Hopf bifurcation of a predator–prey model with time delay and stage structure for the prey

In this paper, a stage-structured predator–prey system with Holling type-III functional response and time delay due to the gestation of the predator is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as ττ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results.