Two Kinds Of Predator - Prey Model Solution Coexistence State Analysis

Since PDE equation established a bridge between mathematical theory and practical issue, it was widely applied in exploring the law of the occurrence and spread of infectious diseases, the relationships among animals, the interaction be-tween animals and environment and many ecological phenomenons. Due to its uni-versality and complexity, the PDE model of interaction between predator-prey in species has been considered intensively, and the researchers have got fruitful results. Based on their estimable work and some existent methods, the coexistence states of two predator-prey models at different boundary conditions are investigated in this paper.In section 1, a prey-predator model subjected to homogeneous Dirichlet bound-ary condition is considered: Where u, v,denote population densities of prey and preator; Q is a bounded domain with smooth boundary (?)Ω; a, b, c, d,α,βare all positive constants.In section 2, a prey-predator model with infectious disease subjected to homo-geneous Neumann boundary condition is studied:Ωis a bounded domain with smooth boundary(?)Ω; the diffusion coefficient d1, d2, d3 are all positive constants. ui(x,0), i=1,2,3 is continuous function. We mainly apply the theories of nonlinear analysis and nonlinear partial differ-ential equations, especially those of parabolic equations and corresponding elliptic equations to study the two mentioned models.In Chapter 1, the existence and stability of positive solution of (01) model is investigated and is divided into two parts. First, the local bifurcation of semi-trivial solution(a;θa,0) is studied by applying spectrum analysis and bifurcation theory, and it is further extended to global bifurcation. Second, asymptotic behavior and stability of positive solutions in this model are given by means of comparison principle, linear stability theory, and some other methods.In Chapter 2, the coexistence states of (02) model are investigated. First, it is proved that the unique positive constant steady state solution is stable by eigenvalue theory. Moreover, a priori estimate of positive steady state solutions is given. And then, the existence of non-constant positive steady state solutions is studied using degree theory. Finally, the positive bifurcation solution of steady-state model is gained using bifurcation theory.