Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type. (English) Zbl 0935.34035

Hsu, Sze-Bi; Hwang, Tzy-Wei

Taiwanese J. Math. 3, No. 1, 35-53 (1999).

The authors study the Hopf bifurcation for the Holling-Tanner predator-prey model. Using Andronov-Hopf bifurcation theorem, they show that for some parameters the bifurcation is subcritical, i.e., there exists a small-amplitude repelling periodic orbit enclosing a stable equilibrium and separating it from another, stable limit cycle. The paper also summarizes earlier results of S.-B. Hsu and T.-W. Hwang [SIAM J. Appl. Math. 55, 763-783 (1995; Zbl 0832.34035)] on global asymptotical stability of the internal equilibrium.

Reviewer: David S.Boukal (České Budějovice)

MSC:

34C23	Bifurcation theory for ordinary differential equations
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
92D25	Population dynamics (general)
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D23	Global stability of solutions to ordinary differential equations

Keywords:

Holling-Tanner model; predator-prey system; Andronov-Hopf bifurcation; multiple limit cycle