Dynamical Behaviors Of The Predator-prey Model In A Random Environment

This work focuse on dynamic behaviors of the predator-prey system with Markov switching. In the predator-prey systems, function response terms reveal that ingestion rate of predators are affected by different disturbances. The Holling ? function re-sponse term is appliable to describe the phenomenon that ingestion rate of predators decreases due to the mutual interference among predators. Moreover, the popula-tion habitats are always subject to environmental noises, and distinguished by the circumstance such as nutrition or rainfall. Therefore, the model described by stochas-tic differential equations(SDEs) with Maxkov switching is more pertinent. The main methods axe Lyapunov function analysis, comparison theorem for stochastic differ-ential equations, Fredholm alternative theorem, and constructing auxiliary stochastic processes etc. Firstly, the existence and uniqueness of the global positive solution for the predator-prey system is obtained. Then bound of moments for the predator and prey and sufficient conditions for stochastic permanence of the prey are yielded. Fur-thermore, extinction of the predator and prey are established and the existence and uniqueness of stationary distribution and stochastic permanence of the predator are also discussed. Finally, a couple of examples and numerical simulations are given to illustrate the results.