Global qualitative analysis for a predator–prey system with delay

In this paper, the dynamics of a predator–prey system with a finite delay is considered. The conditions for the global stability and the existence of Hopf bifurcation at the positive equilibrium of the system are obtained. Explicit algorithms for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are derived, using the normal form theory and center manifold argument [Nussbaum RD. Periodic solutions of some nonlinear autonomous functional equations. Ann Mat Pura Appl 1974;10:263–306]. Numerical simulations supporting the theoretical analysis are also given. Global existence of periodic solutions is established by using a global Hopf bifurcation result of Wu [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981].