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Advisor(s)
Abstract(s)
This paper concerns dynamics and bifurcations properties of a class of continuous-defined one-dimensional maps, in a three-dimensional parameter space: Blumberg'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, associated with the stability of a fixed point. A central point of our investigation is the study of bifurcations structure for this class of functions. We verified that under some sufficient conditions, Blumberg's functions have a particular bifurcations structure: the big bang bifurcations of the so-called "box-within-a-box" type, but for different kinds of boxes. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct attractors. This work contributes to clarify the big bang bifurcation analysis for continuous maps. To support our results, we present fold and flip bifurcations curves and surfaces, and numerical simulations of several bifurcation diagrams.
Description
Keywords
Blumberg's Dynamics Big Bang Bifurcations Allee Effect
Citation
ROCHA, José Leonel Linhares da; FOURNIER-PRUNARET, Daniele; TAHA, Abdel-Kaddous – Big Bang bifurcations and allee effect in Blumberg’s dynamics. Nonlinear Dynamics. ISSN: 0924-090X. Vol. 77, nr. 4 (2014), pp. 1749-1711
Publisher
Springer