Hopf bifurcation and global stability for a delayed predator–prey system with stage structure for predator

In this paper, a delayed predator–prey system with stage structure for predator is proposed and studied. It is found that the time delay is harmless for permanence of the stage-structured system. If αβ < 1, sufficient conditions which guarantee the global stability of positive equilibrium are given. If αβ > 1, we show that the unique positive equilibrium is locally asymptotically stable when time delay τ∗ is sufficiently small, while loss of stability by a Hopf bifurcation can occur as the delay increases.