Modelling physiologically structured populations: renewal equations and partial differential equations

TL;DR

This paper studies the long-term behavior of measure-valued solutions to linear renewal equations modeling structured populations, linking these to PDEs and applying results to biological models of cell growth and immunity.

Contribution

It introduces a regularisation property of the kernel that connects the asymptotic behavior of measure solutions to their absolutely continuous parts, extending understanding of structured population models.

Findings

Large time behavior deduced from kernel regularisation

Connection established between renewal equations and PDEs

Applied to models of cell growth and immunity

Abstract

We analyse the long term behaviour of the measure-valued solutions of a class of linear renewal equations modelling physiologically structured populations. The renewal equations that we consider are characterised by a regularisation property of the kernel. This regularisation property allows to deduce the large time behaviour of the measure-valued solutions from the asymptotic behaviour of their absolutely continuous, with respect to the Lebesgue measure, component. We apply the results to a model of cell growth and fission and to a model of waning and boosting of immunity. For both models we relate the renewal equation (RE) to the partial differential equation (PDE) formulation and draw conclusions about the asymptotic behaviour of the solutions of the PDEs.