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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