Stochastic Reciprocity Model With Switching Mechanism Driven By Lévy Noise

In this article,we are concerned with the following stochastic mutualism model under regime switching with Lévy jumps:with x(0)=x0>0,y(0)= y0>0 and ?(0)?0?S.To begin with,the existence and uniqueness of the global positive solution is proved with any given positive initial value.Secondly,stochastic ultimate bound-edness of the solutions is discussed,and then sufficient conditions for stochastic permanence are established.Then,the critical value between extinction and persis-tence in mean is obtained.In addition,we show that there is a unique stationary distribution for the system without Lévy jumps.The results show that regime switching can contribute to the permanence,while jump noise may suppress the permanence.Finally,two concrete examples are presented to illustrate our results.