Dynamics of a delayed stage structure predator-prey model with predator-dependent prey refuge

In this paper, a delayed stage structure predator-prey model with predator-dependent prey refuge is proposed. The prey population is divided into two parts: juvenile and adult prey, and the population of predators depends on adult prey only. The number of prey in refugia is nonlinear and depends on the number of predators in this work. First, the existence and local stability of all possible equilibria are investigated. By constructing appropriate Lyapunov functions, we get the sufficient conditions for the global stability of the predator-free equilibrium and the positive equilibrium, respectively. Moreover, we introduce the gestation delay of predator to the system. The existence of periodic solutions at the positive equilibrium point via Hopf-bifurcation with respect to delay is established. The stability and direction of Hopf-bifurcation are also analysed by using Normal form theory and Centre manifold theory.