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Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions. (English) Zbl 1220.34068
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:
34C60Qualitative investigation and simulation of models (ODE)
34D20Stability of ODE
34C23Bifurcation (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)
92D25Population dynamics (general)
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