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4 Chemotactic Cell Motion and Biological Pattern Formation
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of production/decay is a multiple of a constant γ which depends on the ‘size’ of
the modeling domain B. To complete the model, we assume that we know the
concentrations of the chemicals at time t = 0

u(., t = 0) = u I and v(., t = 0) = v I (4.7)
and that, starting from the initial point in time, the two chemicals can neither
flow out of the domain B, nor is any of the two chemicals added. In mathematical
terms we express this by the homogeneous Neumann boundary conditions

ν · grad u = ν · grad v = 0on ∂B , (4.8)

where ∂B is the boundary of the domain B and ν is the unit outward normal
vector to this boundary. Note that the formulation of production/decay using the
two functions f (u, v)and g(u, v) allows that both chemicals may influence their
ownproduction/decayaswellasproduction/decayoftheotherone.Nevertheless
fromnowonweshallinterpretthechemicalwithconcentrationuastheactivator,
i.e. it is likely to increase morphogen production, whereas we shall interpret
the chemical associated to the concentration v as the deactivator. Hence the
presence of the latter chemical is likely to reduce production or even stimulates
consumption of the morphogen.
The central idea of Turing instability is to construct systems of chemicals
with the following property: The concentrations of the chemicals are linearly
stable if they are homogeneous. Clearly, variations of concentrations necessarily
lead to diffusive effects. If the diffusion coefficients are different (i.e. d = 1), then
diffusion shall lead to instability (“diffusion driven instability”) in the sense that
initial variations in concentrations get amplified.
This was a novel concept since in the mathematical field of partial differential
equations (PDEs) diffusion is usually considered to be a stabilising process, just
like variations in temperature in general get equilibrated and inhomogeneous
concentrations are expected to smooth out.
The mechanism which in the present case leads to instability works as fol-
lows: Imagine regions where, for whatever reason, there is a high concentration
of activators and where, as a consequence of activation, lots of inhibitors are pro-
duced. The Turing mechanism is based on the fact that the inhibitor substance
diffuses faster than the activator chemical. So in these regions the deactivators
will diffuse away quickly and will not be able to reduce the concentration of
activators.
Most notably in situations, where the size of the modeling domain is finite
and where chemicals can neither leave nor enter the domain, the inhibitors

Fig. 4.5. Zebra coat pattern