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Dirac Concentrations in a Chemostat Model of Adaptive Evolution(In honor of the immense scientific influence of Ham Brezis)

Dirac Concentrations in a Chemostat Model of Adaptive Evolution(In honor of the immense scientific influence of Ham Brezis)
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摘要 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. 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.
机构地区 Sorbonne Universits
出处 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2017年第2期513-538,共26页 数学年刊(B辑英文版)
基金 supported by ANR-13-BS01-0004 funded by the French Ministry of Research(ANR Kibord)
关键词 Adaptive evolution Asymptotic behaviour Chemostat DIRAC concentrations Hamilton-Jacobi equations Lotka-Volterra equations Viscosity solutions Adaptive evolution Asymptotic behaviour Chemostat Dirac concentrations Hamilton-Jacobi equations Lotka-Volterra equations Viscosity solutions
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