Small populations corrections for selection-mutation models

TL;DR

This paper introduces a modification to existing population models that accounts for small populations, addressing abnormal behaviors caused by strong selection and small mutations in structured population dynamics.

Contribution

It proposes a new correction to classical integro-differential models that better captures small population effects and improves model accuracy.

Findings

Addresses abnormal behaviors in classical models due to small populations

Introduces a correction mechanism for small population effects

Enhances the understanding of population concentration phenomena

Abstract

We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of Diekmann, Jabin, Mischler, and Perthame, this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.