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while consumer variability increases as the spectral exponent
β
increases from zero
(white noise) to positive values (colored noise).
4.6 Environmental systems forced by other types of noise
Noise-induced transitions may emerge also in processes forced by other types of
random drivers. These transitions are typically investigated through numerical simu-
lations rather than through exact solutions of stochastic differential equations, either
because of the lack of a theory for the solution of these equations or because of the in-
ability of the dynamics to be expressed through stochastic differential equations. Here
we present the case of noise-induced transitions emerging in the distribution of aver-
age seasonal soil moisture as an effect of interannual rainfall fluctuations (
D'Odorico
et al.
,
2000
). We consider the daily dynamics of soil moisture
s
, expressed by (
4.19
).
To simplify this analysis we assume that no feedback exists between soil moisture and
precipitation (i.e.,
independent of soil moisture). As noted, the analytical solution
of (
4.19
) provides the probability distribution of
s
[Eq. (
4.22
)] and of its moments
(not shown). We use the average value of soil-moisture during the growing season,
λ
depends on the parame-
ters representative of vegetation, soils, and rainfall regime (
Rodriguez-Iturbe et al.
,
1999b
). Most of these parameters either remain constant with time or undergo small
variability, except for those characterizing the rainfall regime, i.e., the parameters
λ
s
, as an indicator of the seasonal soil-water content.
s
expressing the average rainstorm frequency and depth. Interannual climate
fluctuations determine year-to-year changes in these parameters, particularly in arid
and semiarid regions, which are known for the higher variability and unpredictability
of precipitation (
D'Odorico et al.
,
2000
;
D'Odorico and Porporato
,
2006
).
To evaluate the impact of these interannual rainfall fluctuations on the year-to-
year variability of
and
α
s
, we treat the seasonal values of
λ
and
α
as random variables
that vary from year to year. The effect of fluctuations in
λ
and
α
on the distribution
of
is an interesting case of superstatistics (
Beck
,
2004
) that can be solved with
a Monte Carlo procedure, i.e., sampling values of
s
from their respective
distributions (assuming that these two variables are independent) and calculating the
value of
λ
and
α
is obtained
when this procedure is repeated several times. The distribution of these values of
s
associated with those values. A population of values of
s
.
Figure
4.26
shows an example of the results of this analysis: It is observed that for
small variability (i.e., CV) of the parameters
s
is the derived distribution of
s
as a function of the distributions of
λ
and
α
, the distribution is unimodal.
However, as variability increases a bimodal behavior emerges. Because bimodality is
observed when the noise intensity increases above a critical value, this is an example
of noise-induced transition.
The results in Fig.
4.26
show that the system tends to select two preferential
states and to switch between them as a result of interannual fluctuations in rainfall
λ
and
α
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