Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 513–538 | Cite as

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.

Keywords

Adaptive evolution Asymptotic behaviour Chemostat Dirac concentrations Hamilton-Jacobi equations Lotka-Volterra equations Viscosity solutions 

2000 MR Subject Classification

35B25 35K57 47G20 49L25 92D15 

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Alexander Lorz
    • 1
  • Benoît Perthame
    • 1
  • Cécile Taing
    • 1
  1. 1.Laboratoire Jacques-Louis Lions UMR CNRS 7598Sorbonne Universits, UPMC Univ Paris 06, InriaParisFrance

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