Page 74 -

4 Chemotactic Cell Motion and Biological Pattern Formation
67

Fig. 4.7. Australian saltwater crocodile, skin pattern

two concentrations we have zero production and zero decay,
f (u 0 , v 0 ) = g(u 0 , v 0 ) = 0 . (4.10)

In the sequel we shall analyse the equations (4.6) near this steady state. If we
only look at spatially homogeneous, but time-dependent solutions of (4.6) and
if we simplify this problem even more by linearisation at the steady state, then
we conclude that it is (linearly) stable at (u 0 , v 0 )if

f u + g v < 0and f u g v − f v g u > 0 . (4.11)
Note that the partial derivatives of f and g here and in the sequel are evaluated
at (u 0 , v 0 ). Typically, in the theory of ordinary differential equations linear sta-
bility implies local (nonlinear) stability meaning that solutions of the nonlinear
problem (4.6), which are constant in space and which are sufficiently close to
the steady state (u 0 , v 0 ), will not depart from the steady state as time increases.
TheprerequisiteofTuringinstabilityisinstabilityinthepresenceofspatialin-
homogeneity of the concentrations. To understand this we look at eigenfunctions
2
of the Laplacian such that w xx = −k w and such that the boundary conditions
w x (0) = w x (L) = 0 are satisfied. The functions w = w k (x) are then multiples of
cos(nπx/L), where n ∈ Z and k := nπ/L.The value k is called the wave number.