Asymptotic Properties Of A Stochastic Mutualism Model

In the ecosystem, the mutualism population is an important research topic. Many scholars have extensively investigated the deterministic population models and obtained a lot of results. However, over the years, the quality of our living environment has declined, with a growing number of toxicant substances and contaminants entering into the ecosystem. The toxicant in the environment threats the survival of species. The saturation term of the model is significant to the population. On the other hand, the noise is an important part of the ecological system. Species are always subject to environmental noise which has a important effect on species. Therefore, we study a stochastic mutualism model with toxicant substances and saturation term, which can reveal the impact of these factors on species and is more extensive. In this paper, dynamic properties of a stochastic population models with toxicant substances and saturation term are analyzed.Stochastic two-species mutualism model with toxicant substances and saturation term is investigated. We choose a suitable Lyapunov function. By using It? formula, strong law of large numbers of the martingale and other theory of stochastic differential equation, sufficient conditions for some population dynamical properties including the existence of global positive solutions, stochastic boundness, stochastic permanence, stochastic extinction, global attractivity and pathwise estimation are obtained. In addition, we obtain the sufficient condition which grantees the stationary distribution and ergodic property of the model. It is found that the sufficiently large environment noise is unfavorable to the persistence of the stochastic model. It is interesting and surprising that toxicant substances have no effect on the permanence of the stochastic model. Finally, simulations are also carried out to confirm our analytical results.