| Environmental noise,which can affect biological populations in nature,is an important part of the ecosystem.Therefore,it is necessary to take the stochastic perturbations of environmental noise into account in deterministic population models and use stochastic differential equation as a tool to study the important subject in biomathematics.For stochastic population models,persistence,extinction and global asymptotic stability have been studied extensively by domestic and international scholars.In this paper,we mainly study some stochastic population models and their dynamical properties of the solutions.The research content of this paper consists of two parts.The first part is to study the single-species models.The existence and uniqueness of the solution,global asymptotic stability,existence of stationary distribution,stochastic persistence,strong stochastic persistence,uniform continuity and extinction are obtained for the stochastic single-species models.Firstly,by selecting the appropriate Lyapunov function and applying the It(?) formula,we prove the existence and uniqueness of solutions,the global asymptotic stability of the positive sulutions and the existence of stationary distribution.Then we take some inequality techniques and use the knowledge of stochastic differential equations such as the strong law of large numbers to study other properties.Finally,numerical simulations are used to check whether the conclusions are correct.In the second part,we propose and consider two kinds of stochastic predator-prey models.We prove the existence and uniqueness of the solutions of the stochastic predator-prey models.By constructing an appropriate Lyapunov function,we obtain a sufficient criterion to guarantee the global asymptotic stability of the stochastic model.Finally,Milstein’s method is applied to numerical simulations for the discrete models.It is obtained that the positive equilibrium solution of the stochastic model is asymptotically stable if the positive equilibrium solution of the deterministic model is asymptotically stable,when the disturbance intensity of the environmental noise is not high. |
