Two Types Of Predator - Prey Model The Nature Of The Solution

In this paper, we mainly consider the properties of two types of predator-prey model solutions subject to homogeneous Neumann boundary condition. So far. the population ecology has developed into one of the most widely and deeply applied branches, which is also the fastest and mature developed. Population dynamics model is a mathematical model to describe the competition and dynamic relation-ship between one population and another, as well as between population and envi-ronment, which can also be used to describe, predicate, adjust and even control the process and tendency of the population's development. Population dynamical model is widely used in the development and management of recourse'quantification, the assessment and management of environment as well as the prevention and control of catastrophe. In recent years, predator-prey models are widely used, researches on which have attracted the wide attention of mathematicians and ecologists. A lot of important results with practical significance have been studied out.By using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of parabolic equations and corresponding elliptic equa-tions, in this paper we shall study a prey-predator model subject to homogeneous Neumann boundary condition and a three-species predator-prey model with cross-diffusion subject to homogeneous Neumann boundary condition Turing instability and global stability of positive constant steady-state solution are obtained by operator spectrum theory and Turing theory. Using diffusion coefficient d as bifurcation parameter, the bifurcation at positive constant steady-state solution is discussed by perturbation theory and bifurcation theory and the structure of the global bifurcation are investigated. The existence of positive non-constant steady-state solution is obtained by degree theory.The main contents in this paper are as follows:In section 1. a predator-prey model with sparse effect subject to the Neumann boundary is studied. First, Turing instability and global stability of positive constant steady-state solution are obtained by operator spectrum theory and Turing theory. Second, it is proved that some non-constant coexistent solution is possible created when parameters are in some rage. Using diffusion coefficient as bifurcation we parameter, we obtain the bifurcation at (u, v) by perturbation theory and bifurcation theory. Third, the local branch can be extended to a global branch. Forth, we make numerical simulation about one-dimensional space.In section 2. the positive solutions are discussed for a three-species predator-prey model with cross-diffusion and ratio-dependent response functions under ho-mogeneous Neumann boundary condition. First, the prior estimate to the positive solutions of the model is given by means of maximum principle and Harnack inequal-ity. Second, the Leray-Schauder dgree theory is used for discussing the existence of thenon-constant positive solutions...