The Global Dynamic Behavior of the Competition Model of Three Species1

In this paper, we decompose the dynamic behavior of the competitive Lotka]Volterra LV.model ˙xisxi 1yxiyaixiq1ybixiq2., xi 0.)0, ai)0, bi)0, is1, 2, 3, with x4sx1, x5sx2, into the dynamic behavior of two twodimensional manifolds, and completely analyse the global asymptotic behavior of the competitive LV model. We obtain the necessary and sufficient conditions for the equilibrium of the competitive LV model to be positive and globally asymptotically stable in Int Rq3 , the necessary and sufficient conditions for the model having a family of limit cycle solutions and a heteroclinic cycle, both of which are the v-limit set of some other trajectories of the competitive LV model