Competitive Exclusion by Zip Bifurcation
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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- 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).Google Scholar
- 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).Google Scholar
- Hassard, B.D., Kazarinoff, N.D., and Wan, Y.-H. (1981), Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge, UK).Google Scholar
- May, R.M. (1974), Stability and Complexity in Model Ecosystems, p 82 (Princeton University Press, Princeton, NJ).Google Scholar
- May, R.M. (1981). Theoretical Ecology, 2nd edn., p 79 (Sinauer, Sunderland, MA).Google Scholar