Stability and Hopf bifurcation analysis for a two-enterprise interaction model with delays

Abstract

Based on the viewpoint of ecology, the mathematical model describing the dynamic development of two enterprises is improved. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are analyzed by choosing the sum τ of two delays as the bifurcation parameter and employing the Hopf bifurcation theory. It is found that when τ is less than a certain critical value, the positive equilibrium is locally asymptotically stable while it becomes unstable when τ is greater than the above critical value. In addition, the system can bifurcate a family of nontrivial periodic solutions from the positive equilibrium when τ crosses increasingly through a sequence of critical values containing the above critical value. In particular, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained according to the normal form theory and the center manifold theorem for delay differential equations. Finally, some numerical simulations supporting our theoretical predictions are included.