Stability Analysis For A Predator-Prey Model With Cross-Diffusion

In this thesis, using the theories of nonlinear analysis and nonlinear partial differential equations, stability problems for the following predator-prey model with cross-diffusion from biology are investigated, such as the linear instability induced by cross-diffusion, the exis-tence of bifurcation solutions, the type of the bifurcation diagram, and the nonlinear instability analysis for this model. The organization of this paper is follows:Firstly, we discuss the linear stability of the non-negative equilibria for the model （1） with general reaction terms f（u,v）, g（u,v）. Especially, we focus on the effects of cross-diffusion coefficients ρ₁₂ and p₂₁ on the stability properties.Furthermore, using the Crandall-Rabinowitz branch theory, treating the cross-diffusion coefficient ρ21 as branch parameter, we study the existence of positive bifurcating solutions and the directions of the branches near the bifurcation points.Finally, using the embedding theorem, the energy estimates and the bootstrap technique proposed by Y. Guo and W. Strauss, the nonlinear instability is studied in a N-dimensional box in RN, where N≤3. Our results can be interpreted as a rigorous mathematical characterization for the nonlinear evolution of the spatial and temporal patterns.