good accuracy, they can also manifest a serious degradation in accuracy due to stability problems.
An integration factor method for stochastic and stiff reaction–diffusion systems
Ta, Catherine; Wang, Dongyong; Nie, Qing, E-mail: qnie@uci.edu
2015-08-15
Stochastic effects are often present in the biochemical systems involving reactions and diffusions. When the reactions are stiff, existing numerical methods for stochastic reaction diffusion equations require either very small time steps for any explicit schemes or solving large nonlinear systems at each time step for the implicit schemes. Here we present a class of semi-implicit integration factor methods that treat the diffusion term exactly and reaction implicitly for a system of stochastic reaction–diffusion equations. Our linear stability analysis shows the advantage of such methods for both small and large amplitudes of noise. Direct use of the method to solving several linear and nonlinear stochastic reaction–diffusion equations demonstrates good accuracy, efficiency, and stability properties. This new class of methods, which are easy to implement, will have broader applications in solving stochastic reaction–diffusion equations arising from models in biology and physical sciences.
Gan, Qintao; Lv, Tianshi; Fu, Zhenhua
2016-04-01
In this paper, the synchronization problem for a class of generalized neural networks with time-varying delays and reaction-diffusion terms is investigated concerning Neumann boundary conditions in terms of p-norm. The proposed generalized neural networks model includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks as its special cases. By establishing a new inequality, some simple and useful conditions are obtained analytically to guarantee the global exponential synchronization of the addressed neural networks under the periodically intermittent control. According to the theoretical results, the influences of diffusion coefficients, diffusion space, and control rate on synchronization are analyzed. Finally, the feasibility and effectiveness of the proposed methods are shown by simulation examples, and by choosing different diffusion coefficients, diffusion spaces, and control rates, different controlled synchronization states can be obtained.
Linearization of Systems of Nonlinear Diffusion Equations
KANG Jing; QU Chang-Zheng
2007-01-01
We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.
Vogel, Bernhard; Vogel, Heike; Fiedler, Franz
1994-01-01
A model system is presented that takes into account the main physical and chemical processes on the regional scale here in an area of 100x100 sq km. The horizontal gridsize used is 2x2 sq km. For a case study, it is demonstrated how the model system can be used to separate the contributions of the processes advection, turbulent diffusion, and chemical reactions to the diurnal cycle of ozone. In this way, typical features which are visible in observations and are reproduced by the numerical simulations can be interpreted.
Zoumpanioti, M; Parmaklis, P; de María, P Domínguez; Stamatis, H; Sinisterra, J V; Xenakis, A
2008-09-01
Rhizomucor miehei lipase was immobilized in hydroxy(propylmethyl) cellulose or agar gels containing lecithin or AOT microemulsions. The effect of the diffusion of substrates and products to this catalyst was studied, as well as the effect of temperature on the initial rate of ester synthesis. The composition of the gel affects the reaction rate due to mass transport phenomena. The apparent activation energies were higher for the systems based on agar, independently of the microemulsion used, and lower for the systems based on AOT microemulsions, independently of the polymer used.
A reaction-diffusion model of human brain development.
Julien Lefèvre
2010-04-01
Full Text Available Cortical folding exhibits both reproducibility and variability in the geometry and topology of its patterns. These two properties are obviously the result of the brain development that goes through local cellular and molecular interactions which have important consequences on the global shape of the cortex. Hypotheses to explain the convoluted aspect of the brain are still intensively debated and do not focus necessarily on the variability of folds. Here we propose a phenomenological model based on reaction-diffusion mechanisms involving Turing morphogens that are responsible for the differential growth of two types of areas, sulci (bottom of folds and gyri (top of folds. We use a finite element approach of our model that is able to compute the evolution of morphogens on any kind of surface and to deform it through an iterative process. Our model mimics the progressive folding of the cortical surface along foetal development. Moreover it reveals patterns of reproducibility when we look at several realizations of the model from a noisy initial condition. However this reproducibility must be tempered by the fact that a same fold engendered by the model can have different topological properties, in one or several parts. These two results on the reproducibility and variability of the model echo the sulcal roots theory that postulates the existence of anatomical entities around which the folding organizes itself. These sulcal roots would correspond to initial conditions in our model. Last but not least, the parameters of our model are able to produce different kinds of patterns that can be linked to developmental pathologies such as polymicrogyria and lissencephaly. The main significance of our model is that it proposes a first approach to the issue of reproducibility and variability of the cortical folding.
Modeling and finite difference numerical analysis of reaction-diffusion dynamics in a microreactor.
Plazl, Igor; Lakner, Mitja
2010-03-01
A theoretical description with numerical experiments and analysis of the reaction-diffusion processes of homogeneous and non-homogeneous reactions in a microreactor is presented considering the velocity profile for laminar flows of miscible and immiscible fluids in a microchannel at steady-state conditions. A Mathematical model in dimensionless form, containing convection, diffusion, and reaction terms are developed to analyze and to forecast the reactor performance. To examine the performance of different types of reactors, the outlet concentrations for the plug-flow reactor (PFR), and the continuous stirred-tank reactor (CSTR) are also calculated for the case of an irreversible homogeneous reaction of two components. The comparison of efficiency between ideal conventional macroscale reactors and the microreactor is presented for a wide range of operating conditions, expressed as different Pe numbers (0.01 < Pe < 10). The numerical procedure of complex non-linear systems based on an implicit finite-difference method improved by non-equidistant differences is proposed.
The small-voxel tracking algorithm for simulating chemical reactions among diffusing molecules
Seitaridou, Effrosyni
2014-01-01
Simulating the evolution of a chemically reacting system using the bimolecular propensity function, as is done by the stochastic simulation algorithm and its reaction-diffusion extension, entails making statistically inspired guesses as to where the reactant molecules are at any given time. Those guesses will be physically justified if the system is dilute and well-mixed in the reactant molecules. Otherwise, an accurate simulation will require the extra effort and expense of keeping track of the positions of the reactant molecules as the system evolves. One molecule-tracking algorithm that pays careful attention to the physics of molecular diffusion is the enhanced Green's function reaction dynamics (eGFRD) of Takahashi, Tănase-Nicola, and ten Wolde [Proc. Natl. Acad. Sci. U.S.A.141, 2473 (2010)]. We introduce here a molecule-tracking algorithm that has the same theoretical underpinnings and strategic aims as eGFRD, but a different implementation procedure. Called the small-voxel tracking algorithm (SVTA), it combines the well known voxel-hopping method for simulating molecular diffusion with a novel procedure for rectifying the unphysical predictions of the diffusion equation on the small spatiotemporal scale of molecular collisions. Indications are that the SVTA might be more computationally efficient than eGFRD for the problematic class of non-dilute systems. A widely applicable, user-friendly software implementation of the SVTA has yet to be developed, but we exhibit some simple examples which show that the algorithm is computationally feasible and gives plausible results. PMID:25527927
Wu, Zeng-Qiang; Li, Zhong-Qiu; Li, Jin-Yi; Gu, Jing; Xia, Xing-Hua
2016-05-25
The spatial positioning of enzymes and mass transport play crucial roles in the functionality and efficiency of enzyme cascade reactions. To fully understand the mass transport regulating kinetics of enzyme cascade reactions, we investigated the contribution of convective and diffusive transports to a cascade reaction of β-galactosidase (β-Gal)/glucose oxidase (GOx) confined in a microchannel. β-Gal and GOx are assembled on two separated gold films patterned in a polydimethylsiloxane (PDMS) microchannel with a controllable distance from 50 to 100 μm. Experimental results demonstrated that the reaction yield increases with decreasing distance between two enzymes and increasing substrate flow rate. Together with the simulation results, we extracted individual reaction kinetics of the enzyme cascade reaction and found that the reaction rate catalyzed by β-Gal occurred much faster than by GOx, and thus, the β-Gal catalytic reaction showed diffusion controll, whereas the GOx catalytic reaction showed kinetic controll. Since the decrease in the enzymes distance shortens the transport length of intermediate glucose to GOx, the amount of glucose reaching GOx will be increased in the unit time, and in turn, the enzyme cascade reaction yield will be increased with decreasing the gap distance. This phenomenon is similar to the intermediates pool of tricarboxylic acid (TCA) cycle in the metabolic system. This study promotes the understanding of the metabolic/signal transduction processes and active transport in biological systems and promises to design high performance biosensors and biofuel cells systems.
Berlowitz, D.R.
1996-11-01
In the last few decades the negative impact by humans on the thin atmospheric layer enveloping the earth, the basis for life on this planet, has increased steadily. In order to halt, or at least slow down this development, the knowledge and study of these anthropogenic influence has to be increased and possible remedies have to be suggested. An important tool for these studies are computer models. With their help the atmospheric system can be approximated and the various processes, which have led to the current situation can be quantified. They also serve as an instrument to assess short or medium term strategies to reduce this human impact. However, to assure efficiency as well as accuracy, a careful analysis of the numerous processes involved in the dispersion of pollutants in the atmosphere is called for. This should help to concentrate on the essentials and also prevent excessive usage of sometimes scarce computing resources. The basis of the presented work is the EUMAC Zooming Model (ETM), and particularly the component calculating the dispersion of pollutants in the atmosphere, the model MARS. The model has two main parts: an explicit solver, where the advection and the horizontal diffusion of pollutants are calculated, and an implicit solution mechanism, allowing the joint computation of the change of concentration due to chemical reactions, coupled with the respective influence of the vertical diffusion of the species. The aim of this thesis is to determine particularly the influence of the horizontal components of the turbulent diffusion on the existing implicit solver of the model. Suggestions for a more comprehensive inclusion of the full three dimensional diffusion operator in the implicit solver are made. This is achieved by an appropriate operator splitting. A selection of numerical approaches to tighten the coupling of the diffusion processes with the calculation of the applied chemical reaction mechanisms are examined. (author) figs., tabs., refs.
Eskew, Matthew W.; Harrison, Jason; Simoyi, Reuben H.
2016-11-01
Oxidation reactions of thiourea by chlorite in a Hele-Shaw cell are excitable, autocatalytic, exothermic, and generate a lateral instability upon being triggered by the autocatalyst. Reagent concentrations used to develop convective instabilities delivered a temperature jump at the wave front of 2.1 K. The reaction zone was 2 mm and due to normal cooling after the wave front, this generated a spike rather than the standard well-studied front propagation. The reaction front has solutal and thermal contributions to density changes that act in opposite directions due to the existence of a positive isothermal density change in the reaction. The competition between these effects generates thermal plumes. The fascinating feature of this system is the coexistence of plumes and fingering in the same solution which alternate in frequency as the front propagates, generating hot and cold spots within the Hele-Shaw cell, and subsequently spatiotemporal inhomogeneities. The small ΔT at the wave front generated thermocapillary convection which competed effectively with thermogravitational forces at low Eötvös Numbers. A simplified reaction-diffusion-convection model was derived for the system. Plume formation is heavily dependent on boundary effects from the cell dimensions. This work was supported by Grant No. CHE-1056366 from the NSF and a Research Professor Grant from the University of KwaZulu-Natal.
Reaction-diffusion theory in the presence of an attractive harmonic potential
Spendier, K.; Sugaya, S.; Kenkre, V. M.
2013-12-01
Problems involving the capture of a moving entity by a trap occur in a variety of physical situations, the moving entity being an electron, an excitation, an atom, a molecule, a biological object such as a receptor cluster, a cell, or even an animal such as a mouse carrying an epidemic. Theoretical considerations have almost always assumed that the particle motion is translationally invariant. We study here the case when that assumption is relaxed, in that the particle is additionally subjected to a harmonic potential. This tethering to a center modifies the reaction-diffusion phenomenon. Using a Smoluchowski equation to describe the system, we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions. Interesting features emerge depending on the relative location of the trap, the attractive center, and the initial placement of the diffusing particle.
On the Statistics of Reaction-Diffusion Simulations for Molecular Communication
Noel, Adam; Schober, Robert
2015-01-01
A molecule traveling in a realistic propagation environment can experience stochastic interactions with other molecules and the environment boundary. The statistical behavior of some isolated phenomena, such as dilute unbounded molecular diffusion, are well understood. However, the coupling of multiple interactions can impede closed-form analysis, such that simulations are required to determine the statistics. This paper compares the statistics of molecular reaction-diffusion simulation models from the perspective of molecular communication systems. Microscopic methods track the location and state of every molecule, whereas mesoscopic methods partition the environment into virtual containers that hold molecules. The properties of each model are described and compared with a hybrid of both models. Simulation results also assess the accuracy of Poisson and Gaussian approximations of the underlying Binomial statistics.
Liang, Xiao; Wang, Linshan; Wang, Yangfan; Wang, Ruili
2016-09-01
In this paper, we focus on the long time behavior of the mild solution to delayed reaction-diffusion Hopfield neural networks (DRDHNNs) driven by infinite dimensional Wiener processes. We analyze the existence, uniqueness, and stability of this system under the local Lipschitz function by constructing an appropriate Lyapunov-Krasovskii function and utilizing the semigroup theory. Some easy-to-test criteria affecting the well-posedness and stability of the networks, such as infinite dimensional noise and diffusion effect, are obtained. The criteria can be used as theoretic guidance to stabilize DRDHNNs in practical applications when infinite dimensional noise is taken into consideration. Meanwhile, considering the fact that the standard Brownian motion is a special case of infinite dimensional Wiener process, we undertake an analysis of the local Lipschitz condition, which has a wider range than the global Lipschitz condition. Two samples are given to examine the availability of the results in this paper. Simulations are also given using the MATLAB.
Traveling waves in a nonlocal, piecewise linear reaction-diffusion population model
Autry, E. A.; Bayliss, A.; Volpert, V. A.
2017-08-01
We consider an analytically tractable switching model that is a simplification of a nonlocal, nonlinear reaction-diffusion model of population growth where we take the source term to be piecewise linear. The form of this source term allows us to consider both the monostable and bistable versions of the problem. By transforming to a traveling frame and choosing specific kernel functions, we are able to reduce the problem to a system of algebraic equations. We construct solutions and examine the propagation speed and monotonicity of the resulting waves.
Computer prediction system on solid/solid reaction kinetics
无
2001-01-01
A computer software system of kinetic predication of solid/solid reaction, KinPreSSR, was developed using Visual C++ and FoxPro. It includes two main modules, REACTION and DIFFUSION. KinPreSSR deals with the kinetics on the diffusion in solids as well as solid/solid reactions. The REACTION module in KinPreSSR was mainly described, which has organized the commonly recognized kinetic models, parameters, and employed both numerical and graphical methods for data analyses. The proper combination between the kinetic contents and the analytical methods enables users to use KinPreSSR for the evaluation and prediction of solid/solid reactions interested. As an example to show some of functions of KinPreSSR, the kinetics analysis for the reaction between SrCO3 and TiO2 powders to form SrTiO3 with a series of kinetic data from isothermal measurements was demonstrated.
Siebert, Julien; Alonso, Sergio; Bär, Markus; Schöll, Eckehard
2014-05-01
A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.
1D to 3D diffusion-reaction kinetics of defects in crystals
Trinkaus, H.; Heinisch, H.L.; Barashev, A.V.
2002-01-01
Microstructural features evolving in crystalline solids from diffusion-reaction kinetics of mobile components depend crucially on the dimension of the underlying diffusion process which is commonly assumed to be three-dimensional (3D). In metals, irradiation-induced displacement cascades produce ...... and 3D limiting cases. The analytical result is fully confirmed by kinetic Monte Carlo simulations.......Microstructural features evolving in crystalline solids from diffusion-reaction kinetics of mobile components depend crucially on the dimension of the underlying diffusion process which is commonly assumed to be three-dimensional (3D). In metals, irradiation-induced displacement cascades produce...
Apparent Rate Constant for Diffusion-Controlled Three molecular (catalytic) reaction
Burlatsky, S. F.; Moreau, M
1996-01-01
We present simple explicit estimates for the apparent reaction rate constant for three molecular reactions, which are important in catalysis. For small concentrations and $d> 1$, the apparent reaction rate constant depends only on the diffusion coefficients and sizes of the particles. For small concentrations and $d\\le 1$, it is also time -- dependent. For large concentrations, it gains the dependence on concentrations.
Lacitignola, Deborah; Bozzini, Benedetto; Frittelli, Massimo; Sgura, Ivonne
2017-07-01
The present paper deals with the pattern formation properties of a specific morpho-electrochemical reaction-diffusion model on a sphere. The physico-chemical background to this study is the morphological control of material electrodeposited onto spherical particles. The particular experimental case of interest refers to the optimization of novel metal-air flow batteries and addresses the electrodeposition of zinc onto inert spherical supports. Morphological control in this step of the high-energy battery operation is crucial to the energetic efficiency of the recharge process and to the durability of the whole energy-storage device. To rationalise this technological challenge within a mathematical modeling perspective, we consider the reaction-diffusion system for metal electrodeposition introduced in [Bozzini et al., J. Solid State Electr.17, 467-479 (2013)] and extend its study to spherical domains. Conditions are derived for the occurrence of the Turing instability phenomenon and the steady patterns emerging at the onset of Turing instability are investigated. The reaction-diffusion system on spherical domains is solved numerically by means of the Lumped Surface Finite Element Method (LSFEM) in space combined with the IMEX Euler method in time. The effect on pattern formation of variations in the domain size is investigated both qualitatively, by means of systematic numerical simulations, and quantitatively by introducing suitable indicators that allow to assign each pattern to a given morphological class. An experimental validation of the obtained results is finally presented for the case of zinc electrodeposition from alkaline zincate solutions onto copper spheres.
Hybrid approaches for multiple-species stochastic reaction-diffusion models
Spill, Fabian; Alarcon, Tomas; Maini, Philip K; Byrne, Helen
2015-01-01
Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains ...
Chuangxia Huang
2009-01-01
Full Text Available This paper is concerned with pth moment exponential stability of stochastic reaction-diffusion Cohen-Grossberg neural networks with time-varying delays. With the help of Lyapunov method, stochastic analysis, and inequality techniques, a set of new suffcient conditions on pth moment exponential stability for the considered system is presented. The proposed results generalized and improved some earlier publications.
Simulation of reaction-diffusion processes in three dimensions using CUDA
Molnar, Ferenc; Meszaros, Robert; Lagzi, Istvan
2010-01-01
Numerical solution of reaction-diffusion equations in three dimensions is one of the most challenging applied mathematical problems. Since these simulations are very time consuming, any ideas and strategies aiming at the reduction of CPU time are important topics of research. A general and robust idea is the parallelization of source codes/programs. Recently, the technological development of graphics hardware created a possibility to use desktop video cards to solve numerically intensive problems. We present a powerful parallel computing framework to solve reaction-diffusion equations numerically using the Graphics Processing Units (GPUs) with CUDA. Four different reaction-diffusion problems, (i) diffusion of chemically inert compound, (ii) Turing pattern formation, (iii) phase separation in the wake of a moving diffusion front and (iv) air pollution dispersion were solved, and additionally both the Shared method and the Moving Tiles method were tested. Our results show that parallel implementation achieves t...
