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with
g
(
H
)

=−

H
and constant values of
h

=

1. The steady-state pdf of
H
(
D'Odorico

and Fagherazzi
,
2003
)is

=
C
(
H
+
b
)
−
a
−
1
(1

−
H
)
a
−
1

p
(
H
)

,

(4.18)

b
)
b
λ/
a
. We can obtain the deterministic counterpart of this

process by replacing the stochastic rate of erosion with its mean value

where
C

=

(

λ/

a
)(1

+

H
. Thus the

underlying deterministic process has only one stable state and no unstable state within

the [0, 1] interval [Fig.
4.15
(c), dashed curve with

λ

1], and the stochastic dynamics

have a U-shaped distribution [Fig.
4.15
(d)], suggesting that there is a statistically un-

stable state between 0 and 1; 0 and 1 are the stable states of the stochastic process. This

result is noise induced and resembles the case of noise-induced bistability discussed

in the previous two sections. The Poisson noise is able to convert a deterministic

system with only one stable state into a stochastic one with two preferential states and

an intermediate unstable state. This effect is due to the term

α
=

g
(
H
) in the equation,

λα

i.e., to the multiplicative character of noise (Eq.
2.69
).

4.3.4 Stochastic soil-moisture dynamics

Stochastic differential equations driven by white shot noise have been used in recent

years to model the soil-water balance (
Rodriguez-Iturbe et al.
,
1999b
;
Laio et al.
,

2001
), providing a theoretical framework to investigate the effect of climate, soils, and

vegetation on soil-moisture dynamics in the root zone. The probability distributions

of the soil-water content determined with this model have been used in a number

of ecohydrological applications, including the study of water-stress conditions in

vegetation, the modeling of land-atmosphere interactions, and the analysis of the

hydrologic controls on photosynthesis and nutrient cycling (see
Rodriguez-Iturbe and

Porporato
,
2005
). In this subsection we use this model to investigate the effect of

soil-moisture-precipitation feedbacks on the dynamics of soil moisture. Following

Laio et al.
(
2001
), we consider the water balance of a surface soil layer of depth
Z
r

(Fig.
4.16
). The only input to the soil-water balance is due to precipitation, which is

modeled as a stochastic process. We use a Poisson process of rainfall occurrences at

rate

(average storm frequency), with each storm having an exponentially distributed

random depth
h
with mean storm depth

λ

. The output is due to evapotranspiration and

leakage, which are modeled as deterministic functions of soil moisture (Fig.
4.17
). We

express soil-moisture dynamics through a stochastic differential equation expressing

the soil-water balance at a point (
Laio et al.
,
2001
)as

nZ
r
d
s
(
t
)

d
t

α

=
ϕ

[
s
(
t
)

,

t
]

−
χ

[
s
(
t
)]

,

(4.19)

where
s
is soil moisture,
t
is time,
n
is the soil porosity,
Z
r
is the active soil depth (i.e.,

the root zone, which is active in the exchange of water with the overlying atmosphere),

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