Dynamics Analysis On A Competition Model

In this paper, we build a multi-species competition model(see1.1.3) with certain inhibitors, but the system (1.1.3) is a high dimensional system and the equilibrium O(0,0,…,0) is a saddle, so it’s very difficult to investigate its dy-namics clearly. Therefore, we try to discuss the dynamics of the system with one species u and two species v(i.e. m=1, n=2) in this paper. In the first place, we prove that there are at most two positive equilibria in the system (1.3.1), also we obtain the necessary conditions for the system has two(unique) positive equilib-ria and prove it. Furthermore, if the condition H2(see the following) stands, then the necessary conditions also are sufficient. In the second place, we prove that if the system has two positive equilibria, then both of them are hyperbolic. At the same time, we give the necessary and sufficient conditions for the positive equilibrium of the system is non-hyperbolic and hyperbolic respectively. Com-bined with the stability and hyperbolicity of the boundary equilibria, we draw the global portraits when the system has two positive equilibria. When both boundary equilibria and positive equilibria are hyperbolic, we present a classi-fication for the case that the system has a unique positive equilibrium. Also, we draw the global pictures when there is no positive equilibria for the system. We establish the principal results of the paper by the strong monotonicity of the flow, since the system is type-K cooperative and irreducible.