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A negative answer to a conjecture arising in the study of selection–migration models in population genetics

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Abstract

We deal with the study of the evolution of the allelic frequencies, at a single locus, for a population distributed continuously over a bounded habitat. We consider evolution which occurs under the joint action of selection and arbitrary migration, that is independent of genotype, in absence of mutation and random drift. The focus is on a conjecture, that was raised up in literature of population genetics, about the possible uniqueness of polymorphic equilibria, which are known as clines, under particular circumstances. We study the number of these equilibria, making use of topological tools, and we give a negative answer to that question by means of two examples. Indeed, we provide numerical evidence of multiplicity of positive solutions for two different Neumann problems satisfying the requests of the conjecture.

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Acknowledgements

I thank Prof. Reinhard Bürger for inspiring me to work on this problem. I am also very grateful to Profs. Fabio Zanolin, Carlota Rebelo and Alessandro Margheri for providing useful suggestions that greatly improved the manuscript. I wish also to thank two anonymous referees for their valuable comments to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100, Udine, Italy

    Elisa Sovrano

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  1. Elisa Sovrano
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Correspondence to Elisa Sovrano.

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This work was supported by the auspices of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: “Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni”.

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Sovrano, E. A negative answer to a conjecture arising in the study of selection–migration models in population genetics. J. Math. Biol. 76, 1655–1672 (2018). https://doi.org/10.1007/s00285-017-1185-7

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  • DOI: https://doi.org/10.1007/s00285-017-1185-7

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