A Study On The Stationary Distribution And Extinction Of Several Stochastic Population And Epidemic Models

Population dynamics and epidemic dynamics have always been the focus of attention in the long history of biomathematics research.Environmental noise is ubiquitous in real ecosystems,any ecological system is inevitably subject to some degree of stochastic disturbance(e.g.white noise,telegraph noise,etc.).Therefore,taking into account the effects of stochastic perturbations when carrying out biomathematic modeling can provide a more objective insight into the changing patterns of biological systems.In this paper,two stochastic mussel-algae models are constructed by considering the effects of white noise,feedback control and color noise in a deterministic mussel-algae model.In another deterministic SEIVS epidemic models with saturation incidence and temporary immunity,two stochastic SEIVS epidemic models are constructed by considering the effects of higher-order white noise and color noise.By means of It? formula,the basic theory of stochastic differential equation and Lyapunov function,we analyze the dynamic behavior of the models.The details of this study are as follows:Chapter 1 introduces the research background of the stochastic mussel-algae model and the stochastic SEIVS epidemic model,their research significance,the current status of research at home and abroad and the main research content of this paper.Chapter 2 gives related definitions,lemmas and theorems required for the proof of this paper.In Chapter 3,two types of stochastic mussel-algal models are investigated.Firstly,a stochastic mussel-algae model with feedback control is established by considering the influence of white noise and feedback control.By Lyapunov function method,sufficient conditions for the extinction and the existence of stationary distribution of the model are obtained.Whereafter,a stochastic mussel-algal model with white noise and color noise is established.By means of Lyapunov function method and Has’ minskii ergodic theory,sufficient conditions are obtained for mussel extinction,nonpersistence and the existence of a unique ergodic stationary distribution.Finally,numerical examples are given to verify the feasibility of the theoretical results obtained from the two models.In Chapter 4,two types of stochastic SEIVS epidemic models with high-order perturbations are studied.Firstly,a stochastic SEIVS epidemic model with highorder white noise perturbation is established.The stochastic ultimate boundness,stochastic permanence and the existence of the existence of a unique ergodic stationary distribution of the model are obtained via using Has’ minskii ergodic theory and Lyapunov function method.Then another stochastic SEIVS epidemic model disturbed by white noise and color noise.By utilizing Has’ minskii ergodic theory and Markov switch theory,sufficient conditions for the existence of a unique ergodic stationary distribution are given.Finally,numerical examples are shown to certify the feasibility of the theoretical results achieved from the two types of models.