On selection dynamics for a nonlocal phenotype-structured model

This paper is devoted to the analysis of the long-time behavior of a phenotype-structured model in which phenotypic changes do not occur. We present a mathematical description of the process through which the best adapted trait is selected in a given environment created by the total population. It is demonstrated that the long-time limit of the unique solution to the nonlocal equation is represented by a delta Dirac concentrated on the set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> ’s where the peak of the fitness is attained. Furthermore, our numerical experiments provide a sufficient criterion to identify the positions of the peak. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature.