Asymptotic Behavior Of Stochastic Mutualism Systems

Mutualism is an important biological interaction in nature.In recent years,mutualism population models have received extensive attention by academia.There are lots of conclusions about deterministic mutualism population models.However,in the real world,population systems are inevitably subjected to environmental noises.Therefore,a population model with stochastic factors is more reasonable.This paper is devoted to the asymptotic behavior stochastic mutualism systems.The content is as follows:1.Non-autonomous stochastic Gravesa mutualism systems.Firstly,the existence and uniqueness of global positive solution to the system is established for any positive initial value.Secondly,by using the theory of stochastic differential equation and some important inequalities,the ultimate boundedness and uniformly continuity of solution are proved,and then the global attractivity is derived.Then by using the comparison theorem for stochastic differential equations and Lyapunov functions,the sufficient conditions for the permanence and extinction to the system are derived respectively.When coefficients are periodic functions,by using Khasminskii theory of periodic solutions,the existence of the periodic solution is obtained.Finally,some numerical simulations are given to illustrate our theoretical results.2.Stochastic Gopalsamy mutualism systems with Markov switching.Firstly,the existence and uniqueness of global positive solution to the system is established for any positive initial value.Secondly,by using the theory of stochastic differential equations,the stochastically ultimately boundedness and uniform continuity of solution are proved.And sufficient conditions for the global attractivity are established.Then by using Lyapunov functions,the sufficient conditions for the permanence,extinction and existence of invariant distribution to the system are derived respectively.Finally,some numerical simulations are given to illustrate our theoretical results.The results show that the asymptotic behavior of the systems doesn't change when the noise intensity is small;if the noise intensity is large,the survival of the population will be affected.