Abstract
A model that describes the competition of two predator species for a single regenerating prey species was introduced by Hsu et al. (1978 a,b; see also Koch, 1974 a,b) and has been studied since then by several workers, e.g., Butler (1983), Keener (1983), Smith (1982), and Wilken (1982). In this model of a three-dimensional system of ordinary differential equations the prey population is assumed to have a logistic growth rate in the absence of predators, and the predator populations are assumed to obey a Holling-type functional response (Michaelis-Menten kinetics). Butler (1983) has shown that most of the results concerning the model of Hsu et al. can be achieved for a whole class of two-predator-one-prey models whose common feature is that the prey’s growth rate and the predators’ functional response are arbitrary functions satisfying certain natural conditions.
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References
Butler, G.J. (1983), Competitive predator-prey systems and coexistence, in Population Biology Proceedings, Edmonton 1982, Lecture Notes in Biomathematics 52, pp 210–217 (Springer, Berlin).
Butler, G.J. and Waltman, P. (1981), Bifurcation from a limit cycle in a two predator—one prey ecosystem modeled on a chemostat, J. Math. Biol., 12, 295–310.
Farkas, M. (1984), Zip bifurcation in a competition model., Nonlinear Anal. TMA, 8, 1295–1309.
Farkas, M. (1985), A zip bifurcation arising in population dynamics, in 10th Int. Conf. on Nonlinear Oscillations, Varna, 1984, pp. 150–155 (Bulgarian Academy of Science, Sofia).
Hassard, B.D., Kazarinoff, N.D., and Wan, Y.-H. (1981), Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge, UK).
Hsu, S.H., Hubbell, S.P., and Waltman, P. (1978a), Competing predators, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 35, 617–625.
Hsu, S.H., Hubbell, S.P., and Waltman, P. (1978b), A contribution to the theory of competing predators, Ecol. Monogr., 48, 337–349.
Keener, J.P. (1983), Oscillatory coexistence in the chemostat: a codimension two unfolding, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 43, 1005–1018.
Koch, A.L. (1974a), Coexistence resulting from an alteration of density dependent and density independent growth, J. Theor. Biol., 44, 373–386.
Koch, A.L. (1974b), Competitive coexistence of two predators utilizing the same prey under constant environmental conditions, J. Theor. Biol., 44, 387–395.
May, R.M. (1974), Stability and Complexity in Model Ecosystems, p 82 (Princeton University Press, Princeton, NJ).
May, R.M. (1981). Theoretical Ecology, 2nd edn., p 79 (Sinauer, Sunderland, MA).
Smith, H.L. (1982), The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 42, 27–43.
Wilken, D.R. (1982), Some remarks on a competing predators problem, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 42, 895–902.
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Farkas, M. (1987). Competitive Exclusion by Zip Bifurcation. In: Kurzhanski, A.B., Sigmund, K. (eds) Dynamical Systems. Lecture Notes in Economics and Mathematical Systems, vol 287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-00748-8_13
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DOI: https://doi.org/10.1007/978-3-662-00748-8_13
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