The Study Of Solutions To A Reaction-diffusion Equations With The Third Boundary Condition

Reaction-diffusion systems are some of the most widely used models for population dynamics or genetics in situations, where spatial dispersal plays a significant role. The problem of the existence of positive solutions to reaction-diffusion systems has been well investigated. In this paper, we study the dynamical behavior of a prey-predator model with predator saturation and competition under the homo-geneous Robin boundary condition, including the existence of positive solutions, the influence of solutions by reaction parameters and the asymptotic behavior of solutions, some valuable results are obtained. The main contents and results in this paper are as follows:In chapter1, we introduce the background of mathematical ecology models, results and advance in studying. In chapter2, we list some basic theory and classical results of reaction diffusion systems. These are the basic parts that will be very useful in the forthcoming contents. Such as the maximum principle and the eigenvalue problems, the sub-and-super solutions method, the fixed-point index theory in cone. In chapter3, several sufficient conditions for coexistence of the steady-state solutions are given by the standard fixed-point index theory in cone. In chapter4, we prove that the property of the positive steady-state solution while bâ†'0+. In chapter5, we seek for an asymptotic behavior for the coexistence states, that is, the solution of such a system tends to a constant as tâ†'∞. We give some sufficient conditions for one species died.