Permanence, stationary distribution and extinction for one-dimensional stochastic Gilpin-Ayala systems

In this paper, we consider the permanence, extinction and existence of stationary distribution for one-dimensional stochastic Gilpin-Ayala system. The threshold value of the system is obtained under some assumptions. When the intensity of the white noise is less than the threshold value, the system remains and has a unique stationary distribution whilst it becomes extinct when the perturbation is large. In the case of stochastic extinction, the exponential extinction rate is estimated precisely by two parameters of the systems. Nontrivial example is provided to illustrate our results.