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Qualitative analysis of a ratio-dependent Holling-Tanner model. (English) Zbl 1124.34030
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.
MSC:
34C60Qualitative investigation and simulation of models (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)
34C11Qualitative theory of solutions of ODE: growth, boundedness
37C70Attractors and repellers, topological structure
92D25Population dynamics (general)