Asymptotic Properties Of Some Stochastic Single Population Models

In recent decades, many scholars have deeply investigated the deterministic population models and obtained a lot of results, which are applied more and more extensively. In fact, parameters of biological population models are inevitably affected by environmental noises, and the noises are important parts of the ecological models. Therefore, the study of the asymptotic behaviors of population models with stochastic perturbation is very important for revealing the influence of environmental noises on the biological models. In this paper, the dynamic properties of two stochastic single population models under white noise are analyzed.Firstly, the stability of a stochastic Richards growth model is studied. For the non-autonomous stochastic Richards model in the It? sense, we choose a suitable Lyapunov function, use the It? formula, the strong law of large numbers of the martingale and other theory of stochastic differential equation to investigate the existence, uniqueness, stability of the solution of the stochastic model. In the Stratonovic sense, under some conditions, the stability of the zero solution and the positive solution are investigated. And we obtain that the properties of the stochastic model are same as the deterministic case.Secondly, a stochastic logistic model modified to incorporate mating limit ation is studied. The existence, uniqueness of the solution, stochastic boundedness, stochastic permanence, persistence in the mean, global attractivity and extinction are investigated. This paper shows that the stochastic single population model can keep the asymptotic properties of the deterministic population model under the relatively small environmental disturbance, however, the sufficiently large environment noise is unfavorable to the persistence of the stochastic model.