Hopf bifurcation analysis for a ratio-dependent predator-prey system with two delays and stage structure for the predator

The ratio-dependent theory is favored by researchers since it is more suitable for describing the relationship between predator and its prey. In this paper, a ratio-dependent predator-prey system with Holling type II functional response, two time delays and stage structure for the predator is investigated. Firstly, by choosing the two time delays as the bifurcation parameter, the sufficient conditions for the local stability and the existence of Hopf bifurcation with respect to both delays are established. Furthermore, based on the normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Finally, numerical simulations are given to verify the theoretical analysis.