A delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response is proposed and explored in this study. We discuss the positivity and the existence of equilibrium points. By choosing time delay as the bifurcation parameter and analyzing the relevant characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, and the coexistence equilibrium of the system is investigated. In accordance with the normal form method and center manifold theorem, the property analysis of Hopf bifurcation of the system is obtained. Furthermore, for the purpose of protecting the stability of such a biological system, a hybrid control method is presented to control the Hopf bifurcation. Finally, numerical examples are given to verify the theoretical findings.

Hopf bifurcation analysis of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type II functional response is investigated. By analyzing the corresponding characteristic equations, the conditions for the stability and existence of Hopf bifurcation for the system are obtained. In addition, a hybrid control strategy is proposed to postpone the onset of an inherent bifurcation of the system. By utilizing normal form method and center manifold theorem, the explicit formulas that determine the direction of Hopf bifurcation and the stability of bifurcating period solutions of the controlled system are derived. Finally, some numerical simulation examples confirm that the hybrid controller is efficient in controlling Hopf bifurcation.