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.
KeywordsHopf Bifurcation Invariant Manifold Predator Species Predator Population Natural Case
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