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4 Chemotactic Cell Motion and Biological Pattern Formation
69

The above k-interval is called the unstable range. The associated wavelengths
will increase in amplitude during the evolution of the system, whereas other
wavelengths are damped. Bear in mind that in biological applications γ is a mul-
2
tiple of L !
Exemplary choices for the reaction terms, which exhibit Turing instability,
are

u 2
2
f (u, v) = a − bu + and g(u, v) = u − v [15] ,
v
and
f (u, v) = a − u − h(u, v)and g(u, v) = α(b − v)− h(u, v),
ρuv
with h(u, v):= , [16] ,
1+ u + Ku 2
where a, b, α, ρ and K are positive parameters which have to be chosen appro-
priately to satisfy the above conditions.
In animal tissue, these unstable modes are interpreted to characterise the
patterns, which are amplified during the development of the embryo and which
therefore develop spatial inhomogeneity in departing from the homogeneous
stationary state. Since k may only adopt discrete values, there is only a finite
number of amplified wavelengths.
Unbounded domains B correspond to relevant models in situations where
the size of the embryo is much larger than the pattern to be formed and where
therefore the boundaries cannot play a major role in isolating specific wave-
lengths. The analysis is somewhat simpler in this case. In general there is no
finitebandofamplifiedmodes butone specificwave-number whichisassociated
to the largest eigenvalue. Its pattern will finally emerge.
If the domain B is growing as a function in time, let us say during the growth
of the embryo, then the value γ increases and at certain bifurcation points,
either dominant modes turn into damped ones, i.e. they fall out of the unstable
range, or higher wave-numbers turn from stable to unstable. This process is
called mode selection and is a possible explanation for the complex evolution of
patterns during morphogenesis.

Comments on the Images 4.1–4.8 Various modeling approaches for pattern for-
mation in animal coats, with simulation results, can be found in the Ph.D. the-
sis [11]. Also Turing’s reaction-diffusion model is presented there. For a wealth
3
of information on Turing models for morphogenesis we refer to the webpage .
For a cellular automata model, based on Turing’s reaction-diffusion model,
describing the morphogenesis of zebra coat patterns, we refer to [12].
For images of an artistic interpretation of patterns on human bodies we refer
4
to the webpage .
3
http://www.math.wm.edu/∼shij/
4
http://www.pbase.com/gpfoto/bianco_nero