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Singularity theory of fitness functions under dimorphism equivalence

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Abstract

We apply singularity theory to classify monomorphic singular points as they occur in adaptive dynamics. Our approach is based on a new equivalence relation called dimorphism equivalence, which is the largest equivalence relation on strategy functions that preserves ESS singularities, CvSS singularities, and dimorphisms. Specifically, we classify singularities up to topological codimension two and compute their normal forms and universal unfoldings. These calculations lead to the classification of local mutual invasibility plots that can be seen generically in systems with two parameters.

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Acknowledgments

The idea of using singularity theory methods to study adaptive dynamics originated in a conversation with Ulf Dieckmann. We thank Odo Diekmann for suggesting the study of dimorphisms using singularity theory and Ian Hamilton, Yuan Lou, Adrian Lam, and Hans Metz for many helpful discussions. This research was supported in part by the National Science Foundation Grants DMS-1008412 to MG and DMS-0931642 to the Mathematical Biosciences Institute.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Xiaohui Wang

  2. Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, 43210, USA

    Martin Golubitsky

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  1. Xiaohui Wang
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  2. Martin Golubitsky
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Correspondence to Martin Golubitsky.

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Wang, X., Golubitsky, M. Singularity theory of fitness functions under dimorphism equivalence. J. Math. Biol. 73, 525–573 (2016). https://doi.org/10.1007/s00285-015-0958-0

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  • DOI: https://doi.org/10.1007/s00285-015-0958-0

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