Steady distribution of the incremental model for bacteria proliferation

Pierre Gabriel (1) , Hugo Martin (2, 3, 4)

1 LMV - Laboratoire de Mathématiques de Versailles
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 MAMBA - Modelling and Analysis for Medical and Biological Applications
4 SU - Sorbonne Université

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 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.

Mots clés: Integral equation; Structured populations; Cell division; Transport equation; Eigenproblem; Long-time asmptotics

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Gabriel, Hugo Martin. Steady distribution of the incremental model for bacteria proliferation. Networks and Heterogeneous Media, 2019, Special issue on mathematical methods in systems biology, 14 (1), pp.149-171. ⟨10.3934/nhm.2019008⟩. ⟨hal-01742140⟩