Stationary Distribution and Extinction of Stochastic Beddington-DeAngelis Predator-prey Model with Distributed Delay

In this paper, we consider the dynamics of the stochastic Beddington-DeAngelis predator-prey model with distributed delay. First, we adopt the linear chain technique to transfer the stochastic system with strong kernel into an equivalent degenerated stochastic system made up four equations. Then we give the existence and uniqueness of the global positive solution. Next, sufficient conditions for persistence and extinction of two species are obtained. Particularly, the existence of the stationary distribution is established by constructing a suitable Lyapunov function. Finally, numerical simulations illustrate our theoretical results. It shows that the system still maintains the stability for the smaller white noises, but the stronger white noises will lead to the extinction of one or two species.