Delay Differential Equation With Application In Biomathematics

In this paper, a stage-structure predator-prey model with time delay is investigated. Prey population is divided into male and female. Predator only catch the mature prey. We first introduce some basic theory in Hopf bifurcation of auto delay differential equation. Secondly, we consider the local stability of a positive equilibrium and the existence of Hopf bifurcation. The result shows that the small delay will not affect the stability of the equilibrium point. However, when time-delay cross a certain critical value, the stable equilibrium point change to unstable, resulting in Hopf bifurcation. By using the normal form theory and center manifold reduction, we derive explicit formula which determine the stability, direction and other properties of bifurcating periodic solutions.