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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 $xy/\left(ax+y\right)$ 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 of ODE and analogous concepts 92D25 Population dynamics (general) 34D05 Asymptotic stability of ODE 34D20 Stability of ODE