Study On Dynamical Behaviors Of Two Stochastic Differential Equation Models

In this paper,we mainly study the dynamical behaviors of a stochastic.two-species competition chemostat.model and a stochastic SIQR epidemic model.The article includes three chapters.The preface is in chapter 1,we introduce the research background of this article.the main task and some importanrt preliminaries.In Chapter 2,We consider the dynamical behavior of a chemostat model in which two microorganisms compete one nutrient.In the model,we assume that stochastic perturbations are of environment noise which are directly proportional to the concentration of the nutrient and the.microorganisms.Firstly,the global existence and uniqueness of the positive solution of the model are proved by using stochastic Lyapunov function and Ito's formula.Then com-pared with the deterministic chemostat model,there is no equilibrium points in the stochastic chemostat model,so asymptotic behavior and the steady state distributions are established by reconstructing stochastic Lyapunov function and using Ito's formula,Gronwall inequality and stochastic stability theorem of stochastic differential equation.Finally,numerical simulations are given to illustrate the analytical results.In Chapter 3,We discuss a stochastic SIQR epidemic model with bilinear incidences.In the model,we assume that the natural death rate and the effective contact rate are influenced by environment noises.Firstly,the global existence and uniqueness of the positive solution of the model are proved by using stochastic Lyapunov function and Ito's formula.Then the conclusion is obtained that when the noise is small.the stochastic reproduction number R0 determines the persistence and extinction of the disease;on the other hand;the large noise will suppress the disease to prevail by reconstructing stochastic Lyapunov function,using Ito's formula and stochastic stability theorem of stochastic differential equation.Furthermore,the stationary distribution of the stochastic system is studied.At last,some simulations are given to illustrate the analytical results.