Dynamic Behavior Of Stochastic Population Models And Epidemic Model

Population ecology is a science to study the development law of species,which usually pays attention on the quantity and quality of the species.Mathematical models can help us to understand,explain and predict the change of each species so as to govern the species more well.Epidemiology is to predict the development trend of the disease by studying internal rules of the disease,and then give the optimal strategy to control it.However,the development of population/disease is inevitably affected by environmental fluctuations which is an important component in an ecosystem.Therefore,it is meaningful to analyze the effect mechanism of the environmental fluctuations on the survival of population/disease,and propose the effective approaches and strategies for controlling the population/disease.In this paper,we establish some reasonable nonlinear stochastic dynamic models—which ex-plore the effects of environmental fluctuations on the survival of population/disease by use of stochastic analysis—by considering several kinds of deterministic population/epidemic mod-els and choosing suitable stochastic processes?e.g.,Brownian motion,Ornstein-Uhlenbeck process?which is to describe these environmental fluctuations.The content of this paper is as follows:1.We develop and study a stochastic predator-prey system with Holling-type III func-tional response.We firstly obtain that the system admits unique positive global solution starting from the positive initial value.Then,by comparison theorem for stochastic differ-ential equation,sufficient conditions for extinction and persistence in mean are obtained.At last,by constructing some suitable Lyapunov function,we prove that there are unique stationary distribution and they are ergodic.2.A stochastic SIQR epidemic model with saturated incidence and degenerate diffusion is presented.First,we show there exists a unique positive solution of the stochastic model by Lyapunov analysis method.Then,the threshold R₀^sbetween persistence and extinction is established for the disease by the Markov semigroup theory,that is,the disease can be eradicated almost surely if R₀^s<1 and under mild extra conditions;whereas if R₀^s>1,the densities of the distributions of the solution can converge in L¹to an invariant density.3.We develop and analyze the single-species population models with impulsive toxicant input in polluted environments.For the deterministic single-species population model,we carry out the survival analysis,the threshold between persistence and extinction is estab-lished for single-species population.For the stochastic single-species population model,there is a unique positive solution.In addition,the threshold between persistent in the mean and extinct exponentially is established for the single-species.Finally,numerical simulations are carried out to support our theoretical results.