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Normal-form analysis of the cusp-transcritical interaction: applications in population dynamics

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Abstract

Bistability, the presence of alternative stable states, is an important feature of population models as it indicates that long-term predictions are dependent on the current population density. Two distinct kinds of bistability re-occur in population modelling studies, Allee Bistability and Positive Bistability. In this article, we show that a novel codimension-3 bifurcation, the cusp-transcritical interaction, can act as an organising centre for ordinary differential equations that exhibit both Allee Bistability and Positive Bistability. We first show how a normal form for cusp-transcritical interactions emerges from the unfolding of a particular one-dimensional degeneracy. We then illustrate the ecological relevance of the cusp-transcritical interaction. Finally, we provide a comprehensive example of normal-form analysis of an existing population model that demonstrates the occurrence of the codimension-3 bifurcation. We note that Allee Bistability and Positive Bistability may manifest unexpectedly in complex, ecological models, and therefore, this bifurcation-focused approach can provide valuable insight into the behaviour of newly developed ecosystem models.

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Acknowledgements

This publication has emanated from research conducted with the financial support of Science Foundation Ireland under the grant number SFI/13/IA/1923.

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Authors and Affiliations

  1. MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

    John G. Donohue

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Ireland

    Petri T. Piiroinen

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  1. John G. Donohue
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  2. Petri T. Piiroinen
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Correspondence to John G. Donohue.

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Donohue, J.G., Piiroinen, P.T. Normal-form analysis of the cusp-transcritical interaction: applications in population dynamics. Nonlinear Dyn 100, 1741–1753 (2020). https://doi.org/10.1007/s11071-020-05556-z

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  • DOI: https://doi.org/10.1007/s11071-020-05556-z

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