Dynamics of a predator-prey system with stage structure and two delays

A Holling type III predator-prey system with stage structure for the predator and two delays is inves- tigated. At first, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analyzing the distribution of the roots of the associated characteristic equation. Then, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are established by using the normal form method and center manifold argument. Finally, some numerical simulations are carried out to support the main theoretical results.