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In the paper we investigate a mathematical model describing the growth of tumor in the presence of immune response of a host organism. The dynamics of tumour and immune cells is based on the generic Michaelis-Menten kinetics describing interaction and competition between the tumour and the immune system. The appropriate phenomenological equation modeling cell-mediated immune surveillance against cancer is of the predator-prey form and within a given choice of parameters exhibits bistability. Under the influence of spontaneous weak fluctuations, the model may be analyzed in terms of a stochastic differential equation bearing the form of an overdamped Langevin-like dynamics in the external quasi-potential represented by a double well. We analyze properties of the system within the range of parameters for which the potential wells are of the same depth and when the additional perturbation describing a periodic treatment is insufficient to overcome the barrier height ...

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In this paper, we consider the prey-dependent consumption predator-prey (natural enemy-pest) models with age structure for the predator, immature and mature natural enemies are released and pesticide is applied impulsively. We prove that, when the impulsive period is no longer than some threshold, the pest-eradication solution is globally asymptotically stable, or say, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, so we further prove that, when the impulsive period is longer than the threshold, pest population and natural enemy population can coexist, i.e., the system is uniformly permanent. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying ...