Deterministic mechanisms of pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 286

and g ( u ,v ) around ( u 0 ,v 0 ), and seek solutions of the linearized equations in the

form of

u = u + u 0 ,

v = v + v ˆ . (B.23)

0

We obtain

∂ u

p ∂

u

= + v +

x ,

f u u f v ˆ

∂ t ∂

∂ v ˆ

=

g u u

+ v,

g v ˆ (B.24)

∂ t

where f u = ∂ f ( u ,v ) /∂ u , f v = ∂ f ( u ,v ) /∂v , g u = ∂ g ( u ,v ) /∂ u ,and g u = ∂ g ( u ,v ) /

∂v . The solution of equation set ( B.24 ) can be expressed as a sum (or integral sum

in spatially infinite domains) of Fourier modes u k = U k exp( γ t + i k · r )andˆ v k =

V k exp( r ), where U k and V k are the Fourier coefficients of the k th mode.

Because Eqs. ( B.24 ) need to be satisfied for each mode k ,wehave

γ t + i k ·

γ U k = f u U k + f v V k + ipU k k x ,

γ V k = g u U k + g v V k . (B.25)

Nontrivial solutions of equation set ( B.25 ) exist when its determinant is zero:

2

γ − ( f u + g v + ipk x ) γ + f u g v − f v g u + ipk x g v = 0 . (B.26)

γ

Notice how in this case is a complex number. The emergence of instability requires

the real part of γ to be positive. Traveling-wave patterns require the imaginary part

of to be different from zero. It has been noticed ( Rovinsky and Menzinger , 1992 )

that Eq. ( B.26 ) does not lead to the selection of any finite value for the most unstable
wave number in that

γ

is a monotonically increasing function of k . Thus the wave-
number interval of the unstable modes has no upper bound. However, the addition to
Eqs. ( B.25 ) of a diffusion term to either the first or the second equation [or to both, as
in Eqs. ( B.22 )] imposes an upper bound to the range of unstable modes. In this case
the most unstable mode corresponds to a finite value of the wave number.

γ

B.4.1 Case study: A differential-flow ecological model of pattern formation
We present, as an example of differential-flow instability, a model developed to study
the formation of patterns in young mussel beds ( van de Koppel et al. , 2005 ). This
model was used by Borgogno et al. ( 2009 ) to describe a system involving trees
or grasses. Two (dimensionless) state variables, representing nutrient concentration