On measure-valued solutions for a structured population model with transfers

TL;DR

This paper develops a mathematical framework for measure-valued solutions in a structured population model involving trait transfers between cells, establishing existence, uniqueness, and regularity of solutions.

Contribution

It extends the transfer operator to measure spaces with finite second moments and analyzes the regularity and dynamics of solutions in this setting.

Findings

Existence and uniqueness of measure-valued solutions for the transfer model

Extension of transfer operator to non-negative measures with finite second moment

Analysis of regularity of fixed distributions under the transfer operator

Abstract

We consider a transfer operator where two interacting cells carrying non-negative traits transfer a random fraction of their trait to each other. These transfers can lead to population having singular distributions in trait. We extend the definition of the transfer operator to non-negative measures with a finite second moment, and we discuss the regularity of the fixed distributions of that transfer operator. Finally, we consider a dynamic transfer model where an initial population distribution is affected by a transfer operator: we prove the existence and uniqueness of mild measure-valued solutions for that Cauchy problem.