Permanence, Extinction and Periodicity to a Stochastic Competitive Model with Infinite Distributed Delays

In this paper, we study a stochastic Lotka-Volterra competitive model with infinite distributed delays. We obtain sufficient but almost necessary conditions for permanence and extinction of species. Competitive exclusion also takes place in some cases. Besides, criteria are established for the existence and uniqueness of a stationary Markov process. Our results not only cover the traditional Lotka-Volterra competition model without environmental noise, but also reflect the influence of noise to model's dynamics. Moreover there are some challenges to give the existence of periodic solutions for system with periodic coefficients. Via a stochastic comparison approach, we obtain the existence and uniqueness of periodic solutions in distribution.