Qualitative Analysis Of Three Kinds Of Predator - Prey Model

This thesis involves three types of bio-dynamic models, a class of predator-prey model with harvesting term and Holling-II functional response, a class of Gause-type predator-prey model with cross-diffusion and a class of predator-prey model with Holling-IV functional response. Using the knowledge of the nonlinear analysis and nonlinear partial differential equations, we discuss the boundedness, nonexistence and coexistence of solutions of the three models.The main contents are as follows:In chapter1, we introduce the background of predator-prey models. Some research works and results in the related field are also given there.Chapter2is concerned with a predator-prey system with harvesting term and Holling-II type functional response under Dirichlet boundary condition. Firstly, a prior estimate is obtained by using the Maximal Principle and the upper and lower solution method. Secondly, the existence of local bifurcation of semi-trivial solution is proved by applying the spectrum analysis method and bifurcation theory. Finally, we show that the local bifurcation can be extended to global bifurcation, and the continuum goes to infinity.In chapter3, we concern with the generalized Gause-type predator-prey sys-tem with cross-diffusion and homogeneous Neumann boundary condition, where the cross-diffusion is included in such a way that the prey runs away from the predator. We first give a priori estimate for the positive steady states by using the Maximum Principle and the Harnack Inequality. Then the non-existence of the non-constant positive steady states is given by employing the energy integral method. Finally, we investigate the existence of the non-constant positive steady states by using the Leray-Schauder degree theory.In chapter4, the existence of steady-state solutions of a predator-prey model with Holling-IV functional response is studied under homogeneous Neumann bound-ary condition. Firstly, by the spectral analysis method, the stability of the solution is obtained. Secondly, by means of local bifurcation theory. it is proved that the model bifurcations at the trivial solution in the one dimensional case. Thirdly, mak-ing use of global bifurcation theory, it is showed that the local bifurcation can be extend to global bifurcation, and the continuum joins up with infinity.