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state. Thus we conclude that the bimodal behavior of
p
(
s
) does not emerge as a

noise-induced effect but as a result of the positive feedback between soil moisture and

precipitation.

These results differ from those reported in Section
4.6
, in which we show the

emergence of bimodality as a noise-induced effect associated with interannual climate

fluctuations, without invoking feedbacks between soil moisture and precipitation

(
D'Odorico et al.
,
2000
).

4.4 Environmental systems forced by Gaussian white noise

This section presents some examples of noise-induced phenomena in systems forced

by Gaussian white noise:

d

d
t
=

f
(

φ

)

+

g
(

φ

)

ξ

,

(4.26)

gn

where

ξ
gn
is a zero-mean white Gaussian noise with intensity
s
gn
. Because the seminal

work by
Horsthemke and Lefever
(
1984
) concentrated mainly on the case of Gaussian

noise, here we discuss only a few examples that are relevant to the biogeosciences

and point the reader to that treatise for a more detailed discussion of these models.

4.4.1 Harvest process driven by Gaussian white noise

One of the simplest models of population dynamics is expressed by the logistic

equation, or
Verhulst process
(e.g.,
Murray
,
2002
):

d
A

d
t

=

−

,

aA
(
A
c

A
)

(4.27)

where the parameter
a
(also known as the
reproduction rate
) determines the growth

rate and the carrying capacity
A
c
represents themaximum sustainable value of the state

variable
A
.
A
c
typically depends on the available resources (energy, water, nutrients,

or light) and environmental conditions.

To account for the effect of disturbances on the dynamics of
A
, a more general

model, known as a
harvest process
, is often used, which also accounts for the additional

control on population growth exerted by disturbances, emigration, biomass harvesting,

or predators. As noted in Subsection
4.3.2
, harvest models typically assume that the

harvest rate is proportional to
A
(e.g.,
Kot
,
2001
):

d
A

d
t
=

aA
(
A
c
−

A
)

−

kA

.

(4.28)

Equation (
4.28
) has only one stable state,
A

a
. In this subsection we con-

sider the effect of noise on these dynamics. We consider the case of a harvest model

in which parameter
a
is interpreted as a white Gaussian random variable with mean

=

A
c
−

k

/

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