Analysis Of Solutions For Two Types Of Predator-prey Models

Among the studies of population ecology, applying reaction and diffusion mod-els to research the transformations rules of the populations, the mutual functional dynamics between different populations, the interaction relationship between envi-ronment and populations give a certain extent of theoretical meanings in the control of the growth of the population reasonably. In this paper, we analysis the coexistent solutions of two kinds of predator-prey models by means of the bifurcation theory, the degree theory, the nonlinear analysis theory and the theory and methods in reaction and diffusion equation.Chapter 1 is the preface of this paper, here we introduce the related backgrounds and the current developments and research results of predator-prey models and then give a brief introduction of this paper.In Chapter 2, using the bifurcation theory, the perturbation theory of eigenval-ue and the calculation of the fixed index, the following predator-prey model with Holling-III type functional response under Robin boundary condition is researched Firstly, by the Maximal Principle, we get the boundness of the solutions. Then, we prove the existence of the local bifurcation solution which bifurcating from the semi-trivial solution (?,0) by taking the death rate of predator as the bifurcation parameter. Secondly, we show that the local bifurcation solution can be extended to global bifurcation solution and the solution continuum goes to infinity confined in the positive cone using the fixed point index theory. Finally, by the means of the perturbation theory of eigenvalue, the stability of the local bifurcation solution is studied.In Chapter 3, we study the following predator-prey model with Beddington-DeAngelis type and modified Leslie-Gower type functional responses under Dirichlet boundary condition by the use of fixed point index theory, spectrum theory and stability theoryFirstly, by the use of fixed point index theory, the existence conditions and nonexis-tence conditions of positive solution to the model is analysed. Secondly, we research the stability of the positive solutions by the spectrum theory. Finally, by means of Leary-Schauder degree theory, the multiplicity of the positive solution is discussed.