Bifurcation Analysis in a Predator-prey Model with Discrete and Distributed Time Delay

In this paper, a class of predator-prey model with discrete and distributed time delay is considered, where the time delay is regarded as a parameter. It is shown that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By using the normal form theory and center manifold theory, we derive some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are given.