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

Yang Kuang, Edoardo Beretta

Research output: Contribution to journalArticle

391 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)389-406
Number of pages18
JournalJournal of Mathematical Biology
Volume36
Issue number4
StatePublished - Mar 1998

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Predator prey systems
Ratio-dependent
Global Analysis
Predator-prey System
Qualitative Analysis
qualitative analysis
Asymptotic stability
predators
Positive Periodic Solution
Predator-prey Model
predator-prey relationships
Local Asymptotic Stability
Animals
ecologists
Predator-prey
Globally Asymptotically Stable
Global Asymptotic Stability
Local Stability
Behavior of Solutions
Asymptotically Stable

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

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

In: Journal of Mathematical Biology, Vol. 36, No. 4, 03.1998, p. 389-406.

Research output: Contribution to journalArticle

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