# American Institute of Mathematical Sciences

March  2019, 14(1): 149-171. doi: 10.3934/nhm.2019008

## Steady distribution of the incremental model for bacteria proliferation

 1 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France 2 Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Sorbonne université, 4 place Jussieu, 75005 Paris, France

* Corresponding author: hugo.martin@sorbonne-universite.fr

Received  March 2018 Published  January 2019

Fund Project: P. Gabriel has been supported by the ANR project KIBORD, ANR-13-BS01-0004, funded by the French Ministry of Research. H. Martin has been supported by the ERC Starting Grant SKIPPERAD (number 306321)

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 ${\rm{L}} ^1$ 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.

Citation: Pierre Gabriel, Hugo Martin. Steady distribution of the incremental model for bacteria proliferation. Networks & Heterogeneous Media, 2019, 14 (1) : 149-171. doi: 10.3934/nhm.2019008
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##### References:
schematic representation of the variables on an E. coli bacterium
Left: simulation of the function $f$ by the power method with $B(a) = \frac{2}{1+a}{1}_{\{1\leq a\}}$ and $\mu(z) = 2\delta_{\frac{1}{2}}(z).$ Right: level set of the density $N(a,x)$ obtained from this function $f.$ Straight line: the set $\{x = a+1\}.$
Domain of the model, with respect to the choice of variables to describe the bacterium. Grey: domain where the bacteria densities may be positive. Arrows: transport. Left: size increment/size. Right: size increment/birth size. Dashed: location of cells of size $x_1$
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