Stability And Bifurcation Analysis Of A Predator-prey Model With Cross-diffusion

In this paper,we mainly study the following reaction-diffusion predator-prey model with cross-diffusion and homogeneous Neumann boundary conditions:(?)By selecting appropriate parameters and analyzing the corresponding eigenvalue problems in detail,the stability,Turing instability,and Turing bifurcation of the system at the unique constant equilibrium solution are studied.Using time multi-scale analysis,the amplitude equation of the cross-diffusion system near the constant steady-state solution is obtained,and the obtained amplitude equation is used to study different types of Turing patterns that may occur in the original diffusion system.The specific research contents are as follows:The first chapter briefly describes the research background of the system and the main research contents of this paper.In the second chapter,with the help of linearization method and central manifold theorem,through the analysis of the distribution of the characteristic roots of the characteristic equation of the cross-diffusion system,the stability and Turing instability of the cross-diffusion system at the unique constant coexistence equilibrium solution are discussed in detail,and the numerical verification of the obtained results is carried out using MATLAB.In the third chapter,through the analysis of the spatial homogeneous Hopf bifurcation curve and Turing bifurcation curve of the system,the parameter range of Turing mode generated by the system near the constant coexistence equilibrium solution is determined,and the amplitude equation near the Turing critical point is derived by the linearization method and multi-scale analysis method,and the stability and existence of the steady-state solution of the amplitude equation are analyzed.The classification and stability analysis of Turing pattern are given.Finally,the obtained theory is verified by numerical simulation.