Steady distribution of the incremental model for bacteria proliferation

TL;DR

This paper analyzes a mathematical model of bacteria proliferation based on cell size and size increment, establishing existence and uniqueness of solutions, and exploring long-term behavior under linear growth and self-similar division.

Contribution

It introduces a novel approach to construct solutions for a two-variable cell division model from a one-dimensional fixed point problem, advancing understanding of bacterial growth dynamics.

Findings

Existence and uniqueness of the eigenfunction in weighted L^1 space.

Construction of solutions from a one-dimensional fixed point problem.

Insights into the long-term asymptotic behavior of the model.

Abstract

We study the mathematical properties of a model of cell division structured by two variables, the size and the size increment, in the case of a linear growth rate and a self-similar fragmentation kernel. We first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue 1 from a solution of a certain one dimensional fixed point problem. Then we prove the existence and uniqueness of this fixed point in the appropriate $\mathrm{L}^1$ weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.