The Hopf bifurcation and stability of delayed predator-prey system

In this paper, a mathematical model consisting of three populations with discrete time delays is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed using the theory of normal form and center manifold. Discussion with some numerical simulation examples is given to support the theoretical results.