Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator. (English) Zbl 1154.92327

Xu, Rui; Ma, Zhien

Nonlinear Anal., Real World Appl. 9, No. 4, 1444-1460 (2008).

Summary: A predator-prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.

MSC:

92D40	Ecology
34K18	Bifurcation theory of functional-differential equations
34K20	Stability theory of functional-differential equations
37N25	Dynamical systems in biology

Keywords:

stage structure; time delay; local and global stability; Hopf bifurcation

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