Stability and hopf bifurcation analysis for a stage-structured predator-prey model with discrete and distributed delays

We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. by use of a new variable,the original three-dimensional system transforms into an equivalent four-dimensional system. firstly,we study the existence and local stability of positive equilibrium of the new system. and,by choosing the time delay τ as a bifurcation parameter,we show that hopf bifurcation may occur as the time delay τ passes through some critical values. secondly,by use of normal form theory and central manifold argument,we establish the direction and stability of hopf bifurcation. at last,some simple discussion is presented. © 2013 ruiqing shi et al.