The Dynamic Behavior Analysis Of A Predator-prey Model

It is an important topic in the field of biological mathematics to explore the relationship among species in the ecosystem by using second order elliptic equation theory.Scholars construct mathematical equations for a certain phenomena in the ecosystem which is used to reflect the influence of time and space variables on research objects.In addition,combining the numerical simulation technique to verify and perfect it,thus a better understanding of natural phenomena in ecology,so as to maintain effectively the ecological balance.By using the theories and methods proposed by scholars,such as the degree theory of Lerary-Schauder,the method of Lyapunov-Schmidt,the bifurcation theory,the method of spectral analysis,numerical simulation technique.In this paper,under the first boundary,the following predator-prey model is discussed.The conceptual framework of this article is as follows:Chapter 1 is the introduction,which briefly describes the relationship between mathematics and ecology and the research results of the model in recent years,leads into the biological model to be studied in this paper,further explains the significance and function of research questions,summarizes the framework of this paper,moreover,collates some of the related lemmas.Chapter 2 mainly deals with a predator-prey model with toxin and makes care-ful analysis from the following four parts.In the first part,we analyze the uniqueness,non-uniqueness,existence and stability of positive solutions.In the second part,we illustrate that the systems exist bifurcation solutions and the situation of extension.In the third part,we investigate the uniqueness,stability and existence of the bifur-cation solution of double eigenvalues.In the fourth part,we use Matlab software to supplement and explain the conclusions of this chapter.