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4 Chemotactic Cell Motion and Biological Pattern Formation
68

Fig. 4.8. Galapagos giant turtle

We know that linear diffusive systems as (4.6) with zero reaction nonlinear-
ities act (componentwise) on eigenvectors of the Laplacian by damping them
with a certain exponential rate λ < 0. Such solutions are then given by a product
exp(λt)w k (x). Thus after linearizing (4.6) at (u 0 , v 0 )weexpectthatperturba-
c

2
tions of the steady state (u−u 0 , v−v 0 ) can be written as k k exp(λ(k )t)w k (x),
where the components of c k are the Fourier coefficients of u I − u 0 and v I − v 0
respectively.
2
If λ(k ) < 0 then the eigenmode with wave number k is damped, but if for
2
some wave number k = 0wehave λ(k ) > 0, then the respective component
of the solution blows up as t →∞ and we call the the system (4.6) (linearly)
unstable at (u 0 , v 0 ).
2
In fact, we even can compute explicitly the largest valuesλ(k ), which is called
dispersion relation. Doing so it turns out that we only have instability if
2
a> 0and a −4db > 0
where a := df u + g v and b := f u g v − f v g u . Most notably the first inequality implies
2
that the diffusion coefficient satisfies d> 1. Furthermore it turns out that λ(k )
has positive real part only for those wave numbers k which satisfy
√ √
2
a − a −4db a + a −4db
2
γ <k < γ ,
2
2d 2d