A Diffusion Equation for Quantum Adiabatic Systems
Jain, S R
1998-01-01
For ergodic adiabatic quantum systems, we study the evolution of energy distribution as the system evolves in time. Starting from the von Neumann equation for the density operator, we obtain the quantum analogue of the Smoluchowski equation on coarse-graining over the energy spectrum. This result brings out the precise notion of quantum diffusion.
Reaction diffusion in Ni–Al diffusion couples in steady magnetic fields
Li, Chuanjun, E-mail: cjli21@shu.edu.cn [School of Materials Science and Engineering, Shanghai University, Shanghai 200072 (China); Yuan, Zhaojing; Guo, Rui; Xuan, Weidong; Ren, Zhongming; Zhong, Yunbo; Li, Xi [School of Materials Science and Engineering, Shanghai University, Shanghai 200072 (China); Wang, Hui; Wang, Qiuliang [Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190 (China)
2015-08-25
Highlights: • The Ni–Al diffusion couples were prepared by the electrodeposition technique. • The magnetic field reduced the growth rates of product layers in diffusion couples. • The effect of the magnetic field on diffusion depends on its intensity and direction. • The spiral motion of an atom in the magnetic field reduces diffusivity. - Abstract: The effect of a steady magnetic field on reactive diffusion in Ni–Al diffusion couples was investigated. The diffusion couples prepared by the electrodeposition technique were annealed in the temperature range of 530–590 °C with and without the magnetic field of 6 T. Regardless of the magnetic field, two intermetallic compounds, i.e., Ni{sub 2}Al{sub 3} and NiAl{sub 3}, were present in the product layers of diffusion couples. NiAl{sub 3} phase shows island-like structures at relatively lower temperatures while the Ni{sub 2}Al{sub 3} phase forms a typical layered structure. The growth of Ni{sub 2}Al{sub 3} layer was found to be parabolic. When the diffusion direction was perpendicular to the direction of the magnetic field, the external magnetic field reduced the growth rate of the Ni{sub 2}Al{sub 3} phase. Whereas the magnetic field had no obvious effect on the growth rate of Ni{sub 2}Al{sub 3} layers in the diffusion configuration of mutually parallel directions. The magnetic field intensity and direction dependence of growth rate of Ni{sub 2}Al{sub 3} intermetallic layers can be attributed to the change in number of collision of an atom with neighbors during diffusion due to spiral motion under the action of the Lorentz force, which leads to change the frequency factor, not activation energy, for layer growth.
A CLASS OF REACTION-DIFFUSION EQUATIONS WITH HYSTERESIS DIFFERENTIAL OPERATOR
XuLongfeng
2002-01-01
In this paper, the classical and weak derivatives with respect to spatial variable of a class of hysteresis functional are discussed. Some conclusions about solutions of a class of reaction-diffusion equations with hysteresis differential operator are given.
THE CORNER LAYER SOLUTION TO ROBIN PROBLEM FOR REACTION DIFFUSION EQUATION
无
2012-01-01
A class of Robin boundary value problem for reaction diffusion equation is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the corner layer solution to the initial boundary value problem are studied.
A CLASS OF SINGULARLY PERTURBED INITIAL BOUNDARY PROBLEM FOR REACTION DIFFUSION EQUATION
Xie Feng
2003-01-01
The singularly perturbed initial boundary value problem for a class of reaction diffusion equation isconsidered. Under appropriate conditions, the existence-uniqueness and the asymptotic behavior of the solu-tion are showed by using the fixed-point theorem.
Simulating Some Complex Phenomena in Hydrothermal Ore-Forming Processes by Reaction-Diffusion CNN
Xu Deyi; Yu Chongwen; Bao Zhengyu
2003-01-01
Complexity phenomena like dynamic and static patterns, order from disorder, chaos and catastrophe were simulated by the application of 2-D reaction-diffusion CNN of two state variables and two diffusion coefficients transformed from Zhabotinksii model. They revealed somehow the mechanism of hydrothermal ore-forming processes, and answered several questions about the onset of ore forming.
熊岳山; 韦永康
2001-01-01
The sediment reaction and diffusion equation with generalized initial and boundary condition is studied. By using Laplace transform and Jordan lemma , an analytical solution is got, which is an extension of analytical solution provided by Cheng Kwokming James ( only diffusion was considered in analytical solution of Cheng ). Some problems arisen in the computation of analytical solution formula are also analysed.
Abstracts of International Conference on Diffusion and Reactions: From Basis to Applications
NONE
1994-12-31
The conference has been devoted to diffusion of corrosion agents and chemical reactions (sulfidation and oxidation) in metals, alloys and composite materials from the view point of their corrosion mechanism and material resistance in different conditions.The three main topics have been broadly represented at the conference sessions: heterogeneous reactions; high temperature diffusion and corrosion mechanism and current problems and trends in development and characterization of materials.
AUTO-DARBOUX TRANSFORMATION AND EXACT SOLUTIONS OF THE BRUSSELATOR REACTION DIFFUSION MODEL
闫振亚; 张鸿庆
2001-01-01
Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain some authors' results known.Secondly, by using a series of transformations, the model is reduced into a nonlinear reaction diffusion equation and then through using sine- cosine method, more exact solutions are found which contain soliton solutions.
The hydrogen diffusion in disordered systems
Kondratyev, V.V.; Gapontsev, A.V. [Institute of Metal Physics, Ural Branch of Russian Academy of Sciences, Ekaterinburg (Russian Federation); Voloshinskii, A.N.; Obukhov, A.G. [Ural Academy of Mining and Geology, Ekaterinburg (Russian Federation); Timofeyev, N.I. [Ekaterinburg Works for Nonferrous Metals (Russian Federation)
1999-09-01
This paper analyzes the experimental data and presents a critical review of the existing approaches to the description of hydrogen diffusion in disordered amorphous alloys. It is noted that the available theories ignore the role of the short-range order in hydrogen diffusion. A diffusion model, which is based on the approach developed by us earlier, has been proposed for disordered crystalline alloys. In terms of this model specific features of the amorphous state are allowed for by homogeneous local distortions of voids, i.e. a kind of 'frozen' fluctuations of the free volume. General expressions for the chemical diffusion coefficient of hydrogen in amorphous metals and binary alloys having FCC-like short-range crystalline order have been derived. It was shown that the diffusion coefficient may depend on a structural characteristic of disordered systems the mean square of static displacement of atoms in the vicinity of a void and also on the paired radial distribution function of atoms. The analysis of the proposed model suggests that the crystalline disorder causes an increase in the diffusion coefficient, which grows (unlike in crystals) linearly with the hydrogen concentration.
Solutions of fractional reaction-diffusion equations in terms of the H-function
Haubold, H. J.; Mathai, A. M.; Saxena, R. K.
2007-12-01
This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction-diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form.
Relaxation and Diffusion in Complex Systems
Ngai, K L
2011-01-01
Relaxation and Diffusion in Complex Systems comprehensively presents a variety of experimental evidences of universal relaxation and diffusion properties in complex materials and systems. The materials discussed include liquids, glasses, colloids, polymers, rubbers, plastic crystals and aqueous mixtures, as well as carbohydrates, biomolecules, bioprotectants and pharmaceuticals. Due to the abundance of experimental data, emphasis is placed on glass-formers and the glass transition problem, a still unsolved problem in condensed matter physics and chemistry. The evidence for universal properties of relaxation and diffusion dynamics suggests that a fundamental physical law is at work. The origin of the universal properties is traced to the many-body effects of the interaction, rigorous theory of which does not exist at the present time. However, using solutions of simplified models as guides, key quantities have been identified and predictions of the universal properties generated. These predictions from Ngai’...
Barrier heights of hydrogen-transfer reactions with diffusion quantum monte carlo method.
Zhou, Xiaojun; Wang, Fan
2017-04-30
Hydrogen-transfer reactions are an important class of reactions in many chemical and biological processes. Barrier heights of H-transfer reactions are underestimated significantly by popular exchange-correlation functional with density functional theory (DFT), while coupled-cluster (CC) method is quite expensive and can be applied only to rather small systems. Quantum Monte-Carlo method can usually provide reliable results for large systems. Performance of fixed-node diffusion quantum Monte-Carlo method (FN-DMC) on barrier heights of the 19 H-transfer reactions in the HTBH38/08 database is investigated in this study with the trial wavefunctions of the single-Slater-Jastrow form and orbitals from DFT using local density approximation. Our results show that barrier heights of these reactions can be calculated rather accurately using FN-DMC and the mean absolute error is 1.0 kcal/mol in all-electron calculations. Introduction of pseudopotentials (PP) in FN-DMC calculations improves efficiency pronouncedly. According to our results, error of the employed PPs is smaller than that of the present CCSD(T) and FN-DMC calculations. FN-DMC using PPs can thus be applied to investigate H-transfer reactions involving larger molecules reliably. In addition, bond dissociation energies of the involved molecules using FN-DMC are in excellent agreement with reference values and they are even better than results of the employed CCSD(T) calculations using the aug-cc-pVQZ basis set. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Defect reactions in gallium antimonide studied by zinc and self-diffusion
Sunder, Kirsten; Bracht, Hartmut
2007-12-01
Extrinsic diffusion of zinc (Zn) in gallium antimonide (GaSb) under Ga-rich conditions was analyzed on the basis of the kick-out and the dissociative diffusion mechanism. It is concluded that the changeover of interstitial Zn to substitutional gallium (Ga) sites is mainly mediated by Ga interstitials ( IGa). Fitting of the Zn profiles provides the relative contributions of IGa to Ga diffusion. This contribution is lower than the directly measured Ga diffusion coefficient indicating that Ga diffusion in GaSb is rather mediated by Ga vacancies than by Ga interstitials even under Ga-rich conditions. This finding supports transformation reactions between native point defects that are confirmed by first-principles total-energy calculations. In addition Ga and Sb diffusion experiments under H22 atmosphere were performed to reconcile the controversial data on self-diffusion in GaSb published by Weiler et al. and Bracht et al.
KPP reaction-diffusion equations with a non-linear loss inside a cylinder
Giletti, Thomas
2010-01-01
We consider in this paper a reaction-diffusion system in presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a non-linear spacedependent loss term inside the domain. In particular, we will point out the existence of a minimal speed, above which any real value is an admissible speed. We will also give some spreading results for initial conditions decaying exponentially at infinity.
Transport dissipative particle dynamics model for mesoscopic advection- diffusion-reaction problems
Zhen, Li; Yazdani, Alireza; Tartakovsky, Alexandre M.; Karniadakis, George E.
2015-07-07
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic DPD framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between particles, and an analytical formula is proposed to relate the mesoscopic concentration friction to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers.
Diffusive gas transport through flooded rice systems
Bodegom, van P.M.; Groot, T.; Hout, van de B.; Leffelaar, P.A.; Goudriaan, J.
2001-01-01
A fully mechanistic model based on diffusion equations for gas transport in a flooded rice system is presented. The model has transport descriptions for various compartments in the water-saturated soil and within the plant. Plant parameters were estimated from published data and experiments independ
Diffusion-influenced reactions in a hollow nano-reactor with a circular hole
Piazza, Francesco; Traytak, Sergey D.
Hollow nanostructures are paid increasing attention in many nanotechnology-related communities in view of their numerous applications in chemistry and biotechnology, e.g. as smart nanoreactors or drug-delivery systems. In this paper we consider irreversible, diffusion-influenced reactions occurring within a hollow spherical cavity endowed with a circular hole on its surface. Importantly, our model is not limited to small sizes of the aperture. In our scheme, reactants can freely diffuse inside and outside the cavity through the hole, and react at a spherical boundary of given size encapsulated in the chamber and endowed with a given intrinsic rate constant. We work out the solution of the above problem, enabling one to compute the reaction rate constant to any desired accuracy. Remarkably, we show that, in the case of narrow holes, the rate constant is extremely well-approximated by a simple formula that can be derived on the basis of simple physical arguments and that can be readily employed to analyze experimental data.
Luisa Malaguti
2011-01-01
Full Text Available The paper deals with a degenerate reaction-diffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusive-aggregative and to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated, and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed and of the wave profiles on a real parameter is studied, both when the process maintains its diffusion-aggregation nature and when it switches from it to another regime.
Realistic boundary conditions for stochastic simulations of reaction-diffusion processes
Erban, R; Erban, Radek
2006-01-01
Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecul...
A Two-grid Method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations
Wei Liu; Hong-xing Rui; Hui Guo
2011-01-01
Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + hk+1 + H2k+2-d/2) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.
Thermally activated reaction–diffusion-controlled chemical bulk reactions of gases and solids
S. Möller
2015-01-01
Full Text Available The chemical kinetics of the reaction of thin films with reactive gases is investigated. The removal of thin films using thermally activated solid–gas to gas reactions is a method to in-situ control deposition inventory in vacuum and plasma vessels. Significant scatter of experimental deposit removal rates at apparently similar conditions was observed in the past, highlighting the need for understanding the underlying processes. A model based on the presence of reactive gas in the films bulk and chemical kinetics is presented. The model describes the diffusion of reactive gas into the film and its chemical interaction with film constituents in the bulk using a stationary reaction–diffusion equation. This yields the reactive gas concentration and reaction rates. Diffusion and reaction rate limitations are depicted in parameter studies. Comparison with literature data on tokamak co-deposit removal results in good agreement of removal rates as a function of pressure, film thickness and temperature.
Cross-Diffusion-Driven Instability in a Reaction-Diffusion Harrison Predator-Prey Model
Xiaoqin Wang
2013-01-01
Full Text Available We present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with cross-diffusion in a Harrison-type predator-prey model. We analyze the global behaviour of the model by establishing a Lyapunov function. We carry out the analytical study in detail and find out the certain conditions for Turing’s instability induced by cross-diffusion. And the numerical results reveal that, on increasing the value of the half capturing saturation constant, the sequences “spots → spot-stripe mixtures → stripes → hole-stripe mixtures → holes” are observed. The results show that the model dynamics exhibits complex pattern replication controlled by the cross-diffusion.
Voituriez, R.; Moreau, M.; Oshanin, G.
2004-01-01
The validity of two fundamental concepts of classical chemical kinetics - the notion of "Chemical Equilibrium" and the "Law of Mass Action" - are re-examined for reversible \\textit{diffusion-limited} reactions (DLR), as exemplified here by association/dissociation $A+A \\rightleftharpoons B$ reactions. We consider a general model of long-ranged reactions, such that any pair of $A$ particles, separated by distance $\\mu$, may react with probability $\\omega_+(\\mu)$, and any $B$ may dissociate wit...
Flow-Injection Responses of Diffusion Processes and Chemical Reactions
Andersen, Jens Enevold Thaulov
2000-01-01
The technique of Flow-injection Analysis (FIA), now aged 25 years, offers unique analytical methods that are fast, reliable and consuming an absolute minimum of chemicals. These advantages together with its inherent feasibility for automation warrant the future applications of FIA as an attractive...... be used in the resolution of FIA profiles to obtain information about the content of interference’s, in the study of chemical reaction kinetics and to measure absolute concentrations within the FIA-detector cell....
Contribution to an effective design method for stationary reaction-diffusion patterns
Szalai, István; Horváth, Judit [Laboratory of Nonlinear Chemical Dynamics, Institute of Chemistry, Eötvös Loránd University, P.O. Box 32, H-1518 Budapest 112 (Hungary); De Kepper, Patrick [Centre de Recherche Paul Pascal, CNRS, University of Bordeaux, 115, Avenue Schweitzer, F-33600 Pessac (France)
2015-06-15
The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences.
Contribution to an effective design method for stationary reaction-diffusion patterns
Szalai, István; Horváth, Judit; De Kepper, Patrick
2015-06-01
The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences.
Contribution to an effective design method for stationary reaction-diffusion patterns.
Szalai, István; Horváth, Judit; De Kepper, Patrick
2015-06-01
The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences.
Chalupecký, Vladimír; Kruschwitz, Jens; Muntean, Adrian
2012-01-01
We consider a two-scale reaction diffusion system able to capture the corrosion of concrete with sulfates. Our aim here is to define and compute two macroscopic corrosion indicators: typical pH drop and gypsum profiles. Mathematically, the system is coupled, endowed with micro-macro transmission conditions, and posed on two different spatially-separated scales: one microscopic (pore scale) and one macroscopic (sewer pipe scale). We use a logarithmic expression to compute values of pH from the volume averaged concentration of sulfuric acid which is obtained by resolving numerically the two-scale system (microscopic equations with direct feedback with the macroscopic diffusion of one of the reactants). Furthermore, we also evaluate the content of the main sulfatation reaction (corrosion) product---the gypsum---and point out numerically a persistent kink in gypsum's concentration profile. Finally, we illustrate numerically the position of the free boundary separating corroded from not-yet-corroded regions.
Hybrid approaches for multiple-species stochastic reaction-diffusion models.
Spill, Fabian
2015-10-01
Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.
Plante, Ianik; Devroye, Luc
2015-09-01
Several computer codes simulating chemical reactions in particles systems are based on the Green's functions of the diffusion equation (GFDE). Indeed, many types of chemical systems have been simulated using the exact GFDE, which has also become the gold standard for validating other theoretical models. In this work, a simulation algorithm is presented to sample the interparticle distance for partially diffusion-controlled reversible ABCD reaction. This algorithm is considered exact for 2-particles systems, is faster than conventional look-up tables and uses only a few kilobytes of memory. The simulation results obtained with this method are compared with those obtained with the independent reaction times (IRT) method. This work is part of our effort in developing models to understand the role of chemical reactions in the radiation effects on cells and tissues and may eventually be included in event-based models of space radiation risks. However, as many reactions are of this type in biological systems, this algorithm might play a pivotal role in future simulation programs not only in radiation chemistry, but also in the simulation of biochemical networks in time and space as well.
Plante, Ianik, E-mail: ianik.plante-1@nasa.gov [Wyle Science, Technology & Engineering, 1290 Hercules, Houston, TX 77058 (United States); Devroye, Luc, E-mail: lucdevroye@gmail.com [School of Computer Science, McGill University, 3480 University Street, Montreal H3A 0E9 (Canada)
2015-09-15
Several computer codes simulating chemical reactions in particles systems are based on the Green's functions of the diffusion equation (GFDE). Indeed, many types of chemical systems have been simulated using the exact GFDE, which has also become the gold standard for validating other theoretical models. In this work, a simulation algorithm is presented to sample the interparticle distance for partially diffusion-controlled reversible ABCD reaction. This algorithm is considered exact for 2-particles systems, is faster than conventional look-up tables and uses only a few kilobytes of memory. The simulation results obtained with this method are compared with those obtained with the independent reaction times (IRT) method. This work is part of our effort in developing models to understand the role of chemical reactions in the radiation effects on cells and tissues and may eventually be included in event-based models of space radiation risks. However, as many reactions are of this type in biological systems, this algorithm might play a pivotal role in future simulation programs not only in radiation chemistry, but also in the simulation of biochemical networks in time and space as well.
