Abstract
In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.
Highlights
This article was downloaded by: [Arizona State University] On: 06 March 2014, At: 12:40 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
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We study the global dynamics of a discrete two-dimensional competition model
Summary
Mathematical models can provide important insights into the general conditions that permit the coexistence of competing species and the situations that lead to competitive exclusion [8,9]. Exclusion principle fails where two species can coexist through a locally stable period-2 orbit. This phenomenon of coexistence has been observed in many other competition models (e.g., [8,9,18,19]) including system (1)–(2) with a = 0: r1xn , xn + yn (3). Studying sufficient conditions for the relative permanence of the generalization of such biological models can be our future direction
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