Steady distribution of the incremental model for bacteria proliferation

Pierre Gabriel; Hugo Martin

doi:10.3934/nhm.2019008

Article Contents

2019, Volume 14, Issue 1: 149-171. Doi: 10.3934/nhm.2019008

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Steady distribution of the incremental model for bacteria proliferation

Pierre Gabriel^1, and
Hugo Martin^2, ,

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

Received: March 2018

Published: January 2019

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 SKIPPER^AD (number 306321).

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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 $ {\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.

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Mathematics Subject Classification: Primary: 35Q92, 35P05, 45K05, 45P05, 92D25; Secondary: 35A22, 35B40, 35B65.

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Figure 1. schematic representation of the variables on an E. coli bacterium

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Figure 2. 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\}. $

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Figure 3. 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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