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Communications on Pure and Applied Analysis (CPAA)
 

A congestion model for cell migration

Pages: 243 - 260, Volume 11, Issue 1, January 2012      doi:10.3934/cpaa.2012.11.243

 
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Julien Dambrine - MAP5, UFR de Mathématiques et Informatique, Université Paris Descartes, 45 rue des Saints-Pères 75270 Paris cedex 06, France (email)
Nicolas Meunier - MAP5, UFR de Mathématiques et Informatique, Université Paris Descartes, 45 rue des Saints-Pères 75270 Paris cedex 06, France (email)
Bertrand Maury - Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France (email)
Aude Roudneff-Chupin - Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France (email)

Abstract: 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.

Keywords:  Congestion, chemotaxis, aggregation, optimal transport.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: February 2010;      Revised: September 2010;      Available Online: September 2011.

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