Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

Sze Bi Hsu, Tzy Wei Hwang, Yang Kuang

Research output: Contribution to journalArticlepeer-review

327 Scopus citations


The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Kuang and Beretta(1998), where Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while contains many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper. Through a simple change of variable, we transform the Michaelis-Menten-type ratio-dependent model to a better studied Gause-type predator-prey system. As a result, we can obtain a complete classification of the asymptotic behavior of the solutions of the Michaelis-Menten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.

Original languageEnglish (US)
Pages (from-to)489-506
Number of pages18
JournalJournal Of Mathematical Biology
Issue number6
StatePublished - Jun 2001


  • Extinction
  • Global stability
  • Ratio-dependent predator-prey model
  • Uniqueness of limit cycles

ASJC Scopus subject areas

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


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