Measure solutions for some models in population dynamics

TL;DR

This paper establishes well-posedness of solutions for population dynamics models with measure initial data, using optimal transport techniques to handle measure-valued stationary states more naturally.

Contribution

It provides a simplified, unified proof of well-posedness for structured population models with measures, applicable to a broad class of models.

Findings

Well-posedness proven for measure-valued solutions

Techniques based on optimal transport distances

Applicable to various population dynamics models

Abstract

We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and not $L^1$ functions, so the measures are a more natural space to study their dynamics. Our techniques are based on distances between measures appearing in optimal transport and common arguments involving Picard iterations. These tools provide a simplification of previous approaches and are applicable or adaptable to a wide variety of models in population dynamics.