Environmental Engineering Reference

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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:

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