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.
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Albrecht, F., Gatzke, H., Haddad, A., Wax, N.: The dynamics of two interacting populations. J. Math. Anal. Appl. 46, 658–670 (1974)
Freedman, H. I.: Graphical stability, enrichment, and pest control by a natural enemy. Math. Biosci. 31, 207–225 (1976)
Gaus, G. F., Smaragdova, N. P., Witt, A. A.: Further studies of interaction between predators and prey. J. Anim. Ecol. 5, 1–18 (1936)
Goh, B. S.: Global stability in many species systems. Amer. Natur. 111 (977), 135–143 (1977)
Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969
Hastings, A.: Global stability of two species system. J. Math. Biol. 5, 399–403 (1978)
Hsu, S. B.: On global stability of a predator-prey system. Math. Biosci. 39, 1–10 (1978)
Hsu, S. B., Hubbel, S. P., Waltman, P.: Competing predators. SIAM J. Applied Math. 35, 617–625 (1978)
May, R. M.: Stability and complexity in model ecosystems. Princeton, U.P., Princeton, N.J., 1974
Oaten, A., Murdoch, W. W.: Functional response and stability in predator-prey system. Amer. Natur. 109, 289–298 (1975)
Real, L. A.: The kinetics of functional response. Amer. Natur. 111 (1978), 289–300 (1977)
Rosenzweig, M. L.: Why the prey curve has a hump. Amer. Natur. 103, 81–87 (1969)
Rosenzweig, M. L., MacArthur, R. H.: Graphical representation and stability conditions of predator-prey interaction. Amer. Natur. 97, 209–223 (1963)
Rosenzweig, M. L.: Paradox of enrichment: Destabilization of exploitation ecosystem in ecological time. Science 171, 385–387 (1971)
Maynard Smith, J.: Models in ecology. Cambridge: University Press 1974
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Works partially supported by the National Science Council of the Republic of China
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Cheng, KS., Hsu, SB. & Lin, SS. Some results on global stability of a predator-prey system. J. Math. Biol. 12, 115–126 (1982). https://doi.org/10.1007/BF00275207
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DOI: https://doi.org/10.1007/BF00275207