On selection dynamics for a nonlocal phenotype-structured model

TL;DR

This paper analyzes the long-term behavior of a phenotypic-structured model without phenotypic changes, showing that solutions tend to a Dirac mass at the optimal trait, with numerical insights into trait localization.

Contribution

It provides a mathematical analysis of selection dynamics in a nonlocal phenotype model, revealing conditions for convergence to optimal traits and blow-up scenarios.

Findings

Solutions converge to a Dirac mass at the fitness peak.

If the peak is outside the initial support, solutions blow up near the boundary.

Numerical results help identify the position of trait centers.

Abstract

This paper is devoted to the analysis of the long-time behavior of a phenotypic-structured model where phenotypic changes do not occur. We give a mathematical description of the process in which the best adapted trait is selected in a given environment created by the total population. It is exhibited that the long-time limit of the unique solution to the nonlocal equation is given by a Dirac mass centered at the peak of the fitness within or at the boundary of the region where the initial data is positive. Specially, If the peak of the fitness can't be in the support of the solution, then the infinite time blow-up of the solution occurs near the boundary of the region where the solution is positive. Moreover, our numerical results facilitate a deeper understanding of identifying the position of the centers.