Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system.

*(English)*Zbl 0984.92035Summary: The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Y. Kuang and E. Beretta [ibid. 36, No. 4, 389-406 (1998; Zbl 0895.92032)], where a Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while containing many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper.

Through a simple change of variables, we transform the Michaelis-Menten-type ratio-dependent model to a better studied Gause-type predator-prey system. As a result, we obtain a complete classification of the asymptotic behavior of the solutions of the Michaelis-Menten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.

Through a simple change of variables, we transform the Michaelis-Menten-type ratio-dependent model to a better studied Gause-type predator-prey system. As a result, we obtain a complete classification of the asymptotic behavior of the solutions of the Michaelis-Menten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.

##### MSC:

92D40 | Ecology |

34D23 | Global stability of solutions to ordinary differential equations |

37N25 | Dynamical systems in biology |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |