Food chain chaos due to transcritical point

Chaos
Bo Deng, Gwendolen Hines

Abstract

Chaotic dynamics of a classical prey-predator-superpredator ecological model are considered. Although much is known about the behavior of the model numerically, very few results have been proven analytically. A new analytical result is obtained. It is demonstrated that there exists a subset on which a singular Poincare map generated by the model is conjugate to the shift map on two symbols. The existence of such a Poincare map is due to two conditions: the assumption that each species has its own time scale ranging from fast for the prey to slow for the superpredator, and the existence of transcritical points, leading to the classical mathematical phenomenon of Pontryagin's delay of loss of stability. This chaos generating mechanism is new, neither suspected in abstract form nor recognized in numerical experiments in the literature.

References

Jan 1, 1978·Computers in Biology and Medicine·P Hogeweg, B Hesper
Jan 1, 1992·Biomedicine & Pharmacotherapy = Biomédecine & Pharmacothérapie·I Blazsek
May 1, 1996·Mathematical Biosciences·Y A Kuznetsov, S Rinaldi
May 23, 1998·Mathematical Biosciences·O De Feo, S Rinaldi
Jun 5, 2003·Chaos·Bo Deng, Gwendolen Hines
Jan 1, 1989·Trends in Ecology & Evolution·A A Berryman, J A Millstein

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Citations

Apr 7, 2009·Journal of Mathematical Biology·Geertje Hek
Nov 26, 2009·Journal of Integrative Neuroscience·Bo Deng
Jul 8, 2008·Chaos·Jorge DuarteNuno Martins
Oct 2, 2007·Chaos·Bo Deng, Irakli Loladze
Oct 5, 2010·Chaos·Jorge DuarteJosep Sardanyés
Dec 1, 2004·Chaos·Bo Deng
Nov 5, 2011·Journal of Theoretical Biology·Roger Cropp, John Norbury
Apr 4, 2017·Chaos·Bo DengSze-Bi Hsu

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