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Big bang bifurcations and Allee effect in Blumberg’s dynamics

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Abstract

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.

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Acknowledgments

This research has been partially sponsored by the national funds through the Fundação Nacional para a Ciência e Tecnologia, Portugal – FCT, under the project PEst-OE/MAT/UI0006/2014, CEAUL and ISEL. The authors are grateful to the anonymous referees for a careful checking of the details and for helpful comments that improved this paper.

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Authors and Affiliations

  1. Inst. Sup. de Eng. de Lisboa-ISEL, ADM and CEAUL, Rua Conselheiro Em ídio Navarro 1, 1959-007 , Lisboa, Portugal

    J. Leonel Rocha

  2. LAAS-CNRS, INSA, University of Toulouse, 7 avenue du Colonel Roche, 31077 , Toulouse, France

    Danièle Fournier-Prunaret

  3. INSA, University of Toulouse, 135 Avenue du Rangueil, 31077 , Toulouse, France

    Abdel-Kaddous Taha

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  1. J. Leonel Rocha
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  2. Danièle Fournier-Prunaret
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  3. Abdel-Kaddous Taha
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Correspondence to J. Leonel Rocha.

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Rocha, J.L., Fournier-Prunaret, D. & Taha, AK. Big bang bifurcations and Allee effect in Blumberg’s dynamics. Nonlinear Dyn 77, 1749–1771 (2014). https://doi.org/10.1007/s11071-014-1415-0

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  • DOI: https://doi.org/10.1007/s11071-014-1415-0

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