The global dynamic behavior of the competition model of three species

Abstract

In this paper, we decompose the dynamic behavior of the competitive Lotka-Volterra (LV) model ẋi=xi(1-xi-αixi+1-βixi+2), xi(0)0, αi0, βi0, i=1,2,3, with x4=x1, x5=x2, into the dynamic behavior of two two-dimensional 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 IntR3+, the necessary and sufficient conditions for the model having a family of limit cycle solutions and a heteroclinic cycle, both of which are the ω-limit set of some other trajectories of the competitive LV model. © 2000 Academic Press.