Study on a Cross Diffusion Parabolic System
Li Chen; Ling Hsiao; Gerald Warnecke
2007-01-01
This paper considers a kind of strongly coupled cross diffusion parabolic system, which can be used as the multi-dimensional Lyumkis energy transport model in semiconductor science. The global existence and large time behavior are obtained for smooth solution to the initial boundary value problem. When the initial data are a small perturbation of an isothermal stationary solution, the smooth solution of the problem under the insulating boundary condition, converges to that stationary solution exponentially fast as time goes to infinity.
Preconditioned time-difference methods for advection-diffusion-reaction equations
Aro, C.; Rodrigue, G. [Lawrence Livermore National Lab., CA (United States); Wolitzer, D. [California State Univ., Hayward, CA (United States)
1994-12-31
Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions placed on stepsize due to stability. Stability bounds for explicit time differencing methods on advection-diffusion-reaction problems are generally quite severe and implicit methods are used instead. The linear systems arising from these implicit methods are large and sparse so that iterative methods must be used to solve them. In this paper the authors develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods.
Robustness of the Critical Behaviour in a Discrete Stochastic Reaction-Diffusion Medium
Fatès, Nazim; Berry, Hugues
We study the steady states of a reaction-diffusion medium modelled by a stochastic 2D cellular automaton. We consider the Greenberg-Hastings model where noise and topological irregularities of the grid are taken into account. The decrease of the probability of excitation changes qualitatively the behaviour of the system from an "active" to an "extinct" steady state. Simulations show that this change occurs near a critical threshold; it is identified as a nonequilibrium phase transition which belongs to the directed percolation universality class. We test the robustness of the phenomenon by introducing persistent defects in the topology : directed percolation behaviour is conserved. Using experimental and analytical tools, we suggest that the critical threshold varies as the inverse of the average number of neighbours per cell.
Xu Rui [Department of Applied Mathematics, Xi' an Jiaotong University, Xi' an 710049 (China)]. E-mail: rxu88@yahoo.com.cn; Chaplain, M.A.J. [Department of Mathematics, University of Dundee, Dundee DD1 4HN (United Kingdom); Davidson, F.A. [Department of Mathematics, University of Dundee, Dundee DD1 4HN (United Kingdom)
2006-11-15
In this paper, we first investigate a stage-structured competitive model with time delays, harvesting, and nonlocal spatial effect. By using an iterative technique recently developed by Wu and Zou (Wu J, Zou X. Travelling wave fronts of reaction-diffusion systems with delay. J Dynam Differen Equat 2001;13:651-87), sufficient conditions are established for the existence of travelling front solution connecting the two boundary equilibria in the case when there is no positive equilibrium. The travelling wave front corresponds to an invasion by a stronger species which drives the weaker species to extinction. Secondly, we consider a stage-structured competitive model with time delays and nonlocal spatial effect when the domain is finite. We prove the global stability of each of the nonnegative equilibria and demonstrate that the more complex model studied here admits three possible long term behaviors: coexistence, bistability and dominance as is the case for the standard Lotka-Voltera competitive model.
Scalable implicit methods for reaction-diffusion equations in two and three space dimensions
Veronese, S.V.; Othmer, H.G. [Univ. of Utah, Salt Lake City, UT (United States)
1996-12-31
This paper describes the implementation of a solver for systems of semi-linear parabolic partial differential equations in two and three space dimensions. The solver is based on a parallel implementation of a non-linear Alternating Direction Implicit (ADI) scheme which uses a Cartesian grid in space and an implicit time-stepping algorithm. Various reordering strategies for the linearized equations are used to reduce the stride and improve the overall effectiveness of the parallel implementation. We have successfully used this solver for large-scale reaction-diffusion problems in computational biology and medicine in which the desired solution is a traveling wave that may contain rapid transitions. A number of examples that illustrate the efficiency and accuracy of the method are given here; the theoretical analysis will be presented.
Politowicz, P.A.; Kozak, J.J.
1987-12-01
The authors study surface-mediated, diffusion-controlled reactive processes on particles whose overall geometry is homeomorphic to a sphere. Rather than assuming that a coreactant can diffuse freely over the surface of the particle to a target site (reaction center), they consider the case where the coreactant can migrate only among N-1 satellite sites that are networked to the reaction site by means of a number of pathways or reaction channels. Five distinct lattice topologies are considered and they study the reaction efficiency both for the case where the satellite sites are passive and for the case where reaction may occur with finite probability at these sites. The results obtained for this class of surface problems are compared with those obtained by assuming that the reaction-diffusion process takes place on a planar, two-dimensional surface (lattice). The applicability of their results to surface-mediated processes on organizates (cells, vesicles, micelles) and on colloidally dispersed catalyst particles is brought out in the Introduction, and the correspondence between the lattice-based, Markovian approach developed here and Fickian models of surface diffusion, particularly with regard to the exponentiality of the decay, is discussed in the concluding section.
Modelling the impact of an invasive insect via reaction-diffusion.
Roques, Lionel; Auger-Rozenberg, Marie-Anne; Roques, Alain
2008-11-01
An exotic, specialist seed chalcid, Megastigmus schimitscheki, has been introduced along with its cedar host seeds from Turkey to southeastern France during the early 1990s. It is now expanding in plantations of Atlas Cedar (Cedrus atlantica). We propose a model to predict the expansion and impact of this insect. This model couples a time-discrete equation for the ovo-larval stage with a two-dimensional reaction-diffusion equation for the adult stage, through a formula linking the solution of the reaction-diffusion equation to a seed attack rate. Two main diffusion operators, of Fokker-Planck and Fickian types, are tested. We show that taking account of the dependence of the insect mobility with respect to spatial heterogeneity, and choosing the appropriate diffusion operator, are critical factors for obtaining good predictions.
A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation
Chunye Gong
2014-01-01
Full Text Available The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N2M compared with O(NM for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.
Finite volume element method for analysis of unsteady reaction-diffusion problems
Sutthisak Phongthanapanich; Pramote Dechaumphai
2009-01-01
A finite volume element method is developed for analyzing unsteady scalar reaction--diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction--diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the highgradient boundary layers.
Marangoni flows induced by A + B -> C reaction fronts with arbitrary diffusion coefficients
Tiani, Reda; Rongy, Laurence
2016-11-01
We consider horizontal aqueous solutions in contact with air where three reacting species A, B, and C can affect the surface tension of the solution, thereby driving Marangoni flows. When the two reactants A and B, that are initially separated, are brought into contact, a reaction front producing species C is formed and evolves in time due to diffusion, convection and reaction processes. The resulting dynamics is studied by numerically integrating the incompressible Navier-Stokes equations coupled to reaction-diffusion-convection equations for the three chemical species. For equal initial concentrations of reactants and equal diffusion coefficients, we have explained how chemically-driven Marangoni flows can lead to complex dynamics of the front propagation. Here we extend such results for arbitrary values of the diffusion coefficients and initial concentrations of reactants. We give the general classification of the surface tension profiles as a function of the Marangoni numbers quantifying the effect of each species on the surface tension, the ratio of initial concentrations of reactants and the ratios of diffusion coefficients. Such a classification allows us then to study the resulting structure of the convective rolls as well as the nonlinear dynamics of the reaction front. F.R.S.- FNRS, ARC.
Richter, Otto; Moenickes, Sylvia; Suhling, Frank
2012-02-01
The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.
Wang, Kai; Teng, Zhidong; Jiang, Haijun
2012-10-01
In this paper, the adaptive synchronization in an array of linearly coupled neural networks with reaction-diffusion terms and time delays is discussed. Based on the LaSalle invariant principle of functional differential equations and the adaptive feedback control technique, some sufficient conditions for adaptive synchronization of such a system are obtained. Finally, a numerical example is given to show the effectiveness of the proposed synchronization method.
Extension of Newman's method to electrochemical reaction-diffusion in a fuel cell catalyst layer
Duan, Tianping; Weidner, John W.; White, Ralph E.
A numerical technique is developed for solving coupled electrochemical reaction-diffusion equations. Through analyzing the nonlinearity of the problem, a trial and error iterating procedure is constructed. The coefficient matrix is arranged as a tridiagonal form with elements of block matrix and is decomposed to LU form. A compact forward and backward substitution algorithm based on the shift of inversing block matrix by Gauss-Jordan full pivoting is developed. A large number of node points is required to converge the calculation. Computation experiences show that the iteration converges very quickly. The effects of inner diffusion on the electrochemical reaction are analyzed by numerical solutions.
Exact solutions for a diffusion-reaction process in one dimension
Spouge, John L.
1988-03-01
This paper presents a new method for the solution of diffusion-reaction problems in one dimension. The method is used to derive some new exact results for the polymerization (cl-cl aggregation) and annihilation processes on openR and openZ. Through well-known dualities, these results have implications for the T=0 limit of the kinetic Ising model and for two interacting particle processes, the invasion and voter models. Prospectively, the method may be useful in providing one-dimensional verification for speculations in the theory of diffusion reaction.
Time capsule: an autonomous sensor and recorder based on diffusion-reaction.
Gerber, Lukas C; Rosenfeld, Liat; Chen, Yunhan; Tang, Sindy K Y
2014-11-21
We describe the use of chemical diffusion and reaction to record temporally varying chemical information as spatial patterns without the need for external power. Diffusion of chemicals acts as a clock, while reactions forming immobile products possessing defined optical properties perform sensing and recording functions simultaneously. The spatial location of the products reflects the history of exposure to the detected substances of interest. We refer to our device as a time capsule and show an initial proof of principle in the autonomous detection of lead ions in water.
Modeling of advection-diffusion-reaction processes using transport dissipative particle dynamics
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-11-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. In particular, the transport of concentration is modeled by a Fickian flux and a random flux between tDPD particles, and the advection is implicitly considered by the movements of Lagrangian particles. To validate the proposed tDPD model and the boundary conditions, three benchmark simulations of one-dimensional diffusion with different boundary conditions are performed, and the results show excellent agreement with the theoretical solutions. Also, two-dimensional simulations of ADR systems are performed and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, an application of tDPD to the spatio-temporal dynamics of blood coagulation involving twenty-five reacting species is performed to demonstrate the promising biological applications of the tDPD model. Supported by the DOE Center on Mathematics for Mesoscopic Modeling of Materials (CM4) and an INCITE grant.
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-07-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic dissipative particle dynamics (DPD) framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between tDPD particles, and the advection is implicitly considered by the movements of these Lagrangian particles. An analytical formula is proposed to relate the tDPD parameters to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the conventional DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers.
Fluctuations Near Homogeneous States of Chemical Reactions with Diffusion.
1985-11-01
motions obtained by Holley and Stroock [16], Gorostiza [14] and, in the absence of branch- ing (b = d = c = 0) to Martin-Ldf [30] and It8 (18] (cf... GOROSTIZA , L.G. (1983) High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions, Ann. of Probab. 11(2), 374-392.!tt
An analytic algorithm for the space-time fractional reaction-diffusion equation
M. G. Brikaa
2015-11-01
Full Text Available In this paper, we solve the space-time fractional reaction-diffusion equation by the fractional homotopy analysis method. Solutions of different examples of the reaction term will be computed and investigated. The approximation solutions of the studied models will be put in the form of convergent series to be easily computed and simulated. Comparison with the approximation solution of the classical case of the studied modeled with their approximation errors will also be studied.
Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption
Yulan Wang
2012-01-01
Full Text Available We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.
Statistical mechanics of driven diffusive systems
Schmittmann, B
1995-01-01
Far-from-equilibrium phenomena, while abundant in nature, are not nearly as well understood as their equilibrium counterparts. On the theoretical side, progress is slowed by the lack of a simple framework, such as the Boltzmann-Gbbs paradigm in the case of equilibrium thermodynamics. On the experimental side, the enormous structural complexity of real systems poses serious obstacles to comprehension. Similar difficulties have been overcome in equilibrium statistical mechanics by focusing on model systems. Even if they seem too simplistic for known physical systems, models give us considerable insight, provided they capture the essential physics. They serve as important theoretical testing grounds where the relationship between the generic physical behavior and the key ingredients of a successful theory can be identified and understood in detail. Within the vast realm of non-equilibrium physics, driven diffusive systems form a subset with particularly interesting properties. As a prototype model for these syst...
Wang, Jin-Liang; Wu, Huai-Ning; Huang, Tingwen; Ren, Shun-Yan
2016-04-01
Two types of coupled neural networks with reaction-diffusion terms are considered in this paper. In the first one, the nodes are coupled through their states. In the second one, the nodes are coupled through the spatial diffusion terms. For the former, utilizing Lyapunov functional method and pinning control technique, we obtain some sufficient conditions to guarantee that network can realize synchronization. In addition, considering that the theoretical coupling strength required for synchronization may be much larger than the needed value, we propose an adaptive strategy to adjust the coupling strength for achieving a suitable value. For the latter, we establish a criterion for synchronization using the designed pinning controllers. It is found that the coupled reaction-diffusion neural networks with state coupling under the given linear feedback pinning controllers can realize synchronization when the coupling strength is very large, which is contrary to the coupled reaction-diffusion neural networks with spatial diffusion coupling. Moreover, a general criterion for ensuring network synchronization is derived by pinning a small fraction of nodes with adaptive feedback controllers. Finally, two examples with numerical simulations are provided to demonstrate the effectiveness of the theoretical results.
Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction
Arnrich, Steffen; Peletier, Mark A; Savaré, Giuseppe; Veneroni, Marco
2011-01-01
We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\\epsilon}, and in the limit {\\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an ...
The constructive technique and its application in solving a nonlinear reaction diffusion equation
Lai Shao-Yong; Guo Yun-Xi; Qing Yin; Wu Yong-Hong
2009-01-01
A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.
Allergic Contact Dermatitis with Diffuse Erythematous Reaction from Diisopropanolamine in a Compress
Tomoko Rind
2010-04-01
Full Text Available Compresses containing a nonsteroidal antiinflammatory drug (NSAID are commonly used in Japan. However, this treatment may induce both allergic and photoallergic contact dermatitis from the NSAIDs and their ingredients. Here, we describe a case of allergic contact dermatitis with diffuse erythematous reaction due to diisopropanolamine in the applied compress. The absorption of diisopropanolamine might have been enhanced by the occlusive condition.
Allergic Contact Dermatitis with Diffuse Erythematous Reaction from Diisopropanolamine in a Compress
Tomoko Rind; Naoki Oiso; Ayaka Hirao; Akira Kawada
2010-01-01
Compresses containing a nonsteroidal antiinflammatory drug (NSAID) are commonly used in Japan. However, this treatment may induce both allergic and photoallergic contact dermatitis from the NSAIDs and their ingredients. Here, we describe a case of allergic contact dermatitis with diffuse erythematous reaction due to diisopropanolamine in the applied compress. The absorption of diisopropanolamine might have been enhanced by the occlusive condition.
Asymptotic Speed of Wave Propagation for A Discrete Reaction-Diffusion Equation
Xiu-xiang Liu; Pei-xuan Weng
2006-01-01
We deal with asymptotic speed of wave propagation for a discrete reaction-diffusion equation. We find the minimal wave speed c* from the characteristic equation and show that c* is just the asymptotic speed of wave propagation. The isotropic property and the existence of solution of the initial value problem for the given equation are also discussed.
Bifurcation Analysis of Gene Propagation Model Governed by Reaction-Diffusion Equations
Guichen Lu
2016-01-01
Full Text Available We present a theoretical analysis of the attractor bifurcation for gene propagation model governed by reaction-diffusion equations. We investigate the dynamical transition problems of the model under the homogeneous boundary conditions. By using the dynamical transition theory, we give a complete characterization of the bifurcated objects in terms of the biological parameters of the problem.
Ranjit Kumar
2012-09-01
Travelling and solitary wave solutions of certain coupled nonlinear diffusion-reaction equations have been constructed using the auxiliary equation method. These equations arise in a variety of contexts not only in biological, chemical and physical sciences but also in ecological and social sciences.
Ranjit Kumar; R S Kaushal; Awadhesh Prasad
2010-10-01
An auto-Bäcklund transformation derived in the homogeneous balance method is employed to obtain several new exact solutions of certain kinds of nonlinear diffusion-reaction (D-R) equations. These equations arise in a variety of problems in physical, chemical, biological, social and ecological sciences.
LONG-TIME BEHAVIOR OF A CLASS OF REACTION DIFFUSION EQUATIONS WITH TIME DELAYS
无
2006-01-01
The present paper devotes to the long-time behavior of a class of reaction diffusion equations with delays under Dirichlet boundary conditions. The stability and global attractability for the zero solution are provided, and the existence, stability and attractability for the positive stationary solution are also obtained.
Brownian motion in a field of force and the diffusion theory of chemical reactions. II
Brinkman, H.C.
1956-01-01
H. A. Kramers has studied the rate of chemical reactions in view of the Brownian forces caused by a surrounding medium in temperature equilibrium. In a previous paper 3) the author gave a solution of Kramers' diffusion equation in phase space by systematic development. In this paper the general prob
Oxygen Diffusion and Reaction Kinetics in Continuous Fiber Ceramic Matrix Composites
Halbig, Michael C.; Eckel, Andrew J.; Cawley, James D.
1999-01-01
Previous stressed oxidation tests of C/SiC composites at elevated temperatures (350 C to 1500 C) and sustained stresses (69 MPa and 172 MPa) have led to the development of a finite difference cracked matrix model. The times to failure in the samples suggest oxidation occurred in two kinetic regimes defined by the rate controlling mechanisms (i.e. diffusion controlled and reaction controlled kinetics). Microstructural analysis revealed preferential oxidation along as-fabricated, matrix microcracks and also suggested two regimes of oxidation kinetics dependent on the oxidation temperature. Based on experimental results, observation, and theory, a finite difference model was developed. The model simulates the diffusion of oxygen into a matrix crack bridged by carbon fibers. The model facilitates the study of the relative importance of temperature, the reaction rate constant, and the diffusion coefficient on the overall oxidation kinetics.
An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations
Burrage, Kevin
2012-01-01
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator. © 2012 Society for Industrial and Applied Mathematics.
Mix and Inject: Reaction Initiation by Diffusion for Time-Resolved Macromolecular Crystallography
Marius Schmidt
2013-01-01
Full Text Available Time-resolved macromolecular crystallography unifies structure determination with chemical kinetics, since the structures of transient states and chemical and kinetic mechanisms can be determined simultaneously from the same data. To start a reaction in an enzyme, typically, an initially inactive substrate present in the crystal is activated. This has particular disadvantages that are circumvented when active substrate is directly provided by diffusion. However, then it is prohibitive to use macroscopic crystals because diffusion times become too long. With small micro- and nanocrystals diffusion times are adequately short for most enzymes and the reaction can be swiftly initiated. We demonstrate here that a time-resolved crystallographic experiment becomes feasible by mixing substrate with enzyme nanocrystals which are subsequently injected into the X-ray beam of a pulsed X-ray source.
