Dynamic Behavior Of Stochastic Predator-Prey Models And Epidemic Model With Functional Responses

Population ecology and epidemic dynamics are important fields of Biomathematics and have obtained abundant research results.The dynamical properties of population model and infectious disease model are usually studied by using ordinary differential equation theory,impulse differential theory and functional analysis theory.However,environmental noise will inevitably have an impact on population and infectious disease,so it is more practical to use stochastic differential equation to describe the dynamics of population and infectious disease with environmental noise.The environmental pollution is an important social and ecological problems.It is effect on the spread of respiratory disease cannot be ignored,so the introduction of air quality index function is very useful to study the mechanism of the spread of infectious diseases.In this paper,stochastic differential equations and stochastic analysis methods are used to study the dynamic behaviors of several nonlinear populations and epidemics.The specific research contents are as follows:1.We study a stochastic predator-prey model with disease in the prey and BeddingtonDe Angelis functional responses in a polluted environment.We mainly discuss two systems:autonomous and non-autonomous system.In the autonomous system,we firstly obtain that the system has a unique positive global solution by applying the Lyapunov analysis method and the solution is randomly bounded.Then sufficient conditions for the stochastic extinction and persistence of the system are obtained by the comparison theorem of stochastic differential equations.In addition,the sufficient condition is obtained for the existence of a nontrivial positive periodic solution for the nonautonomous periodic system.2.A stochastic predator-prey model with Holling ? increasing function in the predator is presented.This has two models: autonomous and non-autonomous model.In the autonomy model,we demonstrate the existence and uniqueness of the global positive solution.Then we show there are unique stationary distribution which is ergodic by the lyapunov functional method.At last,we obtain respectively sufficient conditions for weakly persistent in the mean and extinction of the prey and extinction of the predator by comparison theorem for stochastic differential equation.In the non-autonomy model,we first prove that the system has a unique global positive solution for any given positive initial value.Then we give some sufficient conditions for the existence of nontrivial positive periodic solutions.Finally,we prove the persistence and extinction of the population.3.We develop and analyze the dynamics of a stochastic SIS model with nonlinear infection rate and random diffusion of air pollutants.The disease transmission coefficient is assumed to be a function of the air quality index in this model.We derive a one-dimensional stochastic differential equation(SDE)model about infected individuals ()according to the statistical properties of stochastic process.Firstly,the existence and uniqueness of positive solutions of SDE model is proved.Then the sufficient conditions for the extinction of the disease and some properties are obtained.Finally,our conclusions are verified by numerical simulation.