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Volume Filling Effect in Modelling Chemotaxis

Published online by Cambridge University Press:  03 February 2010

D. Wrzosek
D. Wrzosek*
Affiliation:
Institute of Applied Mathematics and Mechanics, Warsaw University Banacha 2, 02-097 Warszawa, Poland
*
* E-mail: darekw@mimuw.edu.pl
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Abstract

The oriented movement of biological cells or organisms in response to a chemical gradientis called chemotaxis. The most interesting situation related to self-organizationphenomenon takes place when the cells detect and response to a chemical which is secretedby themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) manyparticularized models have been proposed to describe the aggregation phase of thisprocess. Most of efforts were concentrated, so far, on mathematical models in which theformation of aggregate is interpreted as finite time blow-up of cell density. In recentlyproposed models cells are no more treated as point masses and their finite volume isaccounted for. Thus, arbitrary high cell densities are precluded in such description and athreshold value for cells density is a priori assumed. Different modelingapproaches based on this assumption lead to a class of quasilinear parabolic systems withstrong nonlinearities including degenerate or singular diffusion. We give a survey ofanalytical results on the existence and uniqueness of global-in-time solutions, theirconvergence to stationary states and on a possibility of reaching the density threshold bya solution. Unsolved problems are pointed as well.

Keywords

chemotaxis equationsquasilinear parabolic equationsLyapunov functionaldegenerate diffusionno-flux boundary conditionsemi-group theorycompactness methodattractorconvergence to steady statesnonlinear elliptic problem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 1: Cell migration , 2010 , pp. 123 - 147
Copyright
© EDP Sciences, 2010

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