Coexistence And Qualitative Analysis Of Two Kinds Of Biological Models

This thesis involves two types of bio-dynamic models:A class of predator-prey model with Holling Ⅲ item and a class of predator-prey model with cross-diffusion. Using the knowledge of the nonlinear analysis and nonlinear partial differential equa-tions, especially, the theories and method of the nonlinear elliptic equations, we have further studied the coexistence, multiplicity and stability of solutions of the models.By super-sub solutions method, fixed-point theory and local bifurcation theo-rem, we study a predator-prey model with homogeneous Dirichlet boundary condi-tionsBy local bifurcation theorem and implicit function theorem we discussed the cross-diffusion model with homogeneous Dirichlet boundary conditionsThe main contents in this thesis are as follows:In chapter1, we introduce the background and development situation of predator-prey models and predator-prey models with cross-diffusion. Some research works and results in the related are also given there.In chapter2, a class of ratio-dependent Holling Ⅲ type predator-prey model, is considered. Firstly, according a priori estimate, using Leray-Schauder fixed-point theory, we discussed the existence of positive solutions of the corresponding elliptic equation; secondly, we proved the positive solutions is linearly stable when m is sufficiently large; finally, by local bifurcation theorem we get the bifurcation solution and using fixed-point theory we have a sufficient condition of coexistence of multiple solutions when b is small enough. In chapter3, a class of predator-prey model with cross-diffusion is considered. We give a priori estimate of positive solution, and have a nonexistence of positive solution condition using Poincare inequality; we study the bifurcation phenomena at the semi-trivial solution by means of the bifurcation theory; we give related to the dominant function parameters of solution curve by the implicit function theorem and some function property.