From a discrete model of chemotaxis with volume-filling to a generalized Patlak–Keller–Segel model
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We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak–Keller–Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.
Bubba , F , Lorenzi , T & Macfarlane , F R 2020 , ' From a discrete model of chemotaxis with volume-filling to a generalized Patlak–Keller–Segel model ' , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , vol. 476 , no. 2237 , 20190871 . https://doi.org/10.1098/rspa.2019.0871
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Copyright © 2020 The Author(s). Published by the Royal Society. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1098/rspa.2019.0871
DescriptionFunding: The authors gratefully acknowledge support of the project PICS-CNRS no. 07688. F.B. acknowledges funding from the European Research Council (ERC, grant agreement No. 740623) and the Université Franco-Italienne.
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