Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 249

p φ

S k

2

3.5

2 k

1 φ

1

-1

(b) (c)

(a)

Figure 6.1. Model ( 6.5 ), a =− 1, s gn = 5, D = 50, simulations at t = 100 time units.

(a) Numerical simulations on a 2D square 128 128 lattice, with periodic boundary

conditions. Black and white tones are used for positive and negative values of the field,
respectively. (b) pdf's of

×

(solid line, numerical simulation; dashed curve, standard
mean field; thick dashed curve, corrected mean field). (c) Azimuthal-averaged power
spectrum (solid curve, numerical simulations; dashed curve, structure function).

v

( a is a negative-valued coefficient). The second term expresses the spatial interac-
tions, where D is a parameter indicating the strength of the spatial coupling. The
third term is a random environmental driver (e.g., plant growth or death associated
with fluctuating levels of soil-water availability), which is here modeled as a white (in
time and space) Gaussian noise with zero mean and correlation, t ) ξ

gn ( r

, t ) ξ

gn ( r

, =

t ), where s gn is the noise intensity. The emergence of spatial
patterns from Eq. ( 6.5 ) was discussed in Section 5.3 . In the absence of noise
the dynamics would be expressed by a linear Fisher's equation and would con-
verge toward the homogeneous deterministic steady state,

r | 2 s gn δ ( | r − ) δ ( t −

v = 0, for any initial

conditions.
To assess pattern formation we use two analytical tools: the mean-field analysis and
the structure function (see Chapter 5). The mean-field analysis allows us to obtain the
analytical expression of the pdf at steady state. We use both the standard mean-field
analysis and the corrected version that provides a better approximation (see Box 5.4).
We find that the steady-state probability distribution of

v is Gaussian with zero mean

and variance equal to s gn / 2 D ). The structure function is a prognostic tool able

to assess the presence of a selected wavelength in the spatial field. In the case of
Eq. ( 6.5 ) the steady-state structure function

(2 +

s gn
2( Dk 2

S st ( k ) = (6.6)

− a )

shows that the dynamics do not select any spatial periodicity, but there is a range of
wave numbers close to zero, which compete to produce multiscale patterns.
Figure 6.1 shows the onset of patterns in the dynamics expressed by Eq. ( 6.5 ).
Only the results pertaining to the simulations at t 100 time units are given; other

results for the same model with different parameters are shown in Fig. 5.16 .As

=