Bifurcation and Turing patterns of reaction-diffusion activator-inhibitor model
Wu, Ranchao; Zhou, Yue; Shao, Yan; Chen, Liping
2017-09-01
Gierer-Meinhardt system is one of prototypical pattern formation models. Turing instability could induce various patterns in this system. Hopf bifurcation analysis and its direction are performed on such diffusive model in this paper, by employing normal form and center manifold reduction. The effects of diffusion on the stability of equilibrium point and the bifurcated limit cycle from Hopf bifurcation are investigated. It is found that under some conditions, diffusion-driven instability, i.e, Turing instability, about the equilibrium point and the bifurcated limit cycle will happen, which are stable without diffusion. Those diffusion-driven instabilities will lead to the occurrence of spatially nonhomogeneous solutions. As a result, some patterns, like stripe and spike solutions, will form. The explicit criteria about the stability and instability of the equilibrium point and the limit cycle in the system are derived, which could be readily applied. Further, numerical simulations are presented to illustrate theoretical analysis.
Sayed Ameenuddin Irfan
2017-03-01
Full Text Available A mathematical model for the reaction-diffusion equation is developed to describe the nutrient release profiles and degradation of poly(lactic acid (PLA-coated controlled-release fertilizer. A multi-diffusion model that consists of coupled partial differential equations is used to study the diffusion and chemical reaction (autocatalytic degradation simultaneously. The model is solved using an analytical-numerical method. Firstly, the model equation is transformed using the Laplace transformation as the Laplace transform cannot be inverted analytically. Numerical inversion of the Laplace transform is used by employing the Zakian method. The solution is useful in predicting the nutrient release profiles at various diffusivity, concentration of extraction medium, and reaction rates. It also helps in explaining the transformation of autocatalytic concentration in the coating material for various reaction rates, times of reaction, and reaction-multi diffusion. The solution is also applicable to the other biodegradable polymer-coated controlled-release fertilizers.
Konukoglu, Ender; Clatz, Olivier; Menze, Bjoern H; Stieltjes, Bram; Weber, Marc-André; Mandonnet, Emmanuel; Delingette, Hervé; Ayache, Nicholas
2010-01-01
Reaction-diffusion based tumor growth models have been widely used in the literature for modeling the growth of brain gliomas. Lately, recent models have started integrating medical images in their formulation. Including different tissue types, geometry of the brain and the directions of white matter fiber tracts improved the spatial accuracy of reaction-diffusion models. The adaptation of the general model to the specific patient cases on the other hand has not been studied thoroughly yet. In this paper, we address this adaptation. We propose a parameter estimation method for reaction-diffusion tumor growth models using time series of medical images. This method estimates the patient specific parameters of the model using the images of the patient taken at successive time instances. The proposed method formulates the evolution of the tumor delineation visible in the images based on the reaction-diffusion dynamics; therefore, it remains consistent with the information available. We perform thorough analysis of the method using synthetic tumors and show important couplings between parameters of the reaction-diffusion model. We show that several parameters can be uniquely identified in the case of fixing one parameter, namely the proliferation rate of tumor cells. Moreover, regardless of the value the proliferation rate is fixed to, the speed of growth of the tumor can be estimated in terms of the model parameters with accuracy. We also show that using the model-based speed, we can simulate the evolution of the tumor for the specific patient case. Finally, we apply our method to two real cases and show promising preliminary results.
Elements of a Model State Education Agency Diffusion System.
Mojkowski, Charles
A study, presented to the National Dissemination Conference, provides a conceptualization of a model diffusion system as it might exist within a state education agency (SEA) and places this diffusion model within the context of the SEA's expanding role as an educational service. Five conclusions were reached regarding a model diffusion system.…
Diffusion and particle mobility in 1D system
Borman, V.D. [Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 31, 115409 Moscow (Russian Federation); Johansson, B. [Condensed Matter Theory Group, Department of Physics, Box 530, Uppsala University, S-75121 Uppsala (Sweden); Applied Materials Physics, Department of Materials and Engineering, Royal Institute of Technology (KTH), S-10044 Stockholm (Sweden); Skorodumova, N.V. [Condensed Matter Theory Group, Department of Physics, Box 530, Uppsala University, S-75121 Uppsala (Sweden); Tronin, I.V. [Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 31, 115409 Moscow (Russian Federation)]. E-mail: ivt@rbcmail.ru; Tronin, V.N. [Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 31, 115409 Moscow (Russian Federation); Troyan, V.I. [Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 31, 115409 Moscow (Russian Federation)
2006-12-04
The transport properties of one-dimensional (1D) systems have been studied theoretically. Contradictory experimental results on molecular transport in quasi-1D systems, such as zeolite structures, when both diffusion transport acceleration and the existence of the diffusion mode with lower particle mobility (single-file diffusion (
Diffuse-Illumination Systems for Growing Plants
May, George; Ryan, Robert
2010-01-01
Agriculture in both terrestrial and space-controlled environments relies heavily on artificial illumination for efficient photosynthesis. Plant-growth illumination systems require high photon flux in the spectral range corresponding with plant photosynthetic active radiation (PAR) (400 700 nm), high spatial uniformity to promote uniform growth, and high energy efficiency to minimize electricity usage. The proposed plant-growth system takes advantage of the highly diffuse reflective surfaces on the interior of a sphere, hemisphere, or other nearly enclosed structure that is coated with highly reflective materials. This type of surface and structure uniformly mixes discrete light sources to produce highly uniform illumination. Multiple reflections from within the domelike structures are exploited to obtain diffuse illumination, which promotes the efficient reuse of photons that have not yet been absorbed by plants. The highly reflective surfaces encourage only the plant tissue (placed inside the sphere or enclosure) to absorb the light. Discrete light sources, such as light emitting diodes (LEDs), are typically used because of their high efficiency, wavelength selection, and electronically dimmable properties. The light sources are arranged to minimize shadowing and to improve uniformity. Different wavelengths of LEDs (typically blue, green, and red) are used for photosynthesis. Wavelengths outside the PAR range can be added for plant diagnostics or for growth regulation
Synthesis and materialization of a reaction-diffusion French flag pattern
Zadorin, Anton S.; Rondelez, Yannick; Gines, Guillaume; Dilhas, Vadim; Urtel, Georg; Zambrano, Adrian; Galas, Jean-Christophe; Estevez-Torres, André
2017-10-01
During embryo development, patterns of protein concentration appear in response to morphogen gradients. These patterns provide spatial and chemical information that directs the fate of the underlying cells. Here, we emulate this process within non-living matter and demonstrate the autonomous structuration of a synthetic material. First, we use DNA-based reaction networks to synthesize a French flag, an archetypal pattern composed of three chemically distinct zones with sharp borders whose synthetic analogue has remained elusive. A bistable network within a shallow concentration gradient creates an immobile, sharp and long-lasting concentration front through a reaction-diffusion mechanism. The combination of two bistable circuits generates a French flag pattern whose 'phenotype' can be reprogrammed by network mutation. Second, these concentration patterns control the macroscopic organization of DNA-decorated particles, inducing a French flag pattern of colloidal aggregation. This experimental framework could be used to test reaction-diffusion models and fabricate soft materials following an autonomous developmental programme.
Diffusion in isotopically controlled semiconductor systems
Bracht, H.
1999-12-01
Isotopically controlled heterostructures of 28Si/natSi and Al71GaAs/Al69GaAs/71GaAs have been used to study the self-diffusion process in this elemental and compound semiconductor material. The directly measured Si self-diffusion coefficient is compared with the self-interstitial and vacancy contribution to self-diffusion which were deduced from metal diffusion experiments. The remarkable agreement between the Si self-diffusion coefficients and the individual contributions to self-diffusion shows that both self-interstitials and vacancies mediate Si self-diffusion. The Ga self-diffusion in undoped AlGaAs was found to decrease with increasing Al concentration. The activation enthalpy of Ga and Al diffusion in GaAs and of Ga diffusion in AlGaAs all lie in the range of (3.6±0.1) eV, but with different pre-exponential factors. The doping dependence of Ga self-diffusion reveals a retardation (enhancement) of Ga diffusion under p-type (n-type) doping compared to intrinsic conditions. All experimental results on the group-III atom diffusion are accurately described if vacancies on the group-III sublattice are assumed to mediate the Ga self- and Al-Ga interdiffusion in undoped AlGaAs and the Ga self-diffusion in Be- and Si-doped GaAs with an active dopant concentration of 3×1018 cm-3. The doping dependence of Ga self-diffusion in GaAs provides strong evidence that neutral, singly and doubly charged Ga vacancies govern the self-diffusion process.
Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion
Lei Zhang
2014-01-01
Full Text Available Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing, we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point (u2*,v2*,w2*, and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameter c1, the sequence “holes → holes-stripe mixtures → stripes → spots-stripe mixtures → spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point (u1*,v1*,0, and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.
A reaction-diffusion SIS epidemic model in an almost periodic environment
Wang, Bin-Guo; Li, Wan-Tong; Wang, Zhi-Cheng
2015-12-01
A susceptible-infected-susceptible almost periodic reaction-diffusion epidemic model is studied by means of establishing the theories and properties of the basic reproduction ratio {R0}. Particularly, the asymptotic behaviors of {R0} with respect to the diffusion rate {DI} of the infected individuals are obtained. Furthermore, the uniform persistence, extinction and global attractivity are presented in terms of {R0}. Our results indicate that the interaction of spatial heterogeneity and temporal almost periodicity tends to enhance the persistence of the disease.
Plante, Ianik; Cucinotta, Francis A.
2011-01-01
The irradiation of biological systems leads to the formation of radiolytic species such as H(raised dot), (raised dot)OH, H2, H2O2, e(sup -)(sub aq), etc.[1]. These species react with neighboring molecules, which result in damage in biological molecules such as DNA. Radiation chemistry is there for every important to understand the radiobiological consequences of radiation[2]. In this work, we discuss an approach based on the exact Green Functions for diffusion-influenced reactions which may be used to simulate radiation chemistry and eventually extended to study more complex systems, including DNA.
The mechanism of Turing pattern formation in a positive feedback system with cross diffusion
Yang, Xiyan; Liu, Tuoqi; Zhang, Jiajun; Zhou, Tianshou
2014-03-01
In this paper, we analyze a reaction-diffusion (R-D) system with a double negative feedback loop and find cases where self diffusion alone cannot lead to Turing pattern formation but cross diffusion can. Specifically, we first derive a set of sufficient conditions for Turing instability by performing linear stability analysis, then plot two bifurcation diagrams that specifically identify Turing regions in the parameter phase plane, and finally numerically demonstrate representative Turing patterns according to the theoretical predictions. Our analysis combined with previous studies actually implies an interesting fact that Turing patterns can be generated not only in a class of monostable R-D systems where cross diffusion is not necessary but also in a class of bistable R-D systems where cross diffusion is necessary. In addition, our model would be a good candidate for experimentally testing Turing pattern formation from the viewpoint of synthetic biology.
Ruzi, Mahmut; Anderson, David T
2015-12-17
Our group has been working to develop parahydrogen (pH2) matrix isolation spectroscopy as a method to study low-temperature condensed-phase reactions of atomic hydrogen with various reaction partners. Guided by the well-defined studies of cold atom chemistry in rare-gas solids, the special properties of quantum hosts such as solid pH2 afford new opportunities to study the analogous chemical reactions under quantum diffusion conditions in hopes of discovering new types of chemical reaction mechanisms. In this study, we present Fourier transform infrared spectroscopic studies of the 193 nm photoinduced chemistry of nitric oxide (NO) isolated in solid pH2 over the 1.8 to 4.3 K temperature range. Upon short-term in situ irradiation the NO readily undergoes photolysis to yield HNO, NOH, NH, NH3, H2O, and H atoms. We map the postphotolysis reactions of mobile H atoms with NO and document first-order growth in HNO and NOH reaction products for up to 5 h after photolysis. We perform three experiments at 4.3 K and one at 1.8 K to permit the temperature dependence of the reaction kinetics to be quantified. We observe Arrhenius-type behavior with a pre-exponential factor of A = 0.036(2) min(-1) and Ea = 2.39(1) cm(-1). This is in sharp contrast to previous H atom reactions we have studied in solid pH2 that display definitively non-Arrhenius behavior. The contrasting temperature dependence measured for the H + NO reaction is likely related to the details of H atom quantum diffusion in solid pH2 and deserves further study.
Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Cangiani, Andrea [University of Leicester, Leicester (United Kingdom); Sutton, Oliver [University of Leicester, Leicester (United Kingdom)
2014-10-02
This document describes the conforming formulations for virtual element approximation of the convection-reaction-diffusion equation with variable coefficients. Emphasis is given to construction of the projection operators onto polynomial spaces of appropriate order. These projections make it possible the virtual formulation to achieve any order of accuracy. We present the construction of the internal and the external formulation. The difference between the two is in the way the projection operators act on the derivatives (laplacian, gradient) of the partial differential equation. For the diffusive regime we prove the well-posedness of the external formulation and we derive an estimate of the approximation error in the H^{1}-norm. For the convection-dominated case, the streamline diffusion stabilization (aka SUPG) is also discussed.
A reaction-diffusion model of ROS-induced ROS release in a mitochondrial network.
Lufang Zhou
2010-01-01
Full Text Available Loss of mitochondrial function is a fundamental determinant of cell injury and death. In heart cells under metabolic stress, we have previously described how the abrupt collapse or oscillation of the mitochondrial energy state is synchronized across the mitochondrial network by local interactions dependent upon reactive oxygen species (ROS. Here, we develop a mathematical model of ROS-induced ROS release (RIRR based on reaction-diffusion (RD-RIRR in one- and two-dimensional mitochondrial networks. The nodes of the RD-RIRR network are comprised of models of individual mitochondria that include a mechanism of ROS-dependent oscillation based on the interplay between ROS production, transport, and scavenging; and incorporating the tricarboxylic acid (TCA cycle, oxidative phosphorylation, and Ca(2+ handling. Local mitochondrial interaction is mediated by superoxide (O2.- diffusion and the O2.(--dependent activation of an inner membrane anion channel (IMAC. In a 2D network composed of 500 mitochondria, model simulations reveal DeltaPsi(m depolarization waves similar to those observed when isolated guinea pig cardiomyocytes are subjected to a localized laser-flash or antioxidant depletion. The sensitivity of the propagation rate of the depolarization wave to O(2.- diffusion, production, and scavenging in the reaction-diffusion model is similar to that observed experimentally. In addition, we present novel experimental evidence, obtained in permeabilized cardiomyocytes, confirming that DeltaPsi(m depolarization is mediated specifically by O2.-. The present work demonstrates that the observed emergent macroscopic properties of the mitochondrial network can be reproduced in a reaction-diffusion model of RIRR. Moreover, the findings have uncovered a novel aspect of the synchronization mechanism, which is that clusters of mitochondria that are oscillating can entrain mitochondria that would otherwise display stable dynamics. The work identifies the
Yochelis, Arik; Bar-On, Tomer; Gov, Nir S.
2016-04-01
Unconventional myosins belong to a class of molecular motors that walk processively inside cellular protrusions towards the tips, on top of actin filament. Surprisingly, in addition, they also form retrograde moving self-organized aggregates. The qualitative properties of these aggregates are recapitulated by a mass conserving reaction-diffusion-advection model and admit two distinct families of modes: traveling waves and pulse trains. Unlike the traveling waves that are generated by a linear instability, pulses are nonlinear structures that propagate on top of linearly stable uniform backgrounds. Asymptotic analysis of isolated pulses via a simplified reaction-diffusion-advection variant on large periodic domains, allows to draw qualitative trends for pulse properties, such as the amplitude, width, and propagation speed. The results agree well with numerical integrations and are related to available empirical observations.
Front propagation in cellular flows for fast reaction and small diffusivity
Tzella, Alexandra
2014-01-01
We investigate the influence of fluid flows on the propagation of chemical fronts arising in FKPP type models. For the cellular flows we consider, the front propagation speed can be determined numerically by solving an eigenvalue problem; this is however difficult for small molecular diffusivity and fast reaction, i.e., when the P\\'eclet (Pe) and Damk\\"ohler (Da) numbers are large. Here, we employ a WKB approach to obtain the front speed for a broad range of Pe,Da$\\gg 1$ in terms of a periodic path -- an instanton -- that minimizes a certain functional, and to derive closed-form results for Da$\\ll$Pe and for Da$\\gg$Pe. Our theoretical predictions are compared with (i) numerical solutions of the eigenvalue problem and (ii) simulations of the advection--diffusion--reaction equation.
Front propagation in cellular flows for fast reaction and small diffusivity
Tzella, Alexandra; Vanneste, Jacques
2014-07-01
We investigate the influence of fluid flows on the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large Péclet (Pe) and Damköhler (Da) numbers. The front speed is expressed in terms of a periodic path—an instanton—that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for (logPe)-1≪Da≪Pe and for Da≫Pe. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection-diffusion-reaction equation.
Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions
Bagchi, Partha; Sengupta, Srikumar; Srivastava, Ajit M
2015-01-01
Reaction-diffusion equations with suitable boundary conditions have special propagating solutions which very closely resemble the moving interfaces in a first order transition. We show that the dynamics of chiral order parameter for chiral symmetry breaking transition in heavy-ion collisions, with dissipative dynamics, is governed by one such equation, specifically, the Newell-Whitehead equation. Further, required boundary conditions are automatically satisfied due to the geometry of the collision. The chiral transition is, therefore, completed by a propagating interface, exactly as for a first order transition, even though the transition actually is a crossover for relativistic heavy-ion collisions. Same thing also happens when we consider the initial confinement-deconfinement transition with Polyakov loop order parameter. The resulting equation, again with dissipative dynamics, can then be identified with the reaction-diffusion equation known as the Fitzhugh-Nagumo equation which is used in population genet...
STOCHASTIC CRACKING AND HEALING BEHAVIORS OF THIN FILMS DURING REACTION-DIFFUSION GROWTH
S.L. Zhu; S.L. Yang; Y.M. Xiong; M.S. Li; S.J. Geng; C.S. Hu; Fuhui Wang; W. T. Wu
2001-01-01
The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reaction-diffusion growth. The thermal growthof oxide films follows power law when no cracking occurs. Our results showed that thegrowth kinetics under stochastic cracking and healing conditions was different fromthat without cracking. It might be altered to either pseudo-linear or pseudo-power lawsdependent upon the intensity and frequency of the cracking of the films. When thehoping items dominated, the growth followed pseudo-linear law; when the diffusionalitems dominated, it followed pseudo-power law with the exponentials lower than theintrinsical values. The numerical results were in good agreement with the meassuredkinetics of isothermal and cyclic oxidation of NiAl-0.1 Y (at. %) alloys in air at 1273K.
Phase transition and crossover in diffusion-limited aggregation with reaction times
Nagatani, Takashi; Stanley, H. Eugene
1990-09-01
A generalized diffusion-limited aggregation (DLA) with reaction times that has been proposed by Bunde and Miyazima [Phys. Rev. A 38, 2099 (1988)] is considered. Crossover from the DLA to the diffusion-limited self-avoiding walk (DLSAW) is investigated by using the two-parameter position-space renormalization-group method. The crossover exponent and the crossover radius are calculated. The geometrical phase transition between DLA and DLSAW found by Bunde and Miyajima is analyzed by making use of the three-parameter position-space renormalization-group method. A global flow diagram in the three-parameter space is obtained. Above the percolation threshold all the renormalization flows are merged into the DLA point. Below the threshold all the renormalization flows are merged into the DLSAW point. When the reaction time is large, the double-crossover phenomenon occurs below the threshold.
