Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 248

resources favors plant growth on the uphill side of vegetated patches, thereby securing
more efficient trapping during subsequent rainstorm events. Runoff and erosion are
therefore viewed as a fundamental mechanism tomaintain striped [Figs. 5.1 (a)-5.1(f )]
configurations over hillsides ( Thiery et al. , 1995 ; Dunkerley , 1997a , 1997b ; Okayasu
and Aizawa , 2001 ; Sherrat , 2005 ; D'Odorico et al. , 2007a ; Barbier et al. , 2008 ). Thus
rainfall onto an unvegetated area generates overland flow, which transports water
in the downhill direction until it reaches a vegetated area, where it infiltrates the
ground and is taken up by vegetation. The relatively moist soil on the uphill side
of a stripe creates opportunities for uphill expansion of the vegetation band at the
expenses of the downhill side, which remains deprived of the resources necessary
for vegetation survival. The overall dynamics lead to the uphill migration of vege-
tated bands ( Sherrat , 2005 ). A similar mechanism can explain the banded patterns
of trees in Tierra del Fuego [Argentina; see Fig. 5.1 (h)], where a sawtooth pattern
of tree heights is observed in the wind direction ( Puigdefabregas et al. , 1999 ). Taller
trees provide more protected favorable conditions for seedling establishment and tree
growth in the leeward direction. At the same time, the strong winds uproot and kill
the taller upwind trees, leading to an overall downwind migration of the sawtooth
pattern.

6.4 Patterns induced by additive noise
The ability of noise to induce ordered states in dynamical systems is commonly ex-
plained as an effect of multiplicative noise acting in conjunction with nonlinearities.
However, in Chapter 5 we showed how additive noise can also play a crucial role
by stabilizing short-lived patterns emerging in deterministic dynamics. In this section
we apply a simple stochastic model [based on Eq. ( 5.29 )] of noise-induced pattern
formation driven by additive noise. This model (see also Section 5.3 ) does not in-
voke any nonlinearity. It involves only three linear components: (i) a deterministic
local dynamics term, which linearly damps the system to zero; (ii) an additive noise
able to hamper this tendency of the deterministic dynamics to converge to zero; and
(iii) a linear (diffusive) spatial coupling, which provides spatial coherence. This model
provides a possible noise-induced mechanism for pattern formation in dryland vegeta-
tion. In fact, we can model the dynamics of vegetation biomass as the result of a

linear decrease, acting in conjunction with spatial interactions modeled as linear
diffusion, and an additive stochastic forcing representing a random environmental
driver:

v

∂v
∂

2

t =

a v + D ∇ v + ξ . (6.5) gn

The first term, a , accounts for the local deterministic decay of plant biomass that

would occur in the absence of spatial interactions or environmental fluctuations

v