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We present a discrete model of chemotaxis whereby cells responding to achemoattractant are seen as individual agents whose movement is describedthrough a set of rules that result in a biased random walk. In order to takeinto account possible alterations in cellular motility observed at high celldensities (i.e. volume-filling), we let the probabilities of cell movement bemodulated by a decaying function of the cell density. We formally show that ageneral form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxiscan be formally derived as the appropriate continuum limit of this discretemodel. The family of steady-state solutions of such a generalised PKS model arecharacterised and the conditions for the emergence of spatial patterns arestudied via linear stability analysis. Moreover, we carry out a systematicquantitative comparison between numerical simulations of the discrete model andnumerical solutions of the corresponding PKS model, both in one and in twospatial dimensions. The results obtained indicate that there is excellentquantitative agreement between the spatial patterns produced by the two models.Finally, we numerically show that the outcomes of the two models faithfullyreplicate those of the classical PKS model in a suitable asymptotic regime.

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