Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

Analysis of a model of cell crawling migration

Pages: 1589 – 1611



Thomas Lepoutre (Université Lyon, Inria, Université Claude Bernard, Lyon, France; CNRS UMR5208, Institut Camille Jordan, Villeurbanne, France)

Nicolas Meunier (Laboratoire de Mathématiques et Modélisation d’Évry (LaMME), UMR 8O71 CNRS & Université d’Évry Val d’Essonne, Evry, France)


We introduce and study a model for motility of cells on substrate. The cell is 1D, inextensible and it contains a diffusive back-polarity marker, which satisfies a non-linear and non-local parabolic equation of Fokker–Planck type with attachment/detachment at the boundary. The idea behind the model is a quadratic nonlinear coupling: the marker is advected by the cell velocity, which is itself driven by a front-rear imbalance in marker. We show that it is of bistable type, provided that the coupling between the asymmetry of the marker and the cell velocity is sufficiently strong. In such a case we prove the non-linear stability of the largest steady state, for large initial data. In the weak coupling case we prove the convergence of the molecular concentration towards the Gaussian state.


cell polarisation, cell migration, global existence, asymptotic convergence

2010 Mathematics Subject Classification

35Q92, 92B05, 92C17

Received 13 January 2021

Received revised 1 September 2021

Accepted 17 January 2022

Published 14 September 2022