Abstract
•A stochastic mutualism model with Lévy jumps is proposed and studied.•We show that the positive solution of the system is stochastically ultimate bounded.•We establish sufficient and necessary conditions for the stochastic permanence and extinction of the system.•The results show an important property of the Lévy jumps: they are unfavorable for the permanence of the species.
In this paper, we consider a stochastic mutualism model with Lévy jumps. First of all, we show that the positive solution of the system is stochastically ultimate bounded. Then under a simple assumption, we establish sufficient and necessary conditions for the stochastic permanence and extinction of the system. The results show an important property of the Lévy jumps: they are unfavorable for the permanence of the species. Moreover, when there are no Lévy jumps, we show that there is a unique ergodic stationary distribution of the corresponding system under certain conditions. Some numerical simulations are introduced to validate the theoretical results.