The Research Of Predator-Prey Model With Cross-Diffusion

In recent years, the predator-prey model is widely applied in the population ecology. The predator-prey model has been the important focus by the scholars, and the research has achieved abundant research achievements.In this paper, two predator-prey models with cross-diffusion are studied and some properties are obtained under the homogeneous Dirichlet boundary conditions. Including the priori estimates and the existence of the local bifurcation solutions. Then the local bifurcation solutions are extended to the global bifurcation, the sufficient conditions for the global bifurcation solutions are obtained. Five chapters in this paper as follows.The background and meaning of the topic are introduced in the first chapter, then main research contents are raised.The second chapter outlines some theories, these theories will be used to study the properties of the predator-prey models with cross-diffusion.In the third chapter, the existence of the local bifurcation solutions is studied for a predator-prey model with cross-diffusion under Dirichlet boundary condition. The priori estimates are obtained by using maximum principle and the existence of positive solutions are obtained by the help of the Crandall-Rabinowitz bifurcation theory. The sufficient conditions are achieved for the inexistence of positive solutions, and under certain conditions, the positive solution of the model is bounded, and the sufficient conditions for the positive solutions are obtained form the local bifurcation solutions. Then the local bifurcation solution is extended to the global bifurcation. The sufficient conditions for the positive solutions are obtained from the global bifurcation solutions.In the fourth chapter, the existence of the global bifurcation solutions are studied for a predator-prey model with cross-diffusion under Dirichlet boundary condition. The existence of local bifurcation solutions are obtained by the help of Crandall-Rabinowitz bifurcation theory, then the local bifurcation solution is extended to the global bifurcation.The sufficient conditions for the positive solutions are obtained from the global bifurcation solutions.Some results of this topic are introduced in the chapter five. Finally, some problems that need been researched are presented after summarizing the total paper.