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 boundedness of the solution are discussed. All the feasible equilibrium points are determined. The stability analysis of them are investigated. By employing 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.