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4 Chemotactic Cell Motion and Biological Pattern Formation
66

Fig. 4.6. Crocodile Skin

will finally gather at remote positions with only little activator reactant. There
they will successfully inhibit the rise of the activator concentration. In total this
explains that starting from a small perturbation of a homogeneous situation
a pattern of activators and deactivators can finally emerge and stabilise.
We only remark here that the situation is different if the modeling domain is
not bounded. Then inhibitors are likely to diffuse farther and farther away from
activators, finally to infinity.
We shall analyse now the system (4.6), (4.8), (4.7) in the special case, where
the modeling domain is a bounded (one-dimensional) interval,

B = [0, L]with length L> 0 . (4.9)

Most notably we shall determine the so called Turing space, i.e. the set of param-
eter values (the functions f and g and the constants d and γ)for which Turing
instability can be observed.
We assume that (u 0 , v 0 ) is a homogeneous steady state. Hence u 0 and v 0 are
two constants and looking at (4.6) we observe that for this combination of the