Extinction and stationary distribution of a competition system with distributed delays and higher order coupled noises

A stochastic two-species competition system with saturation effect and distributed delays is formulated, in which two coupling noise sources are incorporated and every noise source has effect on two species' intrinsic growth rates in nonlinear form. By transforming the two-dimensional system with weak kernel into an equivalent four-dimensional system, sufficient conditions for extinction of two species and the existence of a stationary distribution of the positive solutions to the system are obtained. Our main results show that the two coupling noises play a significant role on the long time behavior of system.