Bifurcation analysis for a three-species predator–prey system with two delays

In this paper, a three-species predator–prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.

► The main purpose of this paper is to investigate the effects of the delay. ► We show when the system undergoes a Hopf bifurcation at the positive equilibrium. And obtain the direction of the Hopf bifurcations. ► With the increase of the delay, system will show us the complicated dynamical behaviors.