Tuan, Nguyen Huy; Van Au, Vo; Khoa, Vo Anh; Lesnic, Daniel
2017-05-01
The identification of the population density of a logistic equation backwards in time associated with nonlocal diffusion and nonlinear reaction, motivated by biology and ecology fields, is investigated. The diffusion depends on an integral average of the population density whilst the reaction term is a global or local Lipschitz function of the population density. After discussing the ill-posedness of the problem, we apply the quasi-reversibility method to construct stable approximation problems. It is shown that the regularized solutions stemming from such method not only depend continuously on the final data, but also strongly converge to the exact solution in L 2-norm. New error estimates together with stability results are obtained. Furthermore, numerical examples are provided to illustrate the theoretical results.
Effect of mixing on reaction-diffusion kinetics for protein hydrogel-based microchips.
Zubtsov, D A; Ivanov, S M; Rubina, A Yu; Dementieva, E I; Chechetkin, V R; Zasedatelev, A S
2006-03-09
Protein hydrogel-based microchips are being developed for high-throughput evaluation of the concentrations and activities of various proteins. To shorten the time of analysis, the reaction-diffusion kinetics on gel microchips should be accelerated. Here we present the results of the experimental and theoretical analysis of the reaction-diffusion kinetics enforced by mixing with peristaltic pump. The experiments were carried out on gel-based protein microchips with immobilized antibodies under the conditions utilized for on-chip immunoassay. The dependence of fluorescence signals at saturation and corresponding saturation times on the concentrations of immobilized antibodies and antigen in solution proved to be in good agreement with theoretical predictions. It is shown that the enhancement of transport with peristaltic pump results in more than five-fold acceleration of binding kinetics. Our results suggest useful criteria for the optimal conditions for assays on gel microchips to balance high sensitivity and rapid fluorescence saturation kinetics.
A non-scale-invariant form for coarse-grained diffusion-reaction equations
Ostvar, Sassan; Wood, Brian D.
2016-09-01
The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.
Plante, Ianik; Cucinotta, Francis A.
2011-01-01
Radiolytic species are formed approximately 1 ps after the passage of ionizing radiation through matter. After their formation, they diffuse and chemically react with other radiolytic species and neighboring biological molecules, leading to various oxidative damage. Therefore, the simulation of radiation chemistry is of considerable importance to understand how radiolytic species damage biological molecules [1]. The step-by-step simulation of chemical reactions is difficult, because the radiolytic species are distributed non-homogeneously in the medium. Consequently, computational approaches based on Green functions for diffusion-influenced reactions should be used [2]. Recently, Green functions for more complex type of reactions have been published [3-4]. We have developed exact random variate generators of these Green functions [5], which will allow us to use them in radiation chemistry codes. Moreover, simulating chemistry using the Green functions is which is computationally very demanding, because the probabilities of reactions between each pair of particles should be evaluated at each timestep [2]. This kind of problem is well adapted for General Purpose Graphic Processing Units (GPGPU), which can handle a large number of similar calculations simultaneously. These new developments will allow us to include more complex reactions in chemistry codes, and to improve the calculation time. This code should be of importance to link radiation track structure simulations and DNA damage models.
String theory as a diffusing system
Calcagni, Gianluca
2009-01-01
Recent results on the effective non-local dynamics of the tachyon mode of open string field theory (OSFT) show that approximate solutions can be constructed which obey the diffusion equation. We argue that this structure is inherited from the full theory, where it admits a background-independent formulation. In fact, all known exact OSFT solutions are superpositions of diffusing surface states. In particular, the diffusion equation is a spacetime manifestation of OSFT gauge symmetries.
The two-regime method for optimizing stochastic reaction-diffusion simulations
Flegg, M. B.
2011-10-19
Spatial organization and noise play an important role in molecular systems biology. In recent years, a number of software packages have been developed for stochastic spatio-temporal simulation, ranging from detailed molecular-based approaches to less detailed compartment-based simulations. Compartment-based approaches yield quick and accurate mesoscopic results, but lack the level of detail that is characteristic of the computationally intensive molecular-based models. Often microscopic detail is only required in a small region (e.g. close to the cell membrane). Currently, the best way to achieve microscopic detail is to use a resource-intensive simulation over the whole domain. We develop the two-regime method (TRM) in which a molecular-based algorithm is used where desired and a compartment-based approach is used elsewhere. We present easy-to-implement coupling conditions which ensure that the TRM results have the same accuracy as a detailed molecular-based model in the whole simulation domain. Therefore, the TRM combines strengths of previously developed stochastic reaction-diffusion software to efficiently explore the behaviour of biological models. Illustrative examples and the mathematical justification of the TRM are also presented.
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Bueno-Orovio, Alfonso
2014-04-01
© 2014, Springer Science+Business Media Dordrecht. Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of ℝ. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.
1/f noise: diffusive systems and music
Voss, R.F.
1975-11-01
Measurements of the 1/f voltage noise in continuous metal films are reported. At room temperature, samples of pure metals and bismuth (with a carrier density smaller by 10/sup 5/) of similar volume had comparable noise. The results suggest that the noise arises from equilibrium temperature fluctuations modulating the resistance. Spatial correlation of the noise implied that the fluctuations obey a diffusion equation. The empirical inclusion of an explicit 1/f region and appropriate normalization lead to excellent agreement with the measured noise. If the fluctuations are assumed to be spatially correlated, the diffusion equation can yield an extended 1/f region in the power spectrum. The temperature response of a sample to delta and step function power inputs is shown to have the same shape as the autocorrelation function for uncorrelated and correlated temperature fluctuations, respectively. The spectrum obtained from the cosine transform of the measured step function response is in excellent agreement with the measured 1/f voltage noise spectrum. Spatially correlated equilibrium temperature fluctuations are not the dominant source of 1/f noise in semiconductors and metal films. However, the agreement between the low-frequency spectrum of fluctuations in the mean-square Johnson noise voltage and the resistance fluctuation spectrum measured in the presence of a current demonstrates that in these systems the 1/f noise is also due to equilibrium resistance fluctuations. Loudness fluctuations in music and speech and pitch fluctuations in music also show the 1/f behavior. 1/f noise sources, consequently, are demonstrated to be the natural choice for stochastic composition. 26 figures, 1 table. (auth)
Size-controlled synthesis of Cu2O nanoparticles via reaction-diffusion
Badr, Layla; Epstein, Irving R.
2017-02-01
Copper (I) oxide nanoparticles are synthesized by a simple reaction-diffusion process involving Cu+ ions and sodium hydroxide in gelatin. The mean diameter and the size dispersion of the nanoparticles can be controlled by two experimental parameters, the percent of gelatin in the medium and the hydroxide ion concentration. UV-visible spectroscopy, transmission electron microscopy and X-ray diffraction are used to analyze the size, morphology, and chemical composition of the nanoparticles generated.
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains
Li, Dingshi; Wang, Bixiang; Wang, Xiaohu
2017-02-01
This paper deals with the limiting behavior of stochastic reaction-diffusion equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. We first prove the existence, uniqueness and periodicity of pullback tempered random attractors for the equations in an (n + 1)-dimensional narrow domain, and then establish the upper semicontinuity of these attractors when a family of (n + 1)-dimensional thin domains collapses onto an n-dimensional domain.
Henisch, H K
1991-01-01
Containing illustrations, worked examples, graphs and tables, this book deals with periodic precipitation (also known as Liesegang Ring formation) in terms of mathematical models and their logical consequences, and is entirely concerned with microcomputer analysis and software development. Three distinctive periodic precipitation mechanisms are included: binary diffusion-reaction; solubility modulation, and competitive particle growth. The book provides didactic illustrations of a valuable investigational procedure, in the form of hypothetical experimentation by microcomputer. The development
Solid state diffusion and reaction in ZnO/SiO{sub 2} in thin films
Jakob, A.; Stucki, S.; Schnyder, B.; Koetz, R. [Paul Scherrer Inst. (PSI), Villigen (Switzerland)
1997-06-01
Detoxification of fly ash from waste incineration by evaporating harmful heavy metals is limited by the formation of stable heavy metal-matrix compounds. To study the rate of these heavy metal-matrix reactions, experiments were performed with the diffusion couple ZnO (heavy metal)-SiO{sub 2} (matrix). The atomic concentration profiles after different annealing treatments were analysed by X-ray photoelectron spectroscopy (XPS). (author) 3 figs., 4 refs.
Compact-like kink in a real electrical reaction-diffusion chain
Comte, J.C. [Laboratoire de Physiopathologie des Reseaux Neuronaux du Cycle Veille-Sommeil, CNRS UMR 5167, Faculte de Medecine Laennec 7, Rue Guillaume Paradin, 69372 Lyon Cedex 08 (France)]. E-mail: comtejc@sommeil.univ-lyon1.fr; Marquie, P. [Laboratoire d' Electronique, Informatique et Image (LE2i) UMR CNRS 5158, Aile des Sciences de l' Ingenieur, BP 47870, 21078 Dijon Cedex (France)
2006-07-15
We demonstrate experimentally the compact-like kinks existence in a real electrical reaction-diffusion chain. Our measures show that such entities are strictly localized and consequently present a finite spatial extent. We show equally that the kink velocity is threshold-dependent. A theoretical quantification of the critical coupling under which propagation fails is also achieved and reveals that nonlinear coupling leads to a propagation failure reduction.
Meng-ge Liu; Wei Yu; Chi-xing Zhou
2006-01-01
The kinetic model for diffusion-controlled intermolecular reaction of homogenous polymer under steady shear was theoretically studied. The classic formalism and the concept of conformation ellipsoids were integrated to get a new equation, which directly correlates the rate constant with shear rate. It was found that the rate constant is not monotonic with shear rate. The scale of rate constant is N-1.5 (N is the length of chains), which is in consistent with de Gennes's result.
Xiang-Chao Shi
2016-02-01
Full Text Available The fractional reaction diffusion equation is one of the popularly used fractional partial differential equations in recent years. The fast Adomian decomposition method is used to obtain the solution of the Cauchy problem. Also, the analytical scheme is extended to the fractional one where the Taylor series is employed. In comparison with the classical Adomian decomposition method, the ratio of the convergence is increased. The method is more reliable for the fractional partial differential equations.
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-01-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic dissipative particle dynamics (DPD) framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux a...
Chuangxia Huang
2011-01-01
Full Text Available Stability of reaction-diffusion recurrent neural networks (RNNs with continuously distributed delays and stochastic influence are considered. Some new sufficient conditions to guarantee the almost sure exponential stability and mean square exponential stability of an equilibrium solution are obtained, respectively. Lyapunov's functional method, M-matrix properties, some inequality technique, and nonnegative semimartingale convergence theorem are used in our approach. The obtained conclusions improve some published results.
INFLUENCE OF NOISE AND DELAY ON REACTION-DIFFUSION RECURRENT NEURAL NETWORKS
Li Wu
2006-01-01
In this paper, the influence of the noise and delay upon the stability property of reaction-diffusion recurrent neural networks (RNNs) with the time-varying delay is discussed. The new and easily verifiable conditions to guarantee the mean value exponential stability of an equilibrium solution are derived. The rate of exponential convergence can be estimated by means of a simple computation based on these criteria.
Biktashev, V. N.; Tsyganov, M. A.
2016-08-01
Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasi-solitons, have been observed also in dissipative, “excitable” systems, either at finely tuned parameters (near a bifurcation) or in systems with cross-diffusion. Here we demonstrate that quasi-solitons can be robustly observed in excitable systems with excitable kinetics and with self-diffusion only. This includes quasi-solitons of fixed shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective cross-diffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multi-component, stiff systems by studying simplified, soft systems.
Xu, X; Sumption, M D
2016-01-12
In this work we explore the compositions of non-stoichiometric intermediate phases formed by diffusion reactions: a mathematical framework is developed and tested against the specific case of Nb3Sn superconductors. In the first part, the governing equations for the bulk diffusion and inter-phase interface reactions during the growth of a compound are derived, numerical solutions to which give both the composition profile and growth rate of the compound layer. The analytic solutions are obtained with certain approximations made. In the second part, we explain an effect that the composition characteristics of compounds can be quite different depending on whether it is the bulk diffusion or grain boundary diffusion that dominates in the compounds, and that "frozen" bulk diffusion leads to unique composition characteristics that the bulk composition of a compound layer remains unchanged after its initial formation instead of varying with the diffusion reaction system; here the model is modified for the case of grain boundary diffusion. Finally, we apply this model to the Nb3Sn superconductors and propose approaches to control their compositions.
SYSTEMIC TOXIC REACTIONS TO LOCAL ANESTHETICS
Moore, Daniel C.; Green, John
1956-01-01
The topical use of anesthetic agents involves an element of risk. Systemic toxic reactions are rare, but they do occur and may result in death. When a reaction occurs from a topical application, it usually progresses rapidly to respiratory and cardiovascular collapse, and thus therapy must be instituted with more haste to avoid deaths. Fatal systemic toxic reactions from topically administered anesthetic drugs are, in effect, usually not due to well informed use of the drug but to misuse owing to less than complete understanding of absorption. Emphasis is placed on the causes, prophylaxis and treatment of severe systemic toxic reactions which follow the topical application of local anesthetic drugs. If systemic toxic reactions resulting from a safe dose of a local anesthetic agent are correctly treated, there will usually follow an uneventful recovery rather than a catastrophe. PMID:13343009
Kleinman, Leonid S.; Red, X. B., Jr.
1995-01-01
An algorithm has been developed for time-dependent forced convective diffusion-reaction having convection by a recirculating flow field within the drop that is hydrodynamically coupled at the interface with a convective external flow field that at infinity becomes a uniform free-streaming flow. The concentration field inside the droplet is likewise coupled with that outside by boundary conditions at the interface. A chemical reaction can take place either inside or outside the droplet, or reactions can take place in both phases. The algorithm has been implemented, and for comparison results are shown here for the case of no reaction in either phase and for the case of an external first order reaction, both for unsteady behavior. For pure interphase mass transfer, concentration isocontours, local and average Sherwood numbers, and average droplet concentrations have been obtained as a function of the physical properties and external flow field. For mass transfer enhanced by an external reaction, in addition to the above forms of results, we present the enhancement factor, with the results now also depending upon the (dimensionless) rate of reaction.
Kleinman, Leonid S.; Reed, X. B., Jr.
1995-01-01
An algorithm has been developed for the forced convective diffusion-reaction problem for convection inside and outside a droplet by a recirculating flow field hydrodynamically coupled at the droplet interface with an external flow field that at infinity becomes a uniform streaming flow. The concentration field inside the droplet is likewise coupled with that outside by boundary conditions at the interface. A chemical reaction can take place either inside or outside the droplet or reactions can take place in both phases. The algorithm has been implemented and results are shown here for the case of no reaction and for the case of an external first order reaction, both for unsteady behavior. For pure interphase mass transfer, concentration isocontours, local and average Sherwood numbers, and average droplet concentrations have been obtained as a function of the physical properties and external flow field. For mass transfer enhanced by an external reaction, in addition to the above forms of results, we present the enhancement factor, with the results now also depending upon the (dimensionless) rate of reaction.
Aristotelous, Andreas C; Haider, Mansoor A
2014-08-01
Macroscopic models accounting for cellular effects in natural or engineered tissues may involve unknown constitutive terms that are highly dependent on interactions at the scale of individual cells. Hybrid discrete models, which represent cells individually, were used to develop and apply techniques for modeling diffusive nutrient transport and cellular uptake to identify a nonlinear nutrient loss term in a macroscopic reaction-diffusion model of the system. Flexible and robust numerical methods were used, based on discontinuous Galerkin finite elements in space and a Crank-Nicolson temporal discretization. Scales were bridged via averaging operations over a complete set of subdomains yielding data for identification of a macroscopic nutrient loss term that was accurately captured via a fifth-order polynomial. Accuracy of the identified macroscopic model was demonstrated by direct, quantitative comparisons of the tissue and cellular scale models in terms of three error norms computed on a mesoscale mesh. Copyright © 2014 John Wiley & Sons, Ltd.
Tsibidis, George D
2008-01-01
Temporally and spatially resolved measurements of protein transport inside cells provide important clues to the functional architecture and dynamics of biological systems. Fluorescence Recovery After Photobleaching (FRAP) technique has been used over the past three decades to measure the mobility of macromolecules and protein transport and interaction with immobile structures inside the cell nucleus. A theoretical model is presented that aims to describe protein transport inside the nucleus, a process which is influenced by the presence of a boundary (i.e. membrane). A set of reaction-diffusion equations is employed to model both the diffusion of proteins and their interaction with immobile binding sites. The proposed model has been designed to be applied to biological samples with a Confocal Laser Scanning Microscope (CLSM) equipped with the feature to bleach regions characterised by a scanning beam that has a radially Gaussian distributed profile. The proposed model leads to FRAP curves that depend on the o...
Sato, Makoto; Yasugi, Tetsuo; Minami, Yoshiaki; Miura, Takashi; Nagayama, Masaharu
2016-08-30
Notch-mediated lateral inhibition regulates binary cell fate choice, resulting in salt and pepper patterns during various developmental processes. However, how Notch signaling behaves in combination with other signaling systems remains elusive. The wave of differentiation in the Drosophila visual center or "proneural wave" accompanies Notch activity that is propagated without the formation of a salt and pepper pattern, implying that Notch does not form a feedback loop of lateral inhibition during this process. However, mathematical modeling and genetic analysis clearly showed that Notch-mediated lateral inhibition is implemented within the proneural wave. Because partial reduction in EGF signaling causes the formation of the salt and pepper pattern, it is most likely that EGF diffusion cancels salt and pepper pattern formation in silico and in vivo. Moreover, the combination of Notch-mediated lateral inhibition and EGF-mediated reaction diffusion enables a function of Notch signaling that regulates propagation of the wave of differentiation.
Qiankun Song
2007-06-01
Full Text Available Impulsive bidirectional associative memory neural network model with time-varying delays and reaction-diffusion terms is considered. Several sufficient conditions ensuring the existence, uniqueness, and global exponential stability of equilibrium point for the addressed neural network are derived by M-matrix theory, analytic methods, and inequality techniques. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. The obtained results in this paper are less restrictive than previously known criteria. Two examples are given to show the effectiveness of the obtained results.
Cao Jinde
2007-01-01
Full Text Available Impulsive bidirectional associative memory neural network model with time-varying delays and reaction-diffusion terms is considered. Several sufficient conditions ensuring the existence, uniqueness, and global exponential stability of equilibrium point for the addressed neural network are derived by M-matrix theory, analytic methods, and inequality techniques. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. The obtained results in this paper are less restrictive than previously known criteria. Two examples are given to show the effectiveness of the obtained results.
