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

Yang Kuang, Edoardo Beretta

Research output: Contribution to journalArticle

411 Scopus citations

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
DOIs
StatePublished - 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

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