Qualitative Analysis Of Two Stochastic Predator-prey Models

In the process of exploring practical problems,in order to describe objective problems more accurately,the influence of stochastic factors must be taken into account,which is the reason why stochastic differential equations come into being.Stochastic differential equations are applied in many practical fields,such as biology,oceanography,physics and finance,so the study of stochastic differential equations has important theoretical and practical significance.In recent years,the relationship between predator and prey has provided an important subject for the study of population ecology.At the same time,the disturbance of environmental random factors has an important impact for the population system.Based on this,this paper mainly studies the influence of environmental random factors on the predator-prey model in biology,and discusses the existence and uniqueness of global positive solutions,extinction and non-persistence of the model.Qualitative analysis of the given models is made by applying the basic theory of stochastic differential equation.This paper is divided into four chapters.The specific work is as follows.The first chapter describes the research status of stochastic predator-prey model,and gives the main research contents and methods of this paper.In Chapter 2,for a stochastic predator-prey model with Allee effect,the existence and uniqueness of global positive solutions of the model are proved by constructing appropriate Lyapunov function and applying stochastic analysis theory.Secondly,sufficient conditions for the extinction given by exponential martingale inequality,Borel-Cantelli lemma and strong law of large numbers.Finally,the rationality of the theoretical results is verified by numerical simulations.In Chapter 3,we study a stochastic time-delay predator-prey model in polluted environment.Firstly,we prove the existence and uniqueness of global positive solutions of the model by constructing appropriate Lyapunov function.Secondly,we give sufficient conditions for extinction and non-persistence in the mean by It^o formula and strong law of large numbers.It shows that the numerical simulation results are consistent with the theoretical results.Chapter 4 summarizes the results of the previous two chapters and explains the followup work.