Analysis of a Population Model Structured by the Cells Molecular Content

TL;DR

This paper investigates the mathematical properties of a cell division model structured by internal variables, solving eigenvalue problems and analyzing long-term population behavior, with extensions to complex parameter cases.

Contribution

It introduces a rigorous analysis of a structured cell division model, including eigenvalue solutions, long-time convergence, and handling degeneracies with regularization techniques.

Findings

Eigenvalue problem solutions under various assumptions

Demonstration of long-time convergence of the model

Extension to models with multiple parameters and nonlinear growth

Abstract

We study the mathematical properties of a general model of cell division structured with several internal variables. We begin with a simpler and specific model with two variables, we solve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-time convergence. The main difficulty comes from natural degeneracy of birth terms that we overcome with a regularization technique. We then extend the results to the case with several parameters and recall the link between this simplified model and the one presented in \cite{CBBP1}; an application to the non-linear problem is also given, leading to robust subpolynomial growth of the total population.