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Dirac concentrations in a chemostat model of adaptive evolution

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Abstract

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies.

Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions UMR CNRS 7598, Sorbonne Universits, UPMC Univ Paris 06, Inria, F75005, Paris, France

    Alexander Lorz, Benoît Perthame & Cécile Taing

Authors
  1. Alexander Lorz
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  2. Benoît Perthame
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  3. Cécile Taing
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Corresponding author

Correspondence to Alexander Lorz.

Additional information

In honor of the immense scientific influence of Haïm Brezis

This work was supported by ANR-13-BS01-0004 funded by the French Ministry of Research (ANR Kibord).

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Lorz, A., Perthame, B. & Taing, C. Dirac concentrations in a chemostat model of adaptive evolution. Chin. Ann. Math. Ser. B 38, 513–538 (2017). https://doi.org/10.1007/s11401-017-1081-x

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  • DOI: https://doi.org/10.1007/s11401-017-1081-x

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