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Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure. (English) Zbl 1146.34323
Summary: A ratio-dependent predator-prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the characteristic equations, the local stability of a positive equilibrium and a boundary equilibrium is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium when \(\tau = \tau _{0}\). By using an iteration technique, sufficient conditions are derived for the global attractivity of the positive equilibrium. By comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. Numerical simulations are carried out to illustrate the main results.

MSC:
34D23 Global stability of solutions to ordinary differential equations
34K18 Bifurcation theory of functional-differential equations
37N25 Dynamical systems in biology
92D25 Population dynamics (general)
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