Journal of Mathematical Biology

, Volume 43, Issue 5, pp 377–396

Rich dynamics of a ratio-dependent one-prey two-predators model


  • Sze-Bi Hsu
    • Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C.
  • Tzy-Wei Hwang
    • Department of Mathematics, Kaohsiung Normal University, 802, Kaohsiung, Taiwan, R.O.C.
  • Yang Kuang
    • Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA.

DOI: 10.1007/s002850100100

Cite this article as:
Hsu, S., Hwang, T. & Kuang, Y. J Math Biol (2001) 43: 377. doi:10.1007/s002850100100


The objective of this paper is to systematically study the qualitative properties of a ratio-dependent one-prey two-predator model. We show that the dynamics outcome of the interactions are very sensitive to parameter values and initial data. Specifically, we show the interactions can lead to all the following possible outcomes: 1) competitive exclusion; 2) total extinction, i.e., collapse of the whole system; 3) coexistence in the form of positive steady state; 4) coexistence in the form of oscillatory solutions; and 5) introducing a friendly and better competitor can save a otherwise doomed prey species. These results reveal far richer dynamics compared to similar prey dependent models. Biological implications of these results are discussed.

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© Springer-Verlag Berlin Heidelberg 2001