Stability and Hopf Bifurcation Analysis on a Stage-structured Predator-prey System with Time Delays

In this paper, a stage-structured predator-prey system with time delays is considered, where the time delays are regarded as bifurcation parameters. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. Moreover, we show that Hopf bifurcations occurs when time delay crosses some critical values. By using the normal form method and center manifold theorem, we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is studied. Numerical simulations are carried out to illustrate the theoretical results and they show that the time delays in the feedback of prey’s density in the system under consideration can destroy the stability of the system.