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**Global qualitative analysis of a ratio-dependent predator-prey system.**
*(English)*
Zbl 0895.92032

Summary: Ratio-dependent predator-prey models are favored by many animal ecologists recently as more suitable ones for predator-prey interactions where predation involves searching processes. However, such models are not well studied in the sense that most results are local stability related.

We consider the global behaviors of solutions of ratio-dependent predator-prey systems. Specifically, we shall show that ratio dependent predator-prey models are rich in boundary dynamics, and most importantly, we shall show that if the positive steady state of the so-called Michaelis Menten ratio-dependent predator-prey system is locally asymptotically stable, then the system has no nontrivial positive periodic solutions. We also give sufficient conditions for each of the possible three steady states to be globally asymptotically stable. We note that for ratio-dependent systems, in general, local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system, and therefore does not imply global asymptotic stability. To show that the system has no nontrivial positive periodic solutions we employ the so-called divergency criterion for the stability of limit cycles in planar systems and some critical transformations.

We consider the global behaviors of solutions of ratio-dependent predator-prey systems. Specifically, we shall show that ratio dependent predator-prey models are rich in boundary dynamics, and most importantly, we shall show that if the positive steady state of the so-called Michaelis Menten ratio-dependent predator-prey system is locally asymptotically stable, then the system has no nontrivial positive periodic solutions. We also give sufficient conditions for each of the possible three steady states to be globally asymptotically stable. We note that for ratio-dependent systems, in general, local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system, and therefore does not imply global asymptotic stability. To show that the system has no nontrivial positive periodic solutions we employ the so-called divergency criterion for the stability of limit cycles in planar systems and some critical transformations.

### MSC:

92D40 | Ecology |

34C25 | Periodic solutions to ordinary differential equations |

92D25 | Population dynamics (general) |

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

34D05 | Asymptotic properties of solutions to ordinary differential equations |