Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

Abstract

We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delay $\tau$ as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay $\tau$ passes through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.