# American Institute of Mathematical Sciences

January  2012, 11(1): 243-260. doi: 10.3934/cpaa.2012.11.243

## A congestion model for cell migration

 1 MAP5, UFR de Mathématiques et Informatique, Université Paris Descartes, 45 rue des Saints-Pères 75270 Paris cedex 06, France, France 2 Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex

Received  February 2010 Revised  September 2010 Published  September 2011

This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We are especially interested in two situations: a single active species moves in a passive matrix (cell migration) with a given desired velocity, and a closed-loop Keller-Segel type model, where the desired velocity is the gradient of a self-emitted chemoattractant.
We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.
Citation: Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure & Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243
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##### References:
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