Rich dynamics of a ratio-dependent one-prey two-predators model.

*(English)*Zbl 1007.34054A 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 |