A Study Of Predator-prey Model With Cross Diffusion

By establishing a mathematical model to describe the characteristics of biological systems is an important part in the field of mathematical application. Predator-prey model is an organic composition of mathematical model, attracting many scholars’ attention, and many results are obtained. In reality, the model with cross diffusion can reflect the predator-prey relationships more accurately. At the same time, it also has value for the study of ecology.We mainly study qualitative properties of solutions for two types of predator-prey model with cross-diffusion under homogeneous Dirichlet boundary conditions. One is a predator-prey system with B-D functional response, the other is a predator-prey model with HollingⅣ functional response.By using the theories and methods of the partial differential equations and the reaction diffusion equations, we shall discuss the priori estimates for positive solutions, existence of positive solution, bifurcation and so on.The main contents in this paper are as follows:The first chapter outlines the background, research achievements and progress of ecological mathematical model. Then, for proving the existence of positive solutions, it introduces the preliminary knowledge.The second chapter studies the existence of positive solutions for a predator-prey model with B-D functional response under homogeneous Dirichlet boundary conditions. Firstly, by using the maximum principle, some priori estimates of positive solution are obtained. Secondly, by considering the related eigenvalue problems, two unbounded neutral curves are given. Thirdly, using Crandall-Rabinowitz bifurcation theory, with the growth rate of prey as a bifurcation parameter, the positive solutions emanating from the semi-trivial solutions are derived. Finally, we develop the local bifurcation solution to the global one, thus obtained sufficient conditions of positive solutions. It has shown that the predator and the prey can coexist under certain conditions.The existence of positive solutions for a predator-prey model with cross-diffusion and HollingⅣ functional response under homogeneous Dirichlet boundary conditions is studied in the third chapter. By using Crandall-Rabinowitz bifurcation theory, positive solutions emanating from the semi-trivial solutions are derived. Then, the local bifurcation is extended to the global one, thus obtained sufficient conditions of positive solutions. It has shown that the predator and the prey can coexist under certain conditions. There are 52 references in the paper.