The Study Of Intermittent Diffusion Population Model With Stochastic Selection Patches Under Markovian Switching

Species dispersal phenomena play vital roles in the real world. A large number of deterministic dispersal models have been investigated well by authors. However, there exists random perturbations in the natural world, species can not only be disturbed in the diffusion process by the external stochastic factors, but affected by internal factors (such as earthquake, flood, tsunami ect.). Therefore, based on these study, we focus on the model of species dispersal under the Markovian switching, mainly include:A single-species model with random selection of patches and intermittent diffusion under Markovian switching, a single-species model with selection of patches and intermittent diffusion under random environment and a predator-prey model with random selection of patches and prey intermittent diffusion under Markovian switching.This paper is arranged as follows:In section 1, we introduce the ecological background in this paper. Secondly, we give the current research situation and some main research results about random diffusion model. Finally, we give the major work about the paper.In section 2, The Preliminaries are given about this paper.In section 3, we give a single-species model with random selection patches and inter-mittent diffusion under Markovian switching and a single-species model with selection of patches and intermittent diffusion under random environment. By virtue of constructing appropriate Lyapunov function, applying generalized Ito formula and some assumptions, the global existence of a positive unique solution and the extinction in mean of the sys-tem are obtained. Meanwhile, by using Chebyshev’s inequality, the stochastic ultimate boundedness and stochastic permanence about this system are also got. In the end, some numerical examples are given to confirm our theoretical results.In section 4, we propose a predator-prey model with random selection of patches and intermittent diffusion under Markovian switching. By constructing appropriate Lyapunov function, making full use of generalized Ito formula, Chebyshev’s inequality and some corresponding methods (such as induction, comparison, ect.), the global existence of a positive unique solution, stochastic permanence and the extinction in mean of the system are obtained. Finally, we give some numerical examples to support the analytical results.In section 5, we have a summary about this paper.