Unified path integral approach to theories of diffusion-influenced reactions
Prüstel, Thorsten; Meier-Schellersheim, Martin
2017-08-01
Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we develop a general approach for computational modeling of diffusion-influenced reactions that is capable of capturing not only the classical Smoluchowski picture but also alternative theories, as is here exemplified by a volume reactivity model. In particular, we prove the path decomposition expansion of various Green's functions describing the irreversible and reversible reaction of an isolated pair of molecules. To this end, we exploit a connection between boundary value and interaction potential problems with δ - and δ'-function perturbation. We employ a known path-integral-based summation of a perturbation series to derive a number of exact identities relating propagators and survival probabilities satisfying different boundary conditions in a unified and systematic manner. Furthermore, we show how the path decomposition expansion represents the propagator as a product of three factors in the Laplace domain that correspond to quantities figuring prominently in stochastic spatially resolved simulation algorithms. This analysis will thus be useful for the interpretation of current and the design of future algorithms. Finally, we discuss the relation between the general approach and the theory of Brownian functionals and calculate the mean residence time for the case of irreversible and reversible reactions.
Energy Diffusion in Harmonic System with Conservative Noise
Basile, Giada; Olla, Stefano
2014-06-01
We prove diffusive behaviour of the energy fluctuations in a system of harmonic oscillators with a stochastic perturbation of the dynamics that conserves energy and momentum. The results concern pinned systems in any dimension, or unpinned systems in dimension.
Percolation phenomena in diffusion-controlled polymer matrix systems
徐铜文; 何炳林
1997-01-01
The controlled release of two kinds of drugs,5-fluorouracil (5-FU) and hydrocortisonum (Hydro.) loaded in poly(ethylene-vinylalcohol) (EVAL) was dealt with,of which 5-FU/EVAL and Hydro /EVAL matrix systems are composed.The results were analyzed using the pseudo-steady-diffusion models coupled with the fundamental concepts of percolation theory.The percolation thresholds for the two systems were calculated,which could indicate the contributions of pore diffusion and matrix diffusion.
Thermodynamics of diffusive DM/DE systems
Haba, Z.
2017-04-01
We discuss the energy density, temperature and entropy of dark matter (DM) and dark energy (DE) as functions of the scale factor a in an expanding universe. In a model of non-interacting dark components we repeat a derivation from thermodynamics of the well-known relations between the energy density, entropy and temperature. In particular, the entropy is constant as a consequence of the energy conservation. We consider a model of a DM/DE interaction where the DM energy density increase is proportional to the particle density. In such a model the dependence of the energy density and the temperature on the scale factor a is substantially modified. We discuss (as a realization of the model) DM which consists of relativistic particles diffusing in an environment of DE. The energy gained by the dark matter comes from a cosmological fluid with a negative pressure. We define the entropy and free energy of such a non-equilibrium system. We show that during the universe evolution the entropy of DM is increasing whereas the entropy of DE is decreasing. The total entropy can increase (in spite of the energy conservation) as the DM and DE temperatures are different. We discuss non-equilibrium thermodynamics on the basis of the notion of the relative entropy.
Trinkaus, H.; Singh, Bachu Narain; Golubov, S.I.
In recent years, it has been shown that a number of striking features in the microstructural evolution occurring in metals under cascade damage generating irradiation (e.g. enhanced swelling near grain boundaries, decoration of dislocations with SIA loops, saturation of void growth and void lattice...... formation) can be rationalised in terms of intra-cascade clustering of vacancies and self-interstitial atoms (SIAs), differences in the thermal stability and mobility of the resulting clusters and one-dimensional (1D) glide diffusion of SIA clusters (“production bias model”). The 1D diffusion of SIA...... clusters is generally disturbed by changes between equivalent 1D diffusion paths and by transversal diffusion by self-climb, resulting in diffusion reaction kinetics between the 1D and 3D limiting cases. In this paper, a general treatment of such kinetics operating in systems containing random...
Saxena, R. K.; Mathai, A. M.; Haubold, H. J.
2015-10-01
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function ϕ (x, t) which is a nonlinear function governing reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.
Diffusion and particle mobility in 1D system
Borman, V. D.; Johansson, B.; Skorodumova, N. V.; Tronin, I. V.; Tronin, V. N.; Troyan, V. I.
2006-01-01
The transport properties of one-dimensional (1D) systems have been studied theoretically. Contradictory experimental results on molecular transport in quasi-1D systems, such as zeolite structures, when both diffusion transport acceleration and the existence of the diffusion mode with lower particle
Lund, Halvor; Simonsen, Ingve
2012-01-01
We review and introduce a generalized reaction-diffusion approach to epidemic spreading in a metapopulation modeled as a complex network. The metapopulation consists of susceptible and infected individuals that are grouped in subpopulations symbolising cities and villages that are coupled by human travel in a transportation network. By analytic methods and numerical simulations we calculate the fraction of infected people in the metaopoluation in the long time limit, as well as the relevant parameters characterising the epidemic threshold that separates an epidemic from a non-epidemic phase. Within this model, we investigate the effect of a heterogeneous network topology and a heterogeneous subpopulation size distribution. Such a system is suited for epidemic modeling where small villages and big cities exist simultaneously in the metapopulation. We find that the heterogeneous conditions cause the epidemic threshold to be a non-trivial function of the reaction rates (local parameters), the network's topology ...
The DNA Binding Activity of p53 Displays Reaction-Diffusion Kinetics
Hinow, Peter; Rogers, Carl E.; Barbieri, Christopher E.; Pietenpol, Jennifer A.; Kenworthy, Anne K.; DiBenedetto, Emmanuele
2006-01-01
The tumor suppressor protein p53 plays a key role in maintaining the genomic stability of mammalian cells and preventing malignant transformation. In this study, we investigated the intracellular diffusion of a p53-GFP fusion protein using confocal fluorescence recovery after photobleaching. We show that the diffusion of p53-GFP within the nucleus is well described by a mathematical model for diffusion of particles that bind temporarily to a spatially homogeneous immobile structure with binding and release rates k1 and k2, respectively. The diffusion constant of p53-GFP was estimated to be Dp53-GFP = 15.4 μm2 s−1, significantly slower than that of GFP alone, DGFP = 41.6 μm2 s−1. The reaction rates of the binding and unbinding of p53-GFP were estimated as k1 = 0.3 s−1 and k2 = 0.4 s−1, respectively, values suggestive of nonspecific binding. Consistent with this finding, the diffusional mobilities of tumor-derived sequence-specific DNA binding mutants of p53 were indistinguishable from that of the wild-type protein. These data are consistent with a model in which, under steady-state conditions, p53 is latent and continuously scans DNA, requiring activation for sequence-specific DNA binding. PMID:16603489
Diffuse endocrine system, neuroendocrine tumors and immunity: what's new?
Ameri, Pietro; Ferone, Diego
2012-01-01
During the last two decades, research into the modulation of immunity by the neuroendocrine system has flourished, unravelling significant effects of several neuropeptides, including somatostatin (SRIH), and especially cortistatin (CST), on immune cells. Scientists have learnt that the diffuse neuroendocrine system can regulate the immune system at all its levels: innate immunity, adaptive immunity, and maintenance of immune tolerance. Compelling studies with animal models have demonstrated that some neuropeptides may be effective in treating inflammatory disorders, such as sepsis, and T helper 1-driven autoimmune diseases, like Crohn's disease and rheumatoid arthritis. Here, the latest findings concerning the neuroendocrine control of the immune system are discussed, with emphasis on SRIH and CST. The second part of the review deals with the immune response to neuroendocrine tumors (NETs). The anti-NET immune response has been described in the last years and it is still being characterized, similarly to what is happening for several other types of cancer. In parallel with investigations addressing the mechanisms by which the immune system contrasts NET growth and spreading, ground-breaking clinical trials of dendritic cell vaccination as immunotherapy for metastatic NETs have shown in principle that the immune reaction to NETs can be exploited for treatment.
Korneev, V. G.
2016-11-01
Efficiency of the error control of numerical solutions of partial differential equations entirely depends on the two factors: accuracy of an a posteriori error majorant and the computational cost of its evaluation for some test function/vector-function plus the cost of the latter. In the paper consistency of an a posteriori bound implies that it is the same in the order with the respective unimprovable a priori bound. Therefore, it is the basic characteristic related to the first factor. The paper is dedicated to the elliptic diffusion-reaction equations. We present a guaranteed robust a posteriori error majorant effective at any nonnegative constant reaction coefficient (r.c.). For a wide range of finite element solutions on a quasiuniform meshes the majorant is consistent. For big values of r.c. the majorant coincides with the majorant of Aubin (1972), which, as it is known, for relatively small r.c. (< ch -2 ) is inconsistent and looses its sense at r.c. approaching zero. Our majorant improves also some other majorants derived for the Poisson and reaction-diffusion equations.
High Curie temperature Mn5Ge3 thin films produced by non-diffusive reaction
Assaf, E.; Portavoce, A.; Hoummada, K.; Bertoglio, M.; Bertaina, S.
2017-02-01
Polycrystalline Mn5Ge3 thin films were produced on SiO2 using magnetron sputtering and reactive diffusion (RD) or non-diffusive reaction (NDR). In situ X-ray diffraction and atomic force microscopy were used to determine the layer structures, and magnetic force microscopy, superconducting quantum interference device, and ferromagnetic resonance were used to determine their magnetic properties. RD-mediated layers exhibit similar magnetic properties as molecular beam epitaxy-grown monocrystalline Mn5Ge3 thin films, while NDR-mediated layers show magnetic properties similar to monocrystalline C-doped Mn5Ge3Cx thin films with 0.1 ≤ x ≤ 0.2. NDR appears as a complementary metal oxide semi-conductor-compatible efficient method to produce good magnetic quality high-Curie temperature Mn5Ge3 thin films.
A Lagrangian study of scalar diffusion in isotropic turbulence with chemical reaction
Mitarai, S.; Riley, J. J.; Kosály, G.
2003-12-01
Direct numerical simulations are performed of a single-step, nonpremixed, Arrhenius-type reaction developing in isotropic, incompressible, decaying turbulence, for conditions where flame extinction and re-ignition occur. The Lagrangian characteristics of scalar diffusion, information necessary for modeling approaches such as some implementations of probability density function (PDF) methods, are investigated by tracking fluid particles. Focusing on the mixture fraction and temperature as the scalar variables of interest, fluid particles are characterized as continuously burning or noncontinuously burning based upon their recent time history, and noncontinuously burning particles are further characterized based upon their initial regions relative to the flame zone. The behavior of the mixture fraction and temperature fields is contrasted for the different types of particles characterized. Significant differences among these characterized particles are found, for example, in the unclosed conditional expectations of scalar diffusion appearing in the composition PDF equations.
Hepburn, I.; Chen, W.; De Schutter, E.
2016-08-01
Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-diffusion operator splitting in the literature and discuss their effects on accuracy. We introduce an operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.
A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems
Yang Liu
2013-01-01
Full Text Available A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term ∇·(a(x,t∇u is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical H(div;Ω space and the hyperbolic part d(x(∂u/∂t+c(x,t·∇u is handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in L2- and H1-norms for the scalar unknown u and a priori error estimates in (L22-norm for its gradient λ and its flux σ (the coefficients times the negative gradient are derived. Finally, a numerical example is provided to verify our theoretical results.
Reactions of small molecular systems
Wittig, C. [Univ. of Southern California, Los Angeles, CA (United States)
1993-12-01
This DOE program remains focused on small molecular systems relevant to combustion. Though a number of experimental approaches and machines are available for this research, the authors` activities are centered around the high-n Rydberg time-of-flight (HRTOF) apparatus in this laboratory. One student and one postdoc carry out experiments with this machine and also engage in small intra-group collaborations involving shared equipment. This past year was more productive than the previous two, due to the uninterrupted operation of the HRTOF apparatus. Results were obtained with CH{sub 3}OH, CH{sub 3}SH, Rg-HX complexes, HCOOH, and their deuterated analogs where appropriate. One paper is in print, three have been accepted for publication, and one is under review. Many preliminary results that augur well for the future were obtained with other systems such as HNO{sub 3}, HBr-HI complexes, toluene, etc. Highlights from the past year are presented below that display some of the features of this program.
Liming WU; Zhengliang ZHANG
2006-01-01
We establish Talagrand's T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type's approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to ReactionDiffusion equations are provided.
BHARDWAJ S B; SINGH RAM MEHAR; SHARMA KUSHAL; MISHRA S C
2016-06-01
Attempts have been made to explore the exact periodic and solitary wave solutions of nonlinear reaction diffusion (RD) equation involving cubic–quintic nonlinearity along with timedependent convection coefficients. Effect of varying model coefficients on the physical parameters of solitary wave solutions is demonstrated. Depending upon the parametric condition, the periodic,double-kink, bell and antikink-type solutions for cubic–quintic nonlinear reaction-diffusion equation are extracted. Such solutions can be used to explain various biological and physical phenomena.
Chaotic Diffusion in the Gliese-876 Planetary System
Martí, J G; Beaugé, C
2016-01-01
Chaotic diffusion is supposed to be responsible for orbital instabilities in planetary systems after the dissipation of the protoplanetary disk, and a natural consequence of irregular motion. In this paper we show that resonant multi-planetary systems, despite being highly chaotic, not necessarily exhibit significant diffusion in phase space, and may still survive virtually unchanged over timescales comparable to their age.Using the GJ-876 system as an example, we analyze the chaotic diffusion of the outermost (and less massive) planet. We construct a set of stability maps in the surrounding regions of the Laplace resonance. We numerically integrate ensembles of close initial conditions, compute Poincar\\'e maps and estimate the chaotic diffusion present in this system. Our results show that, the Laplace resonance contains two different regions: an inner domain characterized by low chaoticity and slow diffusion, and an outer one displaying larger values of dynamical indicators. In the outer resonant domain, th...
On Uniform Decay of the Entropy for Reaction–Diffusion Systems
Mielke, Alexander
2014-09-10
This work provides entropy decay estimates for classes of nonlinear reaction–diffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.
Validity conditions for stochastic chemical kinetics in diffusion-limited systems
Gillespie, Daniel T.; Petzold, Linda R.; Seitaridou, Effrosyni
2014-02-01
The chemical master equation (CME) and the mathematically equivalent stochastic simulation algorithm (SSA) assume that the reactant molecules in a chemically reacting system are "dilute" and "well-mixed" throughout the containing volume. Here we clarify what those two conditions mean, and we show why their satisfaction is necessary in order for bimolecular reactions to physically occur in the manner assumed by the CME and the SSA. We prove that these conditions are closely connected, in that a system will stay well-mixed if and only if it is dilute. We explore the implications of these validity conditions for the reaction-diffusion (or spatially inhomogeneous) extensions of the CME and the SSA to systems whose containing volumes are not necessarily well-mixed, but can be partitioned into cubical subvolumes (voxels) that are. We show that the validity conditions, together with an additional condition that is needed to ensure the physical validity of the diffusion-induced jump probability rates of molecules between voxels, require the voxel edge length to have a strictly positive lower bound. We prove that if the voxel edge length is steadily decreased in a way that respects that lower bound, the average rate at which bimolecular reactions occur in the reaction-diffusion CME and SSA will remain constant, while the average rate of diffusive transfer reactions will increase as the inverse square of the voxel edge length. We conclude that even though the reaction-diffusion CME and SSA are inherently approximate, and cannot be made exact by shrinking the voxel size to zero, they should nevertheless be useful in many practical situations.
Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations
Muratov, C B
2012-01-01
We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in $L^2$ under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.
Parabolic inverse convection-diffusion-reaction problem solved using an adaptive parametrization
Deolmi, Giulia
2011-01-01
This paper investigates the solution of a parabolic inverse problem based upon the convection-diffusion-reaction equation, which can be used to estimate both water and air pollution. We will consider both known and unknown source location: while in the first case the problem is solved using a projected damped Gauss-Newton, in the second one it is ill-posed and an adaptive parametrization with time localization will be adopted to regularize it. To solve the optimization loop a model reduction technique (Proper Orthogonal Decomposition) is used.
Asymptotic solution for a class of weakly nonlinear singularly perturbed reaction diffusion problem
TANG Rong-rong
2009-01-01
Under appropriate conditions, with the perturbation method and the theory of differential inequalities, a class of weakly nonlinear singularly perturbed reaction diffusion problem is considered. The existence of solution of the original problem is proved by constructing the auxiliary functions. The uniformly valid asymptotic expansions of the solution for arbitrary mth order approximation are obtained through constructing the formal solutions of the original problem, expanding the nonlinear terms to the power in small parameter e and comparing the coefficient for the same powers of ε. Finally, an example is provided, resulting in the error of O(ε2).
Raftari, Behrouz; Vuik, Kees
2015-01-01
The charging of insulating samples degrades the quality and complicates the interpretation of images in scanning electron microscopy and is important in other applications, such as particle detectors. In this paper we analyze this nontrivial phenomenon on different time scales employing the drift-diffusion-reaction approach augmented with the trapping rate equations and a realistic semi-empirical source function describing the pulsed nature of the electron beam. We consider both the fast processes following the impact of a single primary electron, the slower dynamics resulting from the continuous bombardment of a sample, and the eventual approach to the steady-state regime.
Jingsun Yao; Jiaqi Mo
2005-01-01
The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.
Positional information and reaction-diffusion: two big ideas in developmental biology combine.
Green, Jeremy B A; Sharpe, James
2015-04-01
One of the most fundamental questions in biology is that of biological pattern: how do the structures and shapes of organisms arise? Undoubtedly, the two most influential ideas in this area are those of Alan Turing's 'reaction-diffusion' and Lewis Wolpert's 'positional information'. Much has been written about these two concepts but some confusion still remains, in particular about the relationship between them. Here, we address this relationship and propose a scheme of three distinct ways in which these two ideas work together to shape biological form.
Strong-coupling diffusion in relativistic systems
Georg Wolschin
2003-05-01
Different from the early universe, heavy-ion collisions at very high energies do not reach statistical equilibrium, although thermal models explain many of their features. To account for nonequilibrium strong-coupling effects, a Fokker–Planck equation with time-dependent diffusion coefﬁcient is proposed. A schematic model for rapidity distributions of participant baryons is set up and solved analytically. The evolution from SIS via AGS and SPS to RHIC energies is discussed. Strong-coupling diffusion produces double-peaked spectra in central collisions at the higher SPS momentum of 158 A$\\cdot$GeV/c and beyond.
Aitova, E. V.; Bratsun, D. A.; Kostarev, K. G.; Mizev, A. I.; Mosheva, E. A.
2016-12-01
The development of convective instability in a two-layer system of miscible fluids placed in a narrow vertical gap has been studied theoretically and experimentally. The upper and lower layers are formed with aqueous solutions of acid and base, respectively. When the layers are brought into contact, the frontal neutralization reaction begins. We have found experimentally a new type of convective instability, which is characterized by the spatial localization and the periodicity of the structure observed for the first time in the miscible systems. We have tested a number of different acid-base systems and have found a similar patterning there. In our opinion, it may indicate that the discovered effect is of a general nature and should be taken into account in reaction-diffusion-convection problems as another tool with which the reaction can govern the movement of the reacting fluids. We have shown that, at least in one case (aqueous solutions of nitric acid and sodium hydroxide), a new type of instability called as the concentration-dependent diffusion convection is responsible for the onset of the fluid flow. It arises when the diffusion coefficients of species are different and depend on their concentrations. This type of instability can be attributed to a variety of double-diffusion convection. A mathematical model of the new phenomenon has been developed using the system of reaction-diffusion-convection equations written in the Hele-Shaw approximation. It is shown that the instability can be reproduced in the numerical experiment if only one takes into account the concentration dependence of the diffusion coefficients of the reagents. The dynamics of the base state, its linear stability and nonlinear development of the instability are presented. It is also shown that by varying the concentration of acid in the upper layer one can achieve the occurrence of chemo-convective solitary cell in the bulk of an almost immobile fluid. Good agreement between the
Rate of diffusion-limited reactions for a fractal aggregate of reactive spheres
Tseng, Chin-Yao; Tsao, Heng-Kwong
2002-08-01
We study the reaction rate for a fractal cluster of perfectly absorbing, stationary spherical sinks in a medium containing a mobile reactant. The effectiveness factor eta, which is defined as the ratio of the total reaction rate of the cluster to that without diffusional interactions, is calculated. The scaling behavior of eta is derived for arbitrary fractal dimension based on the Kirkwood-Riseman approximation. The asymptotic as well as the finite size scaling of eta are confirmed numerically by the method of multipole expansion, which has been proven to be an excellent approximation. The fractal assembly is made of N spheres with its dimension varying from D1, eta][approx(ln N)-1 for D=1, and eta][approxN0 for D1 the screening effect of diffusive interactions grows with the size, for Ddecays with decreasing D. The conclusion is also applicable to transport phenomena like dissolution, heat conduction, and sedimentation.
Reaction dynamics in polyatomic molecular systems
Miller, W.H. [Lawrence Berkeley Laboratory, CA (United States)
1993-12-01
The goal of this program is the development of theoretical methods and models for describing the dynamics of chemical reactions, with specific interest for application to polyatomic molecular systems of special interest and relevance. There is interest in developing the most rigorous possible theoretical approaches and also in more approximate treatments that are more readily applicable to complex systems.
Quantum Arnol'd Diffusion in a Simple Nonlinear System
Demikhovskii, V Y; Malyshev, A I
2002-01-01
We study the fingerprint of the Arnol'd diffusion in a quantum system of two coupled nonlinear oscillators with a two-frequency external force. In the classical description, this peculiar diffusion is due to the onset of a weak chaos in a narrow stochastic layer near the separatrix of the coupling resonance. We have found that global dependence of the quantum diffusion coefficient on model parameters mimics, to some extent, the classical data. However, the quantum diffusion happens to be slower that the classical one. Another result is the dynamical localization that leads to a saturation of the diffusion after some characteristic time. We show that this effect has the same nature as for the studied earlier dynamical localization in the presence of global chaos. The quantum Arnol'd diffusion represents a new type of quantum dynamics and can be observed, for example, in 2D semiconductor structures (quantum billiards) perturbed by time-periodic external fields.
Chemical reactions in reverse micelle systems
Matson, Dean W.; Fulton, John L.; Smith, Richard D.; Consani, Keith A.
1993-08-24
This invention is directed to conducting chemical reactions in reverse micelle or microemulsion systems comprising a substantially discontinuous phase including a polar fluid, typically an aqueous fluid, and a microemulsion promoter, typically a surfactant, for facilitating the formation of reverse micelles in the system. The system further includes a substantially continuous phase including a non-polar or low-polarity fluid material which is a gas under standard temperature and pressure and has a critical density, and which is generally a water-insoluble fluid in a near critical or supercritical state. Thus, the microemulsion system is maintained at a pressure and temperature such that the density of the non-polar or low-polarity fluid exceeds the critical density thereof. The method of carrying out chemical reactions generally comprises forming a first reverse micelle system including an aqueous fluid including reverse micelles in a water-insoluble fluid in the supercritical state. Then, a first reactant is introduced into the first reverse micelle system, and a chemical reaction is carried out with the first reactant to form a reaction product. In general, the first reactant can be incorporated into, and the product formed in, the reverse micelles. A second reactant can also be incorporated in the first reverse micelle system which is capable of reacting with the first reactant to form a product.
A hybrid multi-scale computational scheme for advection-diffusion-reaction equation
Karimi, S.; Nakshatrala, K. B.
2016-12-01
Simulation of transport and reaction processes in porous media and subsurface science has become more vital than ever. Over the past few decades, a variety of mathematical models and numerical methodologies for porous media simulations have been developed. As the demand for higher accuracy and validity of the models grows, the issue of disparate temporal and spatial scales becomes more problematic. The variety of reaction processes and complexity of pore geometry poses a huge computational burden in a real-world or reservoir scale simulation. Meanwhile, methods based on averaging or up- scaling techniques do not provide reliable estimates to pore-scale processes. To overcome this problem, development of hybrid and multi-scale computational techniques is considered a promising approach. In these methods, pore-scale and continuum-scale models are combined, hence, a more reliable estimate to pore-scale processes is obtained without having to deal with the tremendous computational overhead of pore-scale methods. In this presentation, we propose a computational framework that allows coupling of lattice Boltzmann method (for pore-scale simulation) and finite element method (for continuum-scale simulation) for advection-diffusion-reaction equations. To capture disparate in time and length events, non-matching grid and time-steps are allowed. Apart from application of this method to benchmark problems, multi-scale simulation of chemical reactions in porous media is also showcased.
Zhang, Xianlong; Wang, Xiaoling; Nie, Kai; Li, Mingpeng; Sun, Qingping
2016-08-01
Various species of bacteria form highly organized spatially-structured aggregates known as biofilms. To understand how microenvironments impact biofilm growth dynamics, we propose a diffusion-reaction continuum model to simulate the formation of Bacillus subtilis biofilm on an agar plate. The extended finite element method combined with level set method are employed to perform the simulation, numerical results show the quantitative relationship between colony morphologies and nutrient depletion over time. Considering that the production of polysaccharide in wild-type cells may enhance biofilm spreading on the agar plate, we inoculate mutant colony incapable of producing polysaccharide to verify our results. Predictions of the glutamate source biofilm’s shape parameters agree with the experimental mutant colony better than that of glycerol source biofilm, suggesting that glutamate is rate limiting nutrient for Bacillus subtilis biofilm growth on agar plate, and the diffusion-limited is a better description to the experiment. In addition, we find that the diffusion time scale is of the same magnitude as growth process, and the common-employed quasi-steady approximation is not applicable here.
Borisenko, Alexander
2016-05-01
During the processes of nucleation and growth of a precipitate cluster from a supersaturated solution, the diffusion flux between the cluster and the solution changes the solute concentration near the cluster-solution interface from its average bulk value. This feature affects the rates of attachment and detachment of solute atoms at the interface, and, therefore, the entire nucleation-growth kinetics is altered. Unless quite obvious, this effect has been ignored in classical nucleation theory. To illustrate the results of this approach, for the case of homogeneous nucleation, we calculate the total solubility and the nucleation rate as functions of two parameters of the model (the reduced interface energy and the inverse second Damköhler number), and we compare these results to the classical ones. One can conclude that discrepancies with classical nucleation theory are great in the diffusion-limited regime, when the rate of bulk diffusion is small compared to the rate of interface reactions, while in the opposite interface-limited case they vanish.
Wiberg, Gustav Karl Henrik; Arenz, Matthias
2015-01-01
We present a study concerning the influence of the diffusion of H+ and OH- ions on the hydrogen and oxygen evolution reactions (HER and OER) in aqueous electrolyte solutions. Using a rotating disk electrode (RDE), it is shown that at certain conditions the observed current, i.e., the reaction rate...
Systemic allergic reaction to pine nuts.
Nielsen, N H
1990-02-01
This case report describes a systemic reaction due to ingestion of pine nuts, confirmed by an open, oral provocation test. Skin prick testing with the aqueous allergen revealed an immediate positive prick test, and histamine release from basophil leukocytes to the aqueous allergen was demonstrated. Radioallergosorbent test demonstrated specific IgE antibodies to pine nuts. In a review of medical literature, we found no reports of either oral provocation tests confirming a systemic reaction due to ingestion of pine nuts or demonstration of specific IgE antibodies.
Lie and conditional symmetries of the three-component diffusive Lotka-Volterra system
Cherniha, Roman; Davydovych, Vasyl'
2013-05-01
Lie and Q-conditional symmetries of the classical three-component diffusive Lotka-Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component nonlinear reaction-diffusion system are constructed for the first time. The results are compared with those derived for the two-component diffusive Lotka-Volterra system. The conditional symmetry obtained for the non-Lie reduction of the three-component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found. Particularly, the exact solution describing different scenarios of competition between three species is constructed.
Dawn Spacecraft Reaction Control System Flight Experience
Mizukami, Masashi; Nakazono, Barry
2014-01-01
The NASA Dawn spacecraft mission is studying conditions and processes of the solar system's earliest epoch by investigating two protoplanets remaining intact since their formations, Ceres and Vesta. Launch was in 2007. Ion propulsion is used to fly to and enter orbit around Vesta, depart Vesta and fly to Ceres, and enter orbit around Ceres. A conventional blowdown hydrazine reaction control system (RCS) is used to provide external torques for attitude control. Reaction wheel assemblies were intended to provide attitude control in most cases. However, the spacecraft experienced one, then two apparent failures of reaction wheels. Also, similar thrusters experienced degradation in a long life application on another spacecraft. Those factors led to RCS being operated in ways completely different than anticipated prior to launch. Numerous mitigations and developments needed to be implemented. The Vesta mission was fully successful. Even with the compromises necessary due to those anomalies, the Ceres mission is also projected to be feasible.
Hallock, Michael J; Stone, John E; Roberts, Elijah; Fry, Corey; Luthey-Schulten, Zaida
2014-05-01
Simulation of in vivo cellular processes with the reaction-diffusion master equation (RDME) is a computationally expensive task. Our previous software enabled simulation of inhomogeneous biochemical systems for small bacteria over long time scales using the MPD-RDME method on a single GPU. Simulations of larger eukaryotic systems exceed the on-board memory capacity of individual GPUs, and long time simulations of modest-sized cells such as yeast are impractical on a single GPU. We present a new multi-GPU parallel implementation of the MPD-RDME method based on a spatial decomposition approach that supports dynamic load balancing for workstations containing GPUs of varying performance and memory capacity. We take advantage of high-performance features of CUDA for peer-to-peer GPU memory transfers and evaluate the performance of our algorithms on state-of-the-art GPU devices. We present parallel e ciency and performance results for simulations using multiple GPUs as system size, particle counts, and number of reactions grow. We also demonstrate multi-GPU performance in simulations of the Min protein system in E. coli. Moreover, our multi-GPU decomposition and load balancing approach can be generalized to other lattice-based problems.
Porcine skin flow-through diffusion cell system.
Baynes, R E
2001-11-01
Porcine Skin Flow-Through Diffusion Cell System (Ronald E. Baynes, North Carolina State University, Raleigh, North Carolina). Porcine skin can be used in a diffusion cell apparatus to study the rate and extent of absorption of topically applied chemicals through the skin. Although the skin of a number of animals can be used in this system, that of the pig most closely approximates human skin anatomically and physiologically.
Production method of raw material dispersion liquid for reaction layer of gas diffusion electrode
Furuya, Choichi; Motoo, Satoshi
1987-10-13
Heretofore, in order to make a raw material dispersion liquid of a reaction layer of a gas diffusion electrode, water repellent carbon, polytetrafluoroethylene, water and a surface active agent are mixed, then a cake is made by filtering this mixed liquid and afterwards the cake is heated and dried before being crushed. Since this crushing is done mechanically, homogeneous fine raw material powders cannot be obtained. Accordingly, even when a reaction layer is made by sintering a mixture of this powder, hydrophilic carbon black or hydrophilic carbon black carrying catalyst, and polytetrafluoroethylene, the hydrophilic part and the water repellent part are not distributed homogeneously and the catalytic performance of the reaction layer declines. In order to solve this, this invention proposes a production method that water repellent carbon black, polyterafluoroethylene, water and a surface active agent are mixed, then this mixture is frozen so that the surface active agent may not become active and homogeneous condensed cores of water repellent carbon black and polytetrafluoroethylene powders may be formed, and afterwards a homogeneous fine raw material dispersion liquid is made from thawing the condensed cores without change by thawing the above frozen mixture.
Chaotic diffusion in the Gliese-876 planetary system
Martí, J. G.; Cincotta, P. M.; Beaugé, C.
2016-07-01
Chaotic diffusion is supposed to be responsible for orbital instabilities in planetary systems after the dissipation of the protoplanetary disc, and a natural consequence of irregular motion. In this paper, we show that resonant multiplanetary systems, despite being highly chaotic, not necessarily exhibit significant diffusion in phase space, and may still survive virtually unchanged over time-scales comparable to their age. Using the GJ-876 system as an example, we analyse the chaotic diffusion of the outermost (and less massive) planet. We construct a set of stability maps in the surrounding regions of the Laplace resonance. We numerically integrate ensembles of close initial conditions, compute Poincaré maps and estimate the chaotic diffusion present in this system. Our results show that, the Laplace resonance contains two different regions: an inner domain characterized by low chaoticity and slow diffusion, and an outer one displaying larger values of dynamical indicators. In the outer resonant domain, the stochastic borders of the Laplace resonance seem to prevent the complete destruction of the system. We characterize the diffusion for small ensembles along the parameters of the outermost planet. Finally, we perform a stability analysis of the inherent chaotic, albeit stable Laplace resonance, by linking the behaviour of the resonant variables of the configurations to the different sub-structures inside the three-body resonance.
Reply to Comment on "Enhanced diffusion of enzymes that catalyze exothermic reactions"
Golestanian, Ramin
2016-01-01
Catalytically active enzymes have recently been observed to exhibit enhanced diffusion. In a recent work [C. Riedel et al., Nature 517, 227 (2015)], it has been suggested that this phenomenon is correlated with the degree of exothermicity of the reaction, and a mechanism was proposed to explain the phenomenon based on channeling the released heat into the center of mass kinetic energy of the enzyme. I addressed this question by comparing four different mechanisms, and concluded that collective heating is the strongest candidate out of those four to explain the phenomenon, and in particular, several orders of magnitude stronger than the mechanism proposed by Riedel et al. In a recent preprint (arXiv:1608.05433), K. Tsekouras, C. Riedel, R. Gabizon, S. Marqusee, S. Presse, and C. Bustamante present a comment on my paper [R. Golestanian, Phys. Rev. Lett. 115, 108102 (2015); arXiv:1508.03219], which I address here in this reply.
Numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory
Ramos, J. I.
1987-01-01
A review of numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory is presented. The methods reviewed include explicit, implicit, quasi-linearization, time linearization, operator-splitting, random walk and finite-element techniques and methods of lines. Adaptive and nonadaptive procedures are also reviewed. These techniques are applied first to solve two model problems which have exact traveling wave solutions with which the numerical results can be compared. This comparison is performed in terms of both the wave profile and computed wave speed. It is shown that the computed wave speed is not a good indicator of the accuracy of a particular method. A fourth-order time-linearized, Hermitian compact operator technique is found to be the most accurate method for a variety of time and space sizes.
The dynamics of nonlinear reaction-diffusion equations with small Lévy noise
Debussche, Arnaud; Imkeller, Peter
2013-01-01
This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.
Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations
Ducrot, Arnaud; Matano, Hiroshi
2012-01-01
We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities (including multistable ones) and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.
THE EXTINCTION BEHAVIOR OF THE SOLUTIONS FOR A CLASS OF REACTION-DIFFUSION EQUATIONS
陈松林
2001-01-01
The methods of Lp estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boudnary values u/ t = Au-λ |u|γ-1u-βu ((x,t) ∈Ω×(0,+∞)),u(x,t) | Ω×(0.+∞) = 0,u(x,0) = uo(x) ∈ H1 0(Ω) ∩ L1+γ(Ω) (x ∈Ω).Sufficient and necessary conditions about the extinction of the solutions is given Here λ＞0, γ＞ 0, β＞ 0 are constants, Ω∈ RN is bounded with smooth boundary Ω At last,it is simulated with a higher order equation by using the present methods.
TRAVELING WAVE SPEED AND SOLUTION IN REACTION-DIFFUSION EQUATION IN ONE DIMENSION
周天寿; 张锁春
2001-01-01
By Painlevé analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.
Grytskyy, Dmytro; Diesmann, Markus; Helias, Moritz
2016-06-01
Self-organized structures in networks with spike-timing dependent synaptic plasticity (STDP) are likely to play a central role for information processing in the brain. In the present study we derive a reaction-diffusion-like formalism for plastic feed-forward networks of nonlinear rate-based model neurons with a correlation sensitive learning rule inspired by and being qualitatively similar to STDP. After obtaining equations that describe the change of the spatial shape of the signal from layer to layer, we derive a criterion for the nonlinearity necessary to obtain stable dynamics for arbitrary input. We classify the possible scenarios of signal evolution and find that close to the transition to the unstable regime metastable solutions appear. The form of these dissipative solitons is determined analytically and the evolution and interaction of several such coexistent objects is investigated.
Grytskyy, Dmytro; Diesmann, Markus; Helias, Moritz
2016-06-01
Self-organized structures in networks with spike-timing dependent synaptic plasticity (STDP) are likely to play a central role for information processing in the brain. In the present study we derive a reaction-diffusion-like formalism for plastic feed-forward networks of nonlinear rate-based model neurons with a correlation sensitive learning rule inspired by and being qualitatively similar to STDP. After obtaining equations that describe the change of the spatial shape of the signal from layer to layer, we derive a criterion for the nonlinearity necessary to obtain stable dynamics for arbitrary input. We classify the possible scenarios of signal evolution and find that close to the transition to the unstable regime metastable solutions appear. The form of these dissipative solitons is determined analytically and the evolution and interaction of several such coexistent objects is investigated.
A reaction-diffusion-based coding rate control mechanism for camera sensor networks.
Yamamoto, Hiroshi; Hyodo, Katsuya; Wakamiya, Naoki; Murata, Masayuki
2010-01-01
A wireless camera sensor network is useful for surveillance and monitoring for its visibility and easy deployment. However, it suffers from the limited capacity of wireless communication and a network is easily overflown with a considerable amount of video traffic. In this paper, we propose an autonomous video coding rate control mechanism where each camera sensor node can autonomously determine its coding rate in accordance with the location and velocity of target objects. For this purpose, we adopted a biological model, i.e., reaction-diffusion model, inspired by the similarity of biological spatial patterns and the spatial distribution of video coding rate. Through simulation and practical experiments, we verify the effectiveness of our proposal.
