Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 259

mechanisms would lead to comparable results. A similar discussion on the role of
random and periodic drivers was presented in Chapter 3 (Subsection 3.2.1.3 )inthe
context of noise-induced transitions in zero-dimensional systems driven by DMN.
The main difference with the stochastic case is that the spatial patterns emerging
from periodic switching are not always time independent but exhibit a pulsating
behavior (i.e., periodic oscillations in time) if the period of the external forcing is of
the same order as the relaxation time to equilibrium in deterministic states 1 and 2
( Buceta and Lindenberg , 2002a ; Buceta et al. , 2002a ).

6.6.2 Random switching between stressed and unstressed conditions in vegetation
In this subsection we follow D'Odorico et al. ( 2006c ) and show how patterns may
emerge as a result of the random switching between two deterministic dynamics,
similar to those discussed in the previous example. The main difference is that in
this case the spatial coupling is expressed by an integral term as in the neural models
discussed in Section 6.2 and Appendix B. We consider in particular the case of dryland
vegetation and show how vegetation patterns could emerge as an effect of random
interannual rainfall fluctuations, which are typically strong in dryland ecosystems
(e.g., Noy-Meir , 1973 ; Nicholson , 1980 ; D'Odorico et al. , 2000 ).
The spatial and temporal variabilities of vegetation biomass (normalized be-

tween 0 and 1) are modeled as a random sequence of two deterministic dynamics
corresponding to (i) drought-induced vegetation decay, and (ii) unstressed vegetation
growth. In both cases, vegetation dynamics at any point r ( x

v

, y ) are expressed as the

sum of two terms accounting for the local dynamics, f 1 , 2 [ v ( r , t )], and spatial inter-

L v , ,v

( r , t )], with the surrounding vegetation existing at all points r in actions, [ ( r t )

the neighborhood of r :
∂v

,

( r t )

( r ,

= f 1 , 2 [ v ( r , t )] + ζL [ v ( r , t ) ,v t )] , (6.15)

∂ t

where is a dimensionless coefficient determining the relative importance of spatial

versus local dynamics. Two different functions, f 1 (

ζ

), are used to describe
the local dynamics: The loss of vegetation occurring in water-stressed conditions is
assumed to be proportional to the existing biomass, and the unstressed growth of

v )and f 2 ( v

v is

v − v

proportional to the existing vegetation biomass and to the available resources 1 :

f 1 [ v ( r , t )] =− α v ( r , t ) , (6.16) 1

f 2 [ v ( r , t )] = α v ( r , t ) [1 − v ( r , t ) , ] , (6.17) 2

where α 1 is the mortality rate per unit v and α 2 is the reproduction rate (e.g., Murray ,

2002 ) of the logistic equation.
Following Lefever and Lejeune ( 1997 ), D'Odorico et al. ( 2006c ) modeled the
spatial interactions as the combined effect of facilitation and competitionmechanisms: