Existence of Positive Solutions for a Three Species Predator-Prey System

Population dynamics is an important branch of biomathematics and has been widely used to study some ecological phenomena. To analyze the interactions between biological population, a three species predator-prey system with Holling Ⅲ Type functional response and homogeneous Neumann boundary condition was studied, in which the two predator species consumed the common prey species. By using linearization method and Routh-Hurwitz criterion, the uniformly asymptotically stable of constant positive solutions was investigated. To research the non-constant positive solutions of the predator-prey system, a priori-estimation of the positive solution was first discussed by using energy method, L^p estimation and Sobolev embedding theorem. Second, the non-existence of non-constant positive solutions was proved by H lder inequality, Young inequality and Poincare inequality. Finally, the existence conditions of the non-constant positive solutions of the predator-prey system were achieved by means of the results obtained before and topological degree theory. Results show that when the predation species has a high diffusion rate and other parameters satisfy certain conditions, the prey-prey system can achieve ecological balance.