Dynamics Research Of Three Stochastic Biological Mathematical Models

Stochastic mathematical model in biology is a way to deepen and generalize the deterministic biological mathematical model.Considering the influences of environmental perturbations for ecosystem,we can establish the mathematical model based on the stochastic differential equations.It can be better adapted to different practical needs and more accurately depict the evolution of the ecosystem,thus it has become a focus of biological mathematics.This dissertation mainly investigates dynamics analysis of three stochastic biological mathematical models.Firstly,we investigate a stochastic non-autonomous Lotka-Volterra predator-prey model with impulsive effects;Secondly,we study a stochastic autonomous SI epidemic model with saturated incidence rate and feedback controls;Finally,we investigate a new nonlinear stochastic SIRS epidemic model with standard incidence rate and saturated treatment function.This paper consists of the following five chapters:In Chapter 1,we introduce the development of stochastic biological mathematics,and introduce some definitions and theorems related to stochastic process,stochastic differential equations and the qualitative and stability theory of ordinary differential equations.In Chapter 2,we propose a stochastic non-autonomous Lotka-Volten-a predator-prey model with impulsive effects and investigate its stochastic dynamics.We first prove that the subsystem of the system has a unique periodic solution which is globally attractive.Furthermore,we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey-predator system.Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species.Finally,we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.In Chapter 3,we consider a stochastic autonomous SI epidemic model with saturated incidence rate and feedback controls and investigate its stochastic dynamics analysis.Firstly,we prove that there is a unique global positive solution of system.Secondly,by using the method of stochastic Lyapunov functions and Ito formula,we investigate the asymptotic behaviors around the disease-free equilibrium and the endemic equilibrium of its deterministic system respectively.The solution of the stochastic system exists a unique stationary distribution,and it also has the ergodic property.Thirdly,conditions for the persistence in the mean and extinction of the system are established.Finally,we use a series of numerical simulations with respect to different stochastic parameters to illustrate the theoretical results.The obtained results show that the stochastic perturbations and feedback controls have crucial effects on the persistence and extinction of population.In Chapter 4,we first propose a new nonlinear stochastic SIRS epidemic model with standard incidence rate and saturated treatment function.The paper mainly investigates the threshold dynamics of the nonlinear stochastic SIRS epidemic model by making use of stochastic inequality techniques.By using Lyapunov methods and Ito formula,we prove the existence and uniqueness of a global positive solution for the corresponding limiting system.Furthermore,we obtain sufficient conditions for the extinction and persistence in mean of the nonlinear stochastic SIRS epidemic model by using the techniques of a series of stochastic inequalities.Finally,we provide some numerical simulations to illustrate the performance of our theoretical findings.In Chapter 5,we summarize the main work of this paper and propose some future research directions.