Dynamic Analysis Of Stochastic Competitive Cooperative Systems

A cooperation-competition system describes that two different populations ex-ist in a relationship in which each benefits from the activity of the other and are both in competition with a third population.There exist some excellent results to explore the qualitative and stability properties of the cooperation-competition system.On the other hand,it is recognition that there are a number of random factors in the natural environment.It has certain practical significance to consider the cooperation-competition system under random perturbations.To the best of our knowledge,there are few studies to analyze dynamics of a stochastic cooperation-competition system.In this paper,we study the dynamics of the stochastic cooperation-competition system.We first establish stochastic persistence and stochastic extinction of species,and analyze the important effect of the noise for species.Secondly,we give a suf-ficient condition for global asymptotic stability of the positive solutions of this s-tochastic system,and prove the existence of a stationary distribution of this system and it has the ergodic property.Finally,we do some numerical simulations to illus-trate or complement our mathematical findings.The final section summarises our findings.