Two Types Of Predator - Prey Coexist State And Qualitative Analysis

The research involves two types of biological dynamics of predator-prey model: A class of Modified Holling-Ⅱ type reaction function Predator-Prey Model and a class of square root of the reaction term Predator-prey model. Primarily using the knowledge of the nonlinear analysis and nonlinear partial differential equations, particularly, the theories and methods of the parabolic equation and the correspond-ing elliptic equation, we have discussed the coexistence, positivity, boundedness and stability of solutions of the two models.By the bifurcation theory, the upper and lower solution method and the energy integral method, we study a predator-prey model with the first modification of the boundary conditions of Modified Holling-Ⅱ type reaction function Predator-Prey ModelWe discuss a predator-prey model with homogeneous Neumann boundary conditions with the square root of the reaction term by using the eigenvalue pertur-bation theory, comparison principle and bifurcation theoryThe main contents in this thesis are as follows:In chapter1, we introduce the biological background and development of the Modified Holling-II type reaction function Predator-Prey Model and the square root of the reaction term predator-prey model, Some research works and results in the related field is also given In chapter2, we introduce a class based on modified Lotka-Volterra predator-prey reaction diffusion model. Firstly, given a priori estimate of the solution of the system and sufficient condition for the existence of positive solutions; Secondly, let d be as the bifurcation parameter,by using the bifurcation theory and Leray-Schauder degree theory and other knowledge, we discuss the bifurcation at positive constant equilibrium by means of the bifurcation theory and the Leray-Schauder degree theory and given the structure of the solution near local points of disagreement; Then the local bifurcation which can be extended to the global bifurcation is proved and the fact that the global bifurcation joins up with infinity in the case of one-dimension is obtained; Lastly, by stability theory, the stability of the equilibrium constant is given.In chapter3, a class of predator-prey model with the square root of the re-action term is considered. Firstly, given the existence of solutions of the model and the non-negative solution bounded by using the maximum principle; Secondly, by use of the stability theory, we discuss the parabolic model equilibrium and the corresponding elliptic equation and equilibrium constant local asymptotic stability, uniform asymptotic stability; Finally, on the bifurcation theory and Leray-Schauder degree theory and other knowledge,we discuss the corresponding equilibrium equa-tion solution in constant equilibrium solution near bifurcation solutions; And local bifurcation can extension into the overall bifurcation and discuss the direction of bifurcation solutions.