Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models

J. Leonel Rocha; Abdel-Kaddous Taha; Danièle Fournier-Prunaret

doi:10.3934/dcdsb.2015.20.3131

Article Contents

2015, Volume 20, Issue 9: 3131-3163. Doi: 10.3934/dcdsb.2015.20.3131

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Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models

1.
Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa
2.
INSA, University of Toulouse, 135 Avenue du Rangueil, 31077 Toulouse
3.
LAAS-CNRS, INSA, University of Toulouse, 7 Avenue du Colonel Roche, 31077 Toulouse

Received: September 30, 2014

Revised: May 31, 2015

Published: October 2015

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Abstract

This work concerns dynamics and bifurcations properties of a new class of continuous-defined one-dimensional maps: Tsoularis-Wallace's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon of extinction. To establish this result we introduce the notions of Allee's functions, Allee's effect region and Allee's bifurcation curve. Another central point of our investigation is the study of bifurcation structures for this class of functions, in a three-dimensional parameter space. We verified that under some sufficient conditions, Tsoularis-Wallace's functions have particular bifurcation structures: the big bang and the double big bang bifurcations of the so-called ``box-within-a-box'' type. The double big bang bifurcations are related to the existence of flip codimension--2 points. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct kinds of boxes. This work contributes to clarify the big bang bifurcation analysis for continuous maps and understand their relationship with explosion birth and extinction phenomena.

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Mathematics Subject Classification: Primary: 92D25, 37H20; Secondary: 37E05.

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