S1 attractor bifurcation analysis for a three-species cooperating model

Abstract

In this paper, the dynamic bifurcation of the three-species cooperating model is considered. It worth noting that the main theory of this paper is the Center manifold reduction and attractor bifurcation theory, which is developed by Ma [1,2]. The main work of this paper shows that if the algebraic multiplicity of the first eigenvalue is 2, there exists an S1 attractor bifurcation, and the number of its singular points can only be eight. Besides, we show that the simplified governing equations bifurcate to an S1 attractor, when the Control parameter λ crosses a critical value λ 0.