Environmental Engineering Reference
In-Depth Information
v
R
R
R
Figure 4.3. Statistically stable states (i.e., modes) as functions of the coefficient of
variation (CV) of the random driver
R
(after
Borgogno et al.
,
2007
).
3
when
are determined in such a way that d
v/
d
t
=−
v
ξ
dn
=
1
and d
v/
d
t
=
v
(1
−
c
+
)when
v
)(
v
−
ξ
dn
=
2
. The transition probabilities between the two states of
ξ
dn
are
k
1
=
(1
−
P
1
)and
k
2
=
P
1
. Moreover, the states
1
and
2
need to satisfy
the condition
dn
to be a zero-mean process. The analytical
solution of (
4.5
) provided in Chapter 2 is here used to calculate the modes of
1
k
2
+
2
k
1
=
0for
ξ
v
and the stochastic potential
) associated with Eq. (
4.5
)(Fig.
4.2
, dotted lines).
This analysis shows that there is a range of values of
V
(
v
in which the stochastic
dynamics have only one preferential state while the stable states
R
v
=
0and
v
=
1of
the deterministic dynamics become unstable.
This finding has important ecological implications because climate fluctuations
are typically considered as a source of disturbance and are believed to control the
transitions between preferential states in bistable dynamics. The emergence of an
intermediate noise-induced stable configuration suggests that rainfall fluctuations
unlock the system from these preferential states and stabilize the dynamics halfway
between bare-soil and full-vegetation cover conditions.
To stress how the emergence of this intermediate stable state can be explained as a
noise-induced effect,
Borgogno et al.
(
2007
) investigated the dependence of the stable
states of the stochastic dynamics on the coefficient of variation (CV), CV
,
of the interannual rainfall fluctuations. Using a model similar to (
4.4
), it was found
that, as the CV increases, the width of the interval of values of
=
σ
R
/
R
R
in which the system
is bistable decreases. For a critical value of CV (CV
15, in Fig.
4.3
) the width of
the bistability range is zero. For larger values of CV, noise-induced stability emerges.
For larger values of the CV, intermediate noise-induced statistically stable states
may even exist within a range of values of
=
0
.
R
wider than the bistability range
[
R
1
,
R
2
], of the underlying deterministic dynamics (Fig.
4.2
). This range broadens
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