The Dynamical Analysis of a Delayed Prey-Predator Model with a Refuge-Stage Structure Prey Population

A mathematical model describing the dynamics  of a  delayed  stage structure prey - predator  system  with  prey  refuge  is  considered.  The  existence,  uniqueness  and bounded- ness  of  the  solution  are  discussed.    All  the  feasibl e  equilibrium  points  are determined.  The   stability  analysis  of  them  are  investigated.  By  employ ing  the time delay as the bifurcation parameter, we observed  the existence of Hopf bifurcation at the positive equilibrium. The stability and direction of the Hopf bifurcation are determined by  utilizing  the  normal  form  method  and  the  center  manifold  reduction.  Numerical simulations are given to support the analytic results.