### 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) |
---|---|

Pages (from-to) | 389-406 |

Number of pages | 18 |

Journal | Journal of Mathematical Biology |

Volume | 36 |

Issue number | 4 |

State | Published - Mar 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

*Journal of Mathematical Biology*,

*36*(4), 389-406.

**Global qualitative analysis of a ratio-dependent predator-prey system.** / Kuang, Yang; Beretta, Edoardo.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 36, no. 4, pp. 389-406.

}

TY - JOUR

T1 - Global qualitative analysis of a ratio-dependent predator-prey system

AU - Kuang, Yang

AU - Beretta, Edoardo

PY - 1998/3

Y1 - 1998/3

N2 - 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.

AB - 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.

KW - Dulac criterion

KW - Global stability

KW - Liapunov function

KW - Limit cycles

KW - Predator-prey system

UR - http://www.scopus.com/inward/record.url?scp=0000252357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000252357&partnerID=8YFLogxK

M3 - Article

VL - 36

SP - 389

EP - 406

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 4

ER -