| Stochastic differential equations developed rapidly in recent years and have been widely applied in various fields.There are lots of conclusions about deterministic biological population or the infectious disease models.However,due to the influence of many uncertain factors in the environment,it is closer to the reality of the change of systems by using stochastic differential equations.This dissertation mainly investigates the dynamic analysis of two stochastic models.Firstly,we investigate a stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response.Then,we study a non-autonomous random infectious disease model with standard infection rate,the condition of extinction is obtained and the existence of positive periodic solution is proved.This paper includes four sections.In Chapter 1,we first introduce some background knowledge of the models and some basic conceptions,definitions and theorems of stochastic differential equation.In Chapter 2,we propose a stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response.Firstly,for the system with white noise perturbation,by analyzing the limit system,the existence of boundary periodic solutions and positive periodic solutions are proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived.Then,for stochastic prey-predator model under regime switching,by the generalized Ito fonnula and ergodicity of Markov,the sufficient conditions for extinction or persistence of such system are obtained.Furthermore,we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is constant.We also give some numerical simulations to illustrate our findings.In Chapter 3,a non-autonomous epidemic model with standard incidence is studied.By using the related theory of stochastic differential equation and some important inequalities,we analyze the sufficient conditions for extinction of the disease,and then the existence of the positive periodic solution of the stochastic system is proved.Finally,we use the numerical simulation to verify the theoretical results.In Chapter 4,we make a summary of the full paper,and propose the next step of the work. |
