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4 Chemotactic Cell Motion and Biological Pattern Formation
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phase-space models replacing the macroscopic Fokker–Planck equation (4.1),
similarly to the framework of the Boltzmann equation of gas dynamics (the
macroscopic Euler or Navier–Stokes equations correspond to the Fokker–Planck
equation (4.1) in this comparison!). A presentation of the corresponding model
hierarchy, the connections of the different PDE models in the hierarchy and
a collection of references on the mathematical analysis of kinetic and macro-
scopic chemotaxis models can be found in [3]. The scaling limit of a phase space
chemotaxis model leading to the Keller–Segel model was rigorously analysed
in [1].
For what follows we consider the classical Keller–Segel model consisting of
(4.1), (4.2) and (4.3) with the additional assumption e = 0(no degradation of
the chemical and d and D 1 very large, such that the parabolic reaction diffusion
equation (4.2) can be approximated by the linear elliptic equation

−ΔS = r (4.4)

n
(after appropriate rescaling). We assume that (4.1) and (4.4) are posed on R ,
n = 1, 2 or 3 and look for solutions such that r decays to 0 as |x| tends to
infinity. This nonlinear, nonlocally coupled elliptic-parabolic system of partial

Fig. 4.2. Kudu coat