Positive Solutions Of A Diffusive Variable-territory Prey-predator Model

The prey-predator model is a quite typical problem in the field of mathematical biology, which has been investigated seriously and significantly by quite a few of people. They set up many mathematics models, and give a lot of widely available theoretical tools.In this paper, on the basis of their work, we investigate a diffusive variable territory prey-predator model (P): where Ω(?)RN is a bounded domain with a smooth boundary (?)Ω, the parameters λ、b are positive numbers, and a(x) is a non-constant, continuous function satisfying a(x)> 0 on Ω. u and v are the respective populations of prey and predator.By studying the properties and image of the solution of the model (P), we can macro scop ically observe and predict, how the population of prey and predator involved in the model (P) changed in the nature, and how will the prey and predator impact on environment.In order to make the model (P) have practical significance, we limit that the population of prey and predator should not identically be zero in the bounded domainΩ. Here, we say(u,v) with u|(?)Ω= v|(?)Ω= 0 is a positive solution of problem (P) if (u,v) is a solution of (P) and u,v> 0 in Ω. We concern the properties of positive solutions of the model (P), for example existence, uniqueness, stability and asymptotic behavior.The main result of this paper is as follows:(1) By studying the change of the parameter A in the model (P), we obtain an existence theorem of positive solutions of the model (P). We give an intuitive explanation of this theorem by eigenvalue theory, the maximum principle and the method of upper and lower solutions of elliptic equations, and then prove this theorem precisery by some results on the fixed point index on cones.(2) By studying the change of the parameter b in the model (P), we obtain the asymptotic behavior of positive solutions of the model (P). We give and prove the theorem on the asymptotic behavior of positive solutions of the model (P) by the regularity theory of elliptic equations, eigenvalue theory, and some results in functional analysis.(3) By limiting the parameters λ、b in the model (P), we obtain that the model (P) has linear stable positive solutions when parameters L 6 satisfy certain conditions. We prove this theorem by the method of upper and lower solutions of elliptic equations and eigenvalue theory.