  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 generalised PKS model are
characterised 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.
