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Rich dynamics of a ratio-dependent one-prey two-predators model. (English) Zbl 1007.34054
A ratio-dependent predator-prey model is the one where the interaction term has the form \(xy/(ax+y)\) where \(x\) is the prey density, \(y\) is the predator density and \(a\) is a constant. The validity of such models is still debatable. Here, a one-prey two-predators model is studied. The authors argue that the ratio dependent model give two important realistic results: The first is the existence of a stable low density prey equilibrium. The second is a simultaneous extinction. The authors derive sufficient conditions for the competitive exclusion (one predator goes extinct). They also derive conditions for a predator to be a system saver, i.e. a predator whose introduction prevents the prey from going extinct. Moreover, conditions for total extinction and for coexistence are derived.
Reviewer: E.Ahmed (Al-Ain)

MSC:
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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