Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka–Volterra system

Abstract

In this paper, we study the Hopf bifurcation for the four-dimensional competitive Lotka–Volterra system, and give an example which can display chaotic dynamics apparent like Rossler’s folded band attractor. This demonstrates that Smale’s conclusions in (J. Math. Biol., 3:5–7, 1976) are true even for the simplest competitive Lotka–Volterra systems when the dimension n is four. We explore the mechanism of occurrence of chaotic behavior for the four-dimensional competitive Lotka–Volterra system. The numerical study indicates that a periodic solution by Hopf bifurcation can undergo successive period-doubling cascades, and a homoclinic orbit can undergo homoclinic bifurcation by Shil’nikov theorem.