Stability and Hopf bifurcation of a ratio-dependent predator-prey model with time delay and stage structure

In this paper, a ratio-dependent predator-prey model described by Holling type II functional response with time delay and stage structure for the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of the coexistence equilibrium of the model is discussed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By using the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By introducing some new lemmas and the comparison theorem, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main results.