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Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2029–2052 | Cite as

Evolutionary Branching via Replicator–Mutator Equations

  • Matthieu AlfaroEmail author
  • Mario Veruete
Article

Abstract

We consider a class of non-local reaction–diffusion problems, referred to as replicator–mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schrödinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation parameter for evolutionary branching to occur.

Keywords

Evolutionary genetics Dynamics of adaptation Branching phenomena Long time behaviour Schrödinger eigenelements 

Mathematics Subject Classification

92B05 92D15 35K15 45K05 

Notes

Acknowledgements

The authors are grateful to Rémi Carles for suggesting Lemma 2.5 and continuous encouragement, and to Bernard Helffer for invaluable comments and essential remarks. They also thank Alexandre Eremenko and Christian Remling for very valuable discussions. Mario Veruete is grateful for the support of the National Council for Science and Technology of Mexico.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMAGUniversité de MontpellierMontpellierFrance

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