## Abstract

Ratio-dependent predator-prey models are favored by many animal ecologists recently as more suitable ones for predator-prey interactions where predation involves searching process. However, such models are not well studied in the sense that most results are local stability related. In this paper, we consider the global behaviors of solutions of a 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.

Original language | English (US) |
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Pages (from-to) | 389-406 |

Number of pages | 18 |

Journal | Journal Of Mathematical Biology |

Volume | 36 |

Issue number | 4 |

DOIs | |

State | Published - Mar 1998 |

## Keywords

- Dulac criterion
- Global stability
- Liapunov function
- Limit cycles
- Predator-prey system

## ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics