Bifurcation and chaos of a delayed predator-prey model with dormancy of predators

In this paper, a delayed predator-prey model with dormancy of predators is investigated. It shows that time delay in the prey-species growth can lead to the occurrence of Hopf bifurcation with stability switches at a coexistence equilibrium. The computing formulas of stability and direction of the Hopf bifurcating periodic solutions are given. Under appropriate conditions, the uniform persistence of this model with time delay is proved. In this simple model, multiple periodic solutions coexist. Through numerical simulation, it is shown that different values of time delay can generate or eliminate chaos. Biologically, our results imply that dynamical behaviors of this system with time delay strongly depend on the initial density of this model and the time delay of the growth of the prey.