A Study Of Predator-prey System With Cross Diffusion

In recent years,the study of predator-prey system under the influence of population diffusion has become a hotspot in the field of biological mathematics.There are many practical results,especially for the predator-prey system with cross-diffusion.In this paper,we study the properties of solutions of three predator-prey models with cross diffusion in homogeneous Dirichlet boundary condition.The first type is B-D functionnal response function,the second type is Holling-IV functional response function,and the third type is Monod-Haldane functional response function.The theory and method of reaction diffusion equation,Crandall-Rabinowitz bifurcation theory and degree theory are mainly used in this paper.The research contents are as follows:In the first chapter,the background and significance of the predator-prey model with cross-diffusion are expounded,the research results and progress are summarized,and some related preliminary knowledge are briefly introduced.In the second chapter,bifurcation of B-D predator-prey model with cross-diffusion and self-diffusion is studied.The existence of positive solutions under homogeneous Dirichlet boundary condition is obtained by analyzing the related eigenvalue problem,and the existence of positive bifurcation solutions is obtained by Crandall-Rabinowitz bifurcation theory,And local positive solutions are extended to obtain global bifurcation solutions.In the third chapter,the bifurcation of Holling-IV predator-prey model with crossdiffusion and self-diffusion is studied.The sufficient conditions for the existence of local bifurcation positive solutions are obtained by means of bifurcation theory,and the local bifurcation positive solutions are extended to obtain the global bifurcation solutions.In the fourth chapter,we discuss the bifurcation of Monod-Haldane predator-prey model with cross diffusion.For the existence of solutions of homogeneous Dirichlet boundary conditions,a priori estimates and the existence and continuation of bifurcate positive solutions are given.It is shown that the predator and the prey can coexist for a long time under certain conditions.Chapter 5 summarizes the research results and problems to be discussed and solved further more are presented.