Steady distribution of the incremental model for bacteria proliferation

Pierre Gabriel; Hugo Martin; Pierre Gabriel; Hugo Martin

doi:10.3934/nhm.2019008

Networks and Heterogeneous Media

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
,},
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

Received: 01 March 2018
Primary: 35Q92, 35P05, 45K05, 45P05, 92D25; Secondary: 35A22, 35B40, 35B65

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.
- Structured populations,
- cell division,
- transport equation,
- eigenproblem,
- long-time asmptotics,
- integral equation
Citation: Pierre Gabriel, Hugo Martin. Steady distribution of the incremental model for bacteria proliferation[J]. Networks and Heterogeneous Media, 2019, 14(1): 149-171. doi: 10.3934/nhm.2019008

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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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