Environmental Engineering Reference

In-Depth Information

(e.g.,
R
lower than a threshold,
R
1
), vegetation establishment is inhibited and only

the bare-soil state (i.e.,

0) is stable; for large values of
R
(e.g.,
R
greater than

another threshold
R
2
), prolonged periods of water stress do not occur, regardless of the

amount of existing biomass; thus the state

v
=

v
=

0 is unstable, whereas

v
=

1isstable.

In intermediate conditions (i.e.,
R
1

R
2
) the system is bistable: Both bare and

completely vegetated soils are stable states of the system. In fact, soil moisture is too

low for the establishment of vegetation in bare soil, whereas in completely vegetated

plots (

≤

R

≤

1) the water available in the soil is sufficient to maintain vegetation cover.

To account for this effect of the feedback, we express the coefficient
c
in Eq. (
4.4
b)

as a function of
R
,
c

v
=

R
1
(Fig.
4.2
).

The cubic polynomial on the right-hand side of Eq. (
4.4
b) induces these three

distinct regimes. It can be shown that this mathematical representation of vegetation

dynamics can be obtained from simple growth-death models of dryland vegetation

driven by soil-moisture dynamics (
D'Odorico et al.
,
2005
;
Borgogno et al.
,
2007
).

To investigate the effect of interannual rainfall fluctuations on vegetation dynamics,

we treat
R
as an uncorrelated random variable, with mean

=

(
R
2
−

R
)

/

(
R
2
−

R
1
)for
R

>

σ
R
,

and gamma distribution
p
(
R
), though the use of other distributions would not alter

the dynamical behavior of the system. As a result of these fluctuations the dynamics

alternate between two different regimes similar to the case discussed in the previous

section: With probability
P
1

R

, standard deviation

=
R
1

0
p
(
R
)d
R
,
R
is lower than
R
1
, and the dynamics

are expressed by Eq. (
4.4
a). With probability (1

−

P
1
),
R
exceeds
R
1
, and the process

is governed by Eq. (
4.4
b) with
c
depending on
R
.

By numerical integration of Eqs. (
4.4
), we find that random interannual fluctua-

tions of
R
stabilize the system around an unstable state of the underlying deterministic

dynamics (region II in Fig.
4.2
).Within a relatively broad range of values of
R
the prob-

ability distribution of

v

exhibits only one mode (Fig.
4.2
) between 0 and 1. This stable

state would not exist without the random forcing, as indicated by the comparison be-

tween the shapes of the deterministic and stochastic potentials
V
(

v

) (inset in Fig.
4.2
).

(Fig.
4.2
, squares and crosses) are the preferential states of the

system and can be also determined analytically through a simplified stochastic model.

To this end, we take
c
as a constant and replace it with its average value
c
+
, conditioned

on
R

The modes of

v

R
1
. Following the approach described in the previous section, we can represent

the temporal dynamics of vegetation by using a stochastic differential equation driven

by DMN:

>

(
c
+
−
v
−

c
+
v

(
c
+
v
+
v
−

c
+
)

d

v

+
v

)

1

dn
v

3

=

f
(

v

)

+

g
(

v

)

ξ

=−
v

+
ξ

,

dn

d

τ

2
−
1

2
−
1

(4.5)

where

ξ

dn
is a zero-mean dichotomous Markov process (see Chapter 2), assuming

values

1
and

2
. As noted in Subsection
2.2.2
, the two functions
g
(

v

)and
f
(

v

)

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