Lattice Boltzmann Method for Diffusion-Reaction-Transport Processes in Heterogeneous Porous Media
XU You-Sheng; ZHONG Yi-Jun; HUANG Guo-Xiang
2004-01-01
Based on the lattice Boltzmann method and general theory of fluids flowing in porous media, a numerical model is presented for the diffusion-reaction-transport (DRT) processes in porous media. As a test, we simulate a DRT process in a two-dimensional horizontal heterogeneous porous medium. The influence of gravitation in this case can be neglected, and the DRT process can be described by a strongly heterogeneous diagnostic test strip or a thin confined piece of soil with stochastically distributing property in horizontal directions. The results obtained for the relations between reduced fluid saturation S, concentration c1, and concentration c2 are shown by using the visualization computing technique. The computational efficiency and stability of the model are satisfactory.
Kanittha Yimnak
2014-01-01
Full Text Available The meshless local Pretrov-Galerkin method (MLPG with the test function in view of the Heaviside step function is introduced to solve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumann boundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK interpolation method for constructing nodal shape function which holds the Kronecker delta property, thereby enhancing the arrangement nodal shape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in two numerical examples with investigating the convergence and the accuracy of numerical results. The numerical experiments revealing the solutions by the developed formulation are stable and more precise.
A soil diffusion-reaction model for surface COS flux: COSSM v1
W. Sun
2015-07-01
Full Text Available Soil exchange of carbonyl sulfide (COS is the second largest COS flux in terrestrial ecosystems. A novel application of COS is the separation of gross primary productivity (GPP from concomitant respiration. This method requires that soil COS exchange is relatively small and can be well quantified. Existing models for soil COS flux have incorporated empirical temperature and moisture functions derived from laboratory experiments, but not explicitly resolved diffusion in the soil column. We developed a 1-D diffusion-reaction model for soil COS exchange that accounts for COS uptake and production, relates source-sink terms to environmental variables, and has an option to enable surface litter layers. We evaluated the model with field data from a wheat field (Southern Great Plains (SGP, OK, USA and an oak woodland (Stunt Ranch Reserve, CA, USA. The model was able to reproduce all observed features of soil COS exchange such as diurnal variations and sink-source transitions. We found that soil COS uptake is strongly diffusion controlled, and limited by low COS concentrations in the soil if there is COS uptake in the litter layer. The model provides novel insights into the balance between soil COS uptake and production: a higher COS production capacity was required despite lower COS emissions during the growing season compared to the post-senescence period at SGP, and unchanged COS uptake capacity despite the dominant role of COS emissions after senescence. Once there is a database of soil COS parameters for key biomes, we expect the model will also be useful to simulate soil COS exchange at regional to global scales.
A soil diffusion-reaction model for surface COS flux: COSSM v1
Sun, W.; Maseyk, K.; Lett, C.; Seibt, U.
2015-10-01
Soil exchange of carbonyl sulfide (COS) is the second largest COS flux in terrestrial ecosystems. A novel application of COS is the separation of gross primary productivity (GPP) from concomitant respiration. This method requires that soil COS exchange is relatively small and can be well quantified. Existing models for soil COS flux have incorporated empirical temperature and moisture functions derived from laboratory experiments but not explicitly resolved diffusion in the soil column. We developed a mechanistic diffusion-reaction model for soil COS exchange that accounts for COS uptake and production, relates source-sink terms to environmental variables, and has an option to enable surface litter layers. We evaluated the model with field data from a wheat field (Southern Great Plains (SGP), OK, USA) and an oak woodland (Stunt Ranch Reserve, CA, USA). The model was able to reproduce all observed features of soil COS exchange such as diurnal variations and sink-source transitions. We found that soil COS uptake is strongly diffusion controlled and limited by low COS concentrations in the soil if there is COS uptake in the litter layer. The model provides novel insights into the balance between soil COS uptake and production: a higher COS production capacity was required despite lower COS emissions during the growing season compared to the post-senescence period at SGP, and unchanged COS uptake capacity despite the dominant role of COS emissions after senescence. Once there is a database of soil COS parameters for key biomes, we expect the model will also be useful to simulate soil COS exchange at regional to global scales.
UNIFORM BOUNDEDNESS AND STABILITY OF SOLUTIONS TO A CUBIC PREDATOR-PREY SYSTEM WITH CROSS-DIFFUSION
无
2009-01-01
Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.
Diffusion studies of dihydroxybenzene isomers in water-alcohol systems.
Codling, Dale J; Zheng, Gang; Stait-Gardner, Tim; Yang, Shu; Nilsson, Mathias; Price, William S
2013-03-01
Nuclear magnetic resonance diffusion studies can be used to identify different compounds in a mixture. However, because the diffusion coefficient is primarily dependent on the effective hydrodynamic radius, it is particularly difficult to resolve compounds with similar size and structure, such as isomers, on the basis of diffusion. Differential solution interactions between species in certain solutions can afford possibilities for separation. In the present study, the self-diffusion of the three isomers of dihydroxybenzene (i.e., (1,2-) catechol, (1,3-) resorcinol, and (1,4-) hydroquinone) was studied in water, aqueous monohydric alcohols (i.e., ethanol, 1-propanol, tert-butanol), and aqueous ethylene glycol. These systems allowed the effects of isomerism and differential solvent interactions on diffusion to be examined. It was found that, while in aqueous solution these isomers had the same diffusion coefficient, in water-monohydric alcohol systems the diffusion coefficient of catechol differed from those of resorcinol and hydroquinone. The separation was found to increase at higher concentrations of monohydric alcohols. The underlying chemical reasons for these differences were investigated.
Local traps as nanoscale reaction-diffusion probes: B clustering in c-Si
Pawlak, B. J., E-mail: bartekpawlak72@gmail.com [Globalfoundries, Kapeldreef 75, B-3001 Leuven (Belgium); Cowern, N. E. B.; Ahn, C. [School of Electrical and Electronic Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU (United Kingdom); Vandervorst, W. [IMEC, Kapeldreef 75, B-3001 Leuven, Belgium and IKS, Department of Physics, KU Leuven, Leuven (Belgium); Gwilliam, R. [Surrey Ion Beam Centre, Nodus Laboratory, University of Surrey, Guildford, Surrey GU2 7XH (United Kingdom); Berkum, J. G. M. van [Philips CFT, Prof. Holstlaan 4, 5656 AA Eindhoven (Netherlands)
2014-12-01
A series of B implantation experiments into initially amorphized and not fully recrystallized Si, i.e., into an existing a/c-Si bi-layer material, have been conducted. We varied B dose, energy, and temperature during implantation process itself. Significant B migration has been observed within c-Si part near the a/c-interface and near the end-of-range region before any activation annealing. We propose a general concept of local trapping sites as experimental probes of nanoscale reaction-diffusion processes. Here, the a/c-Si interface acts as a trap, and the process itself is explored as the migration and clustering of mobile BI point defects in nearby c-Si during implantation at temperatures from 77 to 573 K. We find that at room temperature—even at B concentrations as high as 1.6 atomic %, the key B-B pairing step requires diffusion lengths of several nm owing to a small, ∼0.1 eV, pairing energy barrier. Thus, in nanostructures doped by ion implantation, the implant distribution can be strongly influenced by thermal migration to nearby impurities, defects, and interfaces.
Musho, M.K.; Kozak, J.J.
1984-10-01
A method is presented for calculating exactly the relative width (sigma/sup 2/)/sup 1/2//
A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species.
Peng, Rui; Zhao, Xiao-Qiang
2016-02-01
In this article, we are concerned with a nonlocal reaction-diffusion-advection model which describes the evolution of a single phytoplankton species in a eutrophic vertical water column where the species relies solely on light for its metabolism. The new feature of our modeling equation lies in that the incident light intensity and the death rate are assumed to be time periodic with a common period. We first establish a threshold type result on the global dynamics of this model in terms of the basic reproduction number R0. Then we derive various characterizations of R0 with respect to the vertical turbulent diffusion rate, the sinking or buoyant rate and the water column depth, respectively, which in turn give rather precise conditions to determine whether the phytoplankton persist or become extinct. Our theoretical results not only extend the existing ones for the time-independent case, but also reveal new interesting effects of the modeling parameters and the time-periodic heterogeneous environment on persistence and extinction of the phytoplankton species, and thereby suggest important implications for phytoplankton growth control.
Macroscopic diffusive transport in a microscopically integrable Hamiltonian system.
Prosen, Tomaž; Zunkovič, Bojan
2013-07-26
We demonstrate that a completely integrable classical mechanical model, namely the lattice Landau-Lifshitz classical spin chain, supports diffusive spin transport with a finite diffusion constant in the easy-axis regime, while in the easy-plane regime, it displays ballistic transport in the absence of any known relevant local or quasilocal constant of motion in the symmetry sector of the spin current. This surprising finding should open the way towards analytical computation of diffusion constants for integrable interacting systems and hints on the existence of new quasilocal classical conservation laws beyond the standard soliton theory.
Diffuse Ceiling Inlet Systems and the Room Air Distribution
Nielsen, Peter V.; Jensen, Rasmus Lund; Rong, Li
2010-01-01
A diffuse ceiling inlet system is an air distribution system which is supplying the air through the whole ceiling. The system can remove a large heat load without creating draught in the room. The paper describes measurements in the case of both cooling and heating, and CFD predictions are given ...
Use of perforated acoustic panels as supply air diffusers in diffuse ceiling ventilation systems
Iqbal, Ahsan; Kazemi, Seyed Hossein; Ardkapan, Siamak Rahimi
manufacturing perforated acoustic panels for the last 13 years. The panels can be used not only in applications related to acoustics but also as low pressure drop supply air diffusers, particularly in diffuse ceiling ventilation systems. The present study verifies on a theoretically level the performance......Ventilation is needed for diluting and removing the contaminants, odour and excess heat from the building interior. It is important that the inhabitants perceive the ventilated spaces as comfortable. Therefore, the supply air should reach all parts of the occupied zones. Troldtekt has been...
Use of perforated acoustic panels as supply air diffusers in diffuse ceiling ventilation systems
Iqbal, Ahsan; Kazemi, Seyed Hossein; Ardkapan, Siamak Rahimi
Ventilation is needed for diluting and removing the contaminants, odour and excess heat from the building interior. It is important that the inhabitants perceive the ventilated spaces as comfortable. Therefore, the supply air should reach all parts of the occupied zones. Troldtekt has been...... manufacturing perforated acoustic panels for the last 13 years. The panels can be used not only in applications related to acoustics but also as low pressure drop supply air diffusers, particularly in diffuse ceiling ventilation systems. The present study verifies on a theoretically level the performance...
Simple deterministic dynamical systems with fractal diffusion coefficients
Klages, R
1999-01-01
We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional periodic array of scatterers in which point particles move from cell to cell as defined by a piecewise linear map. The microscopic chaotic scattering process of the map can be changed by a control parameter. This induces a parameter dependence for the macroscopic diffusion coefficient. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match to the ones of the simple phenomenological diffusion equation. Our main result is that the difffusion coefficient exhibits a fractal structure by varying the system parameter. To understand the origin of this fractal structure, we give qualitative and quantitative arguments. These arguments relate the sequence of oscillations in the strength of the parameter-dep...
Manifestation of the Arnol'd Diffusion in Quantum Systems
Demikhovskii, V Y; Malyshev, A I
2002-01-01
We study an analog of the classical Arnol'd diffusion in a quantum system of two coupled non-linear oscillators one of which is governed by an external periodic force with two frequencies. In the classical model this very weak diffusion happens in a narrow stochastic layer along the coupling resonance, and leads to an increase of total energy of the system. We show that the quantum dynamics of wave packets mimics, up to some extent, global properties of the classical Arnol'd diffusion. This specific diffusion represents a new type of quantum dynamics, and may be observed, for example, in 2D semiconductor structures (quantum billiards) perturbed by time-periodic external fields.
Extending Molecular Theory to Steady-State Diffusing Systems
FRINK,LAURA J. D.; SALINGER,ANDREW G.; THOMPSON,AIDAN P.
1999-10-22
Predicting the properties of nonequilibrium systems from molecular simulations is a growing area of interest. One important class of problems involves steady state diffusion. To study these cases, a grand canonical molecular dynamics approach has been developed by Heffelfinger and van Swol [J. Chem. Phys., 101, 5274 (1994)]. With this method, the flux of particles, the chemical potential gradients, and density gradients can all be measured in the simulation. In this paper, we present a complementary approach that couples a nonlocal density functional theory (DFT) with a transport equation describing steady-state flux of the particles. We compare transport-DFT predictions to GCMD results for a variety of ideal (color diffusion), and nonideal (uphill diffusion and convective transport) systems. In all cases excellent agreement between transport-DFT and GCMD calculations is obtained with diffusion coefficients that are invariant with respect to density and external fields.
Thermal Diffusivity Identification of Distributed Parameter Systems to Sea Ice
Liqiong Shi
2013-01-01
Full Text Available A method of optimal control is presented as a numerical tool for solving the sea ice heat transfer problem governed by a parabolic partial differential equation. Taken the deviation between the calculated ice temperature and the measurements as the performance criterion, an optimal control model of distributed parameter systems with specific constraints of thermal properties of sea ice was proposed to determine the thermal diffusivity of sea ice. Based on sea ice physical processes, the parameterization of the thermal diffusivity was derived through field data. The simulation results illustrated that the identified parameterization of the thermal diffusivity is reasonably effective in sea ice thermodynamics. The direct relation between the thermal diffusivity of sea ice and ice porosity is physically significant and can considerably reduce the computational errors. The successful application of this method also explained that the optimal control model of distributed parameter systems in conjunction with the engineering background has great potential in dealing with practical problems.
Weakly bound systems, continuum effects, and reactions
Jaganathen, Y; Ploszajczak, M
2012-01-01
Structure of weakly bound/unbound nuclei close to particle drip lines is different from that around the valley of beta stability. A comprehensive description of these systems goes beyond standard Shell Model and demands an open quantum system description of the nuclear many-body system. We approach this problem using the Gamow Shell Model which provides a fully microscopic description of bound and unbound nuclear states, nuclear decays, and reactions. We present in this paper the first application of the GSM for a description of the elastic and inelastic scattering of protons on 6He.
CHEN LaiWen; WANG JingHua; LEE Chun-Hian
2009-01-01
When hyperthermal atomic oxygen collides with a silicon surface, an ultrathin oxidation regime characterized by fractional atomic-oxygen anions having low diffusive and reactive barriers, along with their enhanced diffusion due to both the electric field and image potential, will form on the surface. In accordance with these properties, an attempt was made in the present study to modify the AlmeidaGoncalves-Baumvol (AGB) model by setting the diffusivity and reaction rate constant to be diffusion-length dependence. According to the modified model, numerical parametric studies for oxidation thin growth were performed. The dependencies of the diffusion coefficient, the reaction rate constant,the attenuation length, and the adjustable parameter upon the translational kinetic energy, flux, temperature, and tangential flux of atomic oxygen were analyzed briefly via the fitting of the experimental data given by Tagawa et al. The numerical results confirmed the rationality of the modified diffusion-reaction model. The model together with the computer code developed in this study would be a useful tool for thickness evaluation of the protective film against the oxidation of atomic oxygen toward spacecraft surface materials in LEO environment.
Vass, József
2016-01-01
A discretization scheme is introduced for a set of convection-diffusion equations with a non-linear reaction term, where the convection velocity is constant for each reactant. This constancy allows a transformation to new spatial variables, which ensures the global stability of discretization. Convection-diffusion equations are notorious for their lack of stability, arising from the algebraic interaction of the convection and diffusion terms. Unexpectedly, our implemented numerical algorithm proves to be faster than computing exact solutions derived for a special case, while remaining reasonably accurate, as demonstrated in our runtime and error analysis.
The swelling behavior of ionic polymers in the presence of diffusion and chemical reactions
Baek, Seungik
During the past few decades, ionic polymers have attracted much attention due to their unique properties, i.e., a stimulus-sensitive swelling behavior. The use of ionic polymers in biotechnology and medicine is increasingly attractive for applications such as controlled drug delivery, biomimetic actuators, and chemical valves. The ability to model and predict the swelling behavior of polymers and mass flux with respect to external environmental conditions is very important in such applications. The focus of this dissertation is on developing a continuum model for the behavior of a swollen solid and diffusion of a fluid through it in the presence of chemical reactions. To model diffusion through an ionic/nonionic polymer in the presence of mechanical deformation, we develop a variational procedure based on a thermodynamic framework for a swollen solid subject to constraints. This thermodynamic framework was mainly developed by Rajagopal and Srinivasa as a general thermodynamic framework for inelastic behavior of materials. First, a model for the mechanical behavior of a rubber-like solid in the presence of mass diffusion is developed using the variational procedure. By assuming a specific form for the Helmholtz potential function, we obtain an equilibrium equation that is identical to the equation suggested by Flory, Huggins, and Rehner. For the non-equilibrium problem, we assume a specific form for the rate of dissipation function and obtain the relation between pressure difference and mass flux. The numerical results show very good agreement with experimental data for the diffusion of organic solvents through a rubber slab. Hysteresis response of the swollen solid due to time-dependent loads is also simulated by the model. The variational procedure is also applied to pH-sensitive ionic polymers, which not only yields the equations for the deformation of a swollen solid and mass flux, but also the equations for chemical equilibrium. As a specific example, an
Marechal, F
1998-02-06
We measured for the first time the elastic and inelastic proton scattering on the {sup 40}S unstable nucleus. The experiment was performed in inverse kinematics at the NSCL AT Michigan State University with a {sup 40}S secondary beam bombarding a CH{sub 2} target at 30 MeV/A. We obtained the elastic scattering angular distribution and two points of the inelastic distribution to the first 2{sup +} excited state found to be located at 860{+-}90 KeV. With a coupled channel analysis, the {beta}{sub 2} quadrupolar deformation parameter is found to be equal to 0.35{+-}0.05. This value can be compared to 0.28{+-}0.02 obtained by coulomb excitation. A macroscopic analysis allowed us to extract the neutron and proton transition matrix element ratio M{sub n}/M{sub p} which is equal to 1.88{+-}0.38. This value, greater than N/Z, could indicate an isovector effect in the first 2{sup +} state excitation which could be due to a difference between the neutron and proton vibrations. The microscopic analysis gives the possibility to test the densities and the transition densities to the first 2{sup +} state. The calculated densities for the {sup 40}S nucleus show a neutron skin. However the microscopic analysis yields a M{sub n}/M{sub p} ratio of 1.40{+-}0.20. A similar elastic and inelastic proton scattering experiment allowed us to get a deformation parameter of 0.25{+-}0.03 for the {sup 43}Ar nucleus. To develop the study of direct reactions induced by radioactive beams at GANIL, we have developed and built, in collaboration with the CEA-Saclay and the CEA-Bruyeres, the new detector MUST.It is based on the silicon strip technology, and is dedicated to the measurement of recoiling light particles emitted in these reactions. The results obtained with a {sup 40}Ar beam at 77 Me V/A, have shown the good performances of the detector for the particle identification as well as for the resolutions, and allow us to consider now a large experimental programme concerning these direct