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Liang, Zhiqing; Pan, Hongwei

Qualitative analysis of a ratio-dependent Holling-Tanner model. (English) Zbl 1124.34030

J. Math. Anal. Appl. 334, No. 2, 954-964 (2007).
The authors consider a so-called ratio-dependent Holling-Tanner predator-prey model, where both components contain coefficients that depend on the ratio of predator and prey densities. By rescaling, the original six positive parameters are reduced to three, which are restricted in such a way that there exists a unique equilibrium \(E\) in the open positive quadrant. Generically, \(E\) is an attractor or repeller. In the first case, an extra condition on the parameters allows the construction of a Lyapunov function showing that \(E\) is globally attractive. If \(E\) is a repeller, the Poincaré-Bendixson theorem yields the existence of a limit cycle. Its uniqueness is, moreover, proved by transforming the given system into a Liénard system and using known results on the uniqueness of limit cycles for such systems.
Reviewer: Josef Hainzl (Freiburg)

Cited in 44 Documents

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
92D25 Population dynamics (general)

Keywords:

Holling-Tanner model; Lyapunov function; global attractor; limit cycle; uniqueness
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\textit{Z. Liang} and \textit{H. Pan}, J. Math. Anal. Appl. 334, No. 2, 954--964 (2007; Zbl 1124.34030)
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