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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
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 t =
f (
g (
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 )
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 )
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
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