Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 250

expected from the structure-function analysis (i.e., no dominant wavelength different
from zero exists), no clear periodicity is visible but many wavelengths are present to
produce multiscale patterns, which become steady after a few time units. Moreover,
no phase transition occurs because the pdf remains unimodal and with zero mean.
In Fig. 6.1 we compare the numerical simulations with the analytical results for the
steady-state structure function and pdf of

φ

from mean-field analysis. For the pdf, the
corrected mean-field prediction, obtained with a halved diffusion coefficient (see Box
5.4) (thick dashed curve overlaid by the solid curve) is more precise than the results
of the standard mean-field analysis (thin dashed curve), which tends to underestimate
the variance of the distribution. The numerical simulations show that the multiscale
patterns emerging from Eq. ( 6.5 ) persist in the steady state of the system and do not
disappear. We stress again that in these dynamics pattern formation is only due to the
presence of an additive noise acting in conjunction to a spatial coupling. Indeed, the
deterministic local dynamics tend to damp the field variable to zero, and the spatial
coupling is unable to destabilize this uniform steady state of the system. Thus at
steady state the deterministic dynamics underlying Eq. ( 6.5 ) do not exhibit pattern
formation. The properties of stochastic dynamics ( 6.5 ) were discussed in detail in
Chapter 5 (Section 5.3 ). It is interesting to notice that the cooperation between noise
and spatial coupling leads to the emergence of temporally stable spatial coherence, as
indicated by the power-spectrum structure, which strongly differs from the spatially
uncorrelated initial condition. We recall that in these dynamics additive noise is able
to keep the system away from the homogeneous state, 0, and the spatial coupling

induces spatial coherence. Therefore, in this system, pattern formation is clearly noise
induced and arises from a synergism between additive noise and spatial coupling.
The mechanism presented in this section differs from the effect of noise-induced
stabilization of transient patterns investigated by Butler and Goldenfeld ( 2009 ). In
fact, Butler and Goldenfeld ( 2009 ) focused on a predator-prey system exhibiting
Turing instability and showed how additive noise can act on the system as a precursor
of a phase transition, which expands the region of the parameter space inwhich pattern-
forming Turing instability may occur. Conversely, in the dynamics presented in this
section additive noise does not play the role of a precursor of a phase transition close
to a (deterministic) bifurcation point. In fact, no bifurcation exists in the deterministic
counterpart of these dynamics (see Chapter 5). Rather, in this case the additive noise
unveils the ability of the deterministic system to induce transient periodic patterns,
and hampers their disappearance.
To apply Eq. ( 6.5 ) to the case of vegetation dynamics we consider a system forced by
an additive noise with mean different from zero (to account for random environmental
drivers with means different from zero):

φ =

∂v
∂

2 2

v + ξ gn ,

t =− v +

D ∇ v + ξ + μ =− v + D ∇ (6.7) gn