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4 Chemotactic Cell Motion and Biological Pattern Formation
56

leads to the reaction-diffusion equation:
S t = div(D 1 grad S)+ g(r, S) . (4.2)
Here D 1 stands for the (positive) diffusivity of the chemical and g for its pro-
duction/degradation rate, i.e. g> 0 describes production of the chemical and
g< 0 its degradation. The Keller–Segel model is thus comprised of the coupled
parabolic system (4.1) and (4.2), supplemented by appropriate conditions for r
and S on the boundary of the modeling domain B (e.g. the Petri dish bound-
aries) and by initial conditions for r and S. The classical Keller–Segel model
refers to equations (4.1) and (4.2), with constant and positive diffusivities and
chemotactic sensitivity and with the linear production/degradation model:
g(r, S):= dr − eS , (4.3)

where d and e are positive constants. This classical Keller-Segel model, with
appropriately fitted parameters, is often sufficient to describe real chemotactic
processes with good qualitative and reasonable quantitative agreement.
In many cases, however, it is of great importance to include specific fea-
tures of individual cells, to deal with stochasticity [8] or to employ microscopic

Fig. 4.1. Giraffe fur pattern