Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

Abstract

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at time $t$ are estimated by $r_i\left(t\right)+{\sigma }_i\left(t\right){\dot{B}}_i(t),\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2,}$ respectively. Viewing the different intensities of the noises ${\sigma }_i(t)$, $i=1,2$ as two parameters at time $t$, we conclude that there exists a global positive solution and the $p$th moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system.