From a discrete model of chemotaxis with volume-filling to a generalized Patlak–Keller–Segel model

Federica Bubba, Tommaso Lorenzi*, Fiona R. Macfarlane

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
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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.

Original languageEnglish
Article number20190871
Number of pages19
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2237
Early online date13 May 2020
Publication statusPublished - 27 May 2020


  • Chemotaxis
  • Discrete models
  • Generalized Patlak–Keller–Segel model
  • Volume-filling


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