Journal of Mathematical Biology

, Volume 42, Issue 6, pp 489–506 | Cite as

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

  • Sze-Bi Hsu
  • Tzy-Wei Hwang
  • Yang Kuang


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.

Key words or phrases: Ratio-dependent predator-prey model – Global stability – Uniqueness of limit cycles – Extinction 
Mathematics Subject Classification (2000): 34D05, 34D20, 92D25 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Sze-Bi Hsu
    • 1
  • Tzy-Wei Hwang
    • 2
  • Yang Kuang
    • 3
  1. 1.Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. Research supported by National Council of Science, Republic of ChinaCN
  2. 2.Department of Mathematics, Kaohsiung Normal University, 802, Kaohsiung, Taiwan, R.O.C. Research supported by National Council of Science, Republic of ChinaCN
  3. 3.Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA. e-mail:

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