Environmental Engineering Reference
In-Depth Information
random pulses. Typical examples include insect outbreaks, epidemics, landslides
(
Benda andDunne
,
1997
;
D'Odorico and Fagherazzi
,
2003
), fires (
Daly and Porporato
,
2006
), and rainfall (
Todorovic and Woolhiser
,
1975
). Accounting for the intermittent
character of these occurrences may be crucial to the correct understanding of the
dynamics of environmental systems at time scales smaller than (or comparable to) the
recurrence time of the random episodic events. In this case, the random forcing needs
to be modeled as a temporal sequence of intermittent events occurring at random
discrete times
t
i
(with
i
=
1
,
2
,...
). When the recurrence times
τ
i
=
t
i
+
1
−
t
i
are
exponentially distributed random variables with mean
λ
, this sequence is known as
white Poisson noise
of rate
(see Chapter 2). In several applications each event has a
random intensity
h
with given distribution
p
(
h
) and mean
λ
; in this case the random
sequence is known as a
marked Poisson process
, and, when
h
is an exponentially
distributed random variable, it is called
white shot noise
. In the case of Poisson noise,
Eq. (
4.1
) can be rewritten in the form
α
d
d
t
=
f
(
φ
)
+
g
(
φ
)
ξ
sn
,
(4.9)
where
ξ
sn
(
t
) is the white-shot-noise term.
4.3.1 Harvest process: Fire-induced tree-grass coexistence in savannas
Harvest processes are a class of simple univariate dynamics (e.g.,
Kot
,
2001
)in
which a population
A
undergoing logistic growth is harvested proportionally to its
abundance:
d
A
d
t
=
aA
(
A
c
−
A
)
−
kA
,
(4.10)
where
A
is, for example, the population's density,
A
c
is the carrying capacity of
A
(i.e., the maximum sustainable value of
A
allowed by the available resources),
a
is the reproduction rate of the logistic growth, and
k
is the harvest rate. If we
assume that
A
is a positive-valued variable and
A
c
>
k
/
a
, these deterministic dy-
namics have two equilibria, namely
A
=
0 (unstable state) and
A
=
A
c
−
k
/
a
(stable
state).
In a number of natural processes the harvest rate has a random and episodic
character. For instance, fires are clear examples of episodic random disturbances
that could be modeled as Poisson noise to account for their intermittent and random
character. In this subsection we investigate the properties of a
Poisson harvest process
,
i.e., of a stochastic harvest process in which the harvest rate
k
is modeled as white
shot noise and the parameters
λ
and
α
of the white shot noise
ξ
sn
represent the average
frequency and magnitude of the harvest events, respectively.
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