Long-time behaviors of a stochastic cooperative Lotka-Volterra system with distributed delay

This article focuses on a stochastic cooperative Lotka-Volterra system with distributed delay. We first transfer the stochastic system with weak kernel into a degenerate stochastic system made up of four equations. For the deterministic system, global stability of the positive equilibrium is investigated. For the stochastic system with distributed delay, sharp sufficient conditions for the persistence of two species are established. What is more, we obtain the existence and uniqueness of the stationary distribution by constructing suitable Lyapunov function and proving the global attraction of the positive solution. The results show that, the weaker white noises can ensure the existence of a unique stationary distribution and the stronger white noises can result in the extinction of one or two species, though the positive equilibrium is globally stable without white noises.