Dynamic Studies Of Several Stochastic Biological Models

In recent years,the research of applying the stochastic analysis theory to ecosystem has become quite popular.Compared with deterministic dynamical systems,random noise in the random stochastic system can better depict the uncertainties exited in practical issues.These random noise often influence the dynamic behavior of system,sometimes even play a decisive role.Thus the behavior can change fundamentally.All of these means that the research of stochastic ecogical models is a very important and meaningful work.Therefore,we need to discuss further about the influence of random noise to dynamic behavior of the ecological model in order to apply the theory to practice.This essay mainly investigate dynamical properties of the epidemic models with Brown white noise and stochastic delay differential equations.Also,we discuss the influence of the dynamic behaviors of stochastic population models with jumps.This essay can be divided into six chapters:The first chapter is introduction,mainly introduces the background,research and status of this study as well as some basic concepts and relative results.Chapters two to five consists of the main research contents of this eassy.In the second chapter,white noise is added to this SIRS models with general nonlinear incidence rate to obtain the stochastic epidemic models.For the deterministic epidemic model:by constructing Lyapunov function,we obtain the basic reproduction number R₀ is a threshold value,which determine the spread of disease or extinction of disease;when R₀?1,the disease will die out;when R₀>1,the disease will be prevalent and form "endemic disease".The main conclusions of stochastic epidemic model are as follows:1?The global existence and positivity of the solution is studied by using Lyapunov functionals;2?When R₀?1,we obtain the pth-moment exponential stability and almost surely asymptopic stability with the multi-dimensional Ito formula and Young inequality.3?When R₀>1,we conclude the solution of stochastic model is stochastically asymptotically stable and the sufficient conditions for the persistent in mean;we show that the large white noise may lead the disease to extinction by local martingale and the strong law of large numbers.In the third chapter,the dynamics of a stochastic SIS epidemic model with multiple parameters under random interference is investigated.First,we show that the model admits a unique positive global solution starting from the positive initial value.Then,the long-time asymptotic behavior of the model is studied:when R₀?1,we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions;when R₀>1,we show that the stochastic model has a stationary distribution under certain parametric restrictions by Diffusion thoery and Hasmiskii's theorem.In particular,we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence.In the fourth chapter,we first discuss the deterministic SIS epidemic model with vaccination and nonlinear incidence rate.For the model,we show the free equilibrium and endemic equilibrium are globally asymptotically stable.White noise is added to the model to obtain a class of stochastic delay differential equations.For the stochastic SIS epidemic model with the delay:first,the global existence and positivity of the solution are obtained by means of Lyapunov function and Ito formula;then the exponentially mean square stable of the linearized system without delay and with delay are studied.Also,we obtain the stability in probability of the linearized system with delay.The above result shows that:the white noise has a certain interference on the linearized system with delay;when the white noise is small,the stochastic SIS epidemic model with delay has still its exponentially mean square stable and stability in probability.That is to say,noise with small intensity fails to change the long-time dynamic behavior of the epidemic model.In the fifth chapter,we mainly discuss the dynamics of stochastic Holling-Tanner predator-prey model with jumps.The Levy jumps can be considered as flood,tsunami,earthquake,drought,large-scale infectious disease,volcanic eruptions and climate warming etc,and they can produce sudden,discrete,large stochastic perturbations.First,we show the stochastic Holling-Tanner predator-prey model with jumps admits a unique positive global solution starting from the positive initial value,in other words,the jump process can prevent the solution of explosion in a limited time.Then,sufficient conditions of persistence in time average and extinction of the model are obtained by making of the stochastic comparison theorem and martingale theory etc.The above conclusions show that Levy jumps have significant effects on the persistence and extinction of the population,which can even make the long-tim dynamic behavior of the population model change fundamentally.Finally,numerical simulations are carried out to illustrate the influence of the Levy jumps on the population.The last chapter is the conclusion work of this essay and the outlook of future work.