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Zbl 1220.34068
Lan, K.Q.; Zhu, C.R.
Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions. (English)
[J] Nonlinear Anal., Real World Appl. 12, No. 4, 1961-1973 (2011). ISSN 1468-1218

Summary: The phase portraits, existence and uniqueness of stable limit cycles and Hopf bifurcations of the well-known Holling-Tanner model for predator-prey interactions are studied. The ranges of the parameters involved are provided under which the unique interior equilibrium can be determined to be a stable (or an unstable) node or focus. The Hopf bifurcations and the existence and uniqueness of stable limit cycles of the models are obtained by computing the Lyapunov number involved. Our results confirm some previous results observed and suggested from real ecological systems.

MSC 2000:: *34C60 Applications of qualitative theory of ODE
34D20 Lyapunov stability of ODE
34C23 Bifurcation (periodic solutions)
34C05 Qualitative theory of some special solutions of ODE
92D25 Population dynamics

Keywords: Holling-Tanner model; predator-prey system; phase portrait; limit cycle; Hopf bifurcation; Lyapunov number

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