Journal of Mathematical Biology

, Volume 12, Issue 1, pp 115–126

Some results on global stability of a predator-prey system

  • Kuo-Shung Cheng
  • Sze-Bi Hsu
  • Song-Sun Lin
Article

Abstract

In this paper we derive some results to ensure the global stability of a predator-prey system. The results cover most of the models which have been proposed in the ecological literature for predator-prey systems. The first result is very geometric and it is very easy to check from the graph of prey and predator isoclines. The second one is purely algebraic, however, it covers the defects of the first one especially in dealing with Holling's type-3 functional response in some sense. We also discuss the global stability of Kolmogorov's model. Some examples are presented in the discussion section.

Key words

Global stability Predator-prey system 

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References

  1. 1.
    Albrecht, F., Gatzke, H., Haddad, A., Wax, N.: The dynamics of two interacting populations. J. Math. Anal. Appl. 46, 658–670 (1974)Google Scholar
  2. 2.
    Freedman, H. I.: Graphical stability, enrichment, and pest control by a natural enemy. Math. Biosci. 31, 207–225 (1976)Google Scholar
  3. 3.
    Gaus, G. F., Smaragdova, N. P., Witt, A. A.: Further studies of interaction between predators and prey. J. Anim. Ecol. 5, 1–18 (1936)Google Scholar
  4. 4.
    Goh, B. S.: Global stability in many species systems. Amer. Natur. 111 (977), 135–143 (1977)Google Scholar
  5. 5.
    Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969Google Scholar
  6. 6.
    Hastings, A.: Global stability of two species system. J. Math. Biol. 5, 399–403 (1978)Google Scholar
  7. 7.
    Hsu, S. B.: On global stability of a predator-prey system. Math. Biosci. 39, 1–10 (1978)Google Scholar
  8. 8.
    Hsu, S. B., Hubbel, S. P., Waltman, P.: Competing predators. SIAM J. Applied Math. 35, 617–625 (1978)Google Scholar
  9. 9.
    May, R. M.: Stability and complexity in model ecosystems. Princeton, U.P., Princeton, N.J., 1974Google Scholar
  10. 10.
    Oaten, A., Murdoch, W. W.: Functional response and stability in predator-prey system. Amer. Natur. 109, 289–298 (1975)Google Scholar
  11. 11.
    Real, L. A.: The kinetics of functional response. Amer. Natur. 111 (1978), 289–300 (1977)Google Scholar
  12. 12.
    Rosenzweig, M. L.: Why the prey curve has a hump. Amer. Natur. 103, 81–87 (1969)Google Scholar
  13. 13.
    Rosenzweig, M. L., MacArthur, R. H.: Graphical representation and stability conditions of predator-prey interaction. Amer. Natur. 97, 209–223 (1963)Google Scholar
  14. 14.
    Rosenzweig, M. L.: Paradox of enrichment: Destabilization of exploitation ecosystem in ecological time. Science 171, 385–387 (1971)Google Scholar
  15. 15.
    Maynard Smith, J.: Models in ecology. Cambridge: University Press 1974Google Scholar

Copyright information

© Springer-Verlag GmbH & Co KG 1981

Authors and Affiliations

  • Kuo-Shung Cheng
    • 1
  • Sze-Bi Hsu
    • 1
  • Song-Sun Lin
    • 1
  1. 1.Department of Applied MathematicsNational Chiao Tung UniversityHsin-ChuTaiwan 300, Republic of China

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