Environmental Engineering Reference

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following two different approaches, depending on the nature of the physical system

under examination. In the functional approach, we start from the analysis of the

deterministic dynamics d

φ/

d
t

=

f
(

φ

) and investigate the effect of multiplicative

noise on these dynamics:

d

d
t
=

f
(

φ

)

+

g
(

φ

)

ξ
dn
(
t
)

.

(3.1)

In this case the steady-state pdf is given by Eq. (
2.32
); we easily find the deterministic

counterpart of the process by setting

ξ
dn
(
t
)

=

0inEq.(
3.1
). Thus the deterministic

steady states

0.

In the mechanistic approach, the dynamics switch between the two deterministic

processes,