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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