Coexistence Analysis Of Two Kinds Of Biological Diffusion Model

Population dynamics relationship of predator-prey is hot issues for biology workers. Due to the different forms of biological phenomena, diverse models are used to study the interaction. Reaction-diffusion equation is an important part of it. People can explain and predict natural phenomenon by studying reaction-diffusion equation, thus promote the development of natural science. We study the property of two kinds of predator-prey modal, a Monod-Haldancee prey-predator model subject to Dirichlet boundary condition and a Sigmoidal prey-predator model subject to Dirichlet boundary conditionIn chapter1, we introduce the background of the two types of predator-prey models. Some research works and results in the related field are also given there.In chapter2, the bifurcation solution from a double eigenvalue and its stability are studied by the Lyapunov-Schmidt method.In chapter3, Firstly, the background of local bifurcation and global bifurcation are given. Secondly, the priori estimates of positive solution are given by using the way of upper and lower solutions、variation principle and Harnack inequality. Thirdly, taking r as the bifurcation parameter, the structure of local bifurcation solution is given by bifurcation theory. Next, we research the structure of global bifurcation solution by using global bifurcation theory and prove that the local branch can be extended to the global branch. Eventually, the asymptotic and the stability of solution of the predator-prey model are studied by the comparison principle and stability theory.