Analysis of a model of cell crawling migration

International audienceWe 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