Dirac concentrations in a chemostat model of adaptive evolution

Abstract : We consider parabolic systems of Lotka-Volterra type that describe 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. We review different methods aimed at showing the convergence of the solutions to a moving Dirac mass. Setting first two frameworks based on weak or strong regularity assumptions in which we study the concentration of the solution, we state BV estimates in time on appropriate quantities and derive a constrained Hamilton-Jacobi equation to identify the Dirac locations.
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Alexander Lorz, Benoît Perthame, Cécile Taing. Dirac concentrations in a chemostat model of adaptive evolution. Chinese Annals of Mathematics - Series B, Springer Verlag, 2017. ⟨hal-01255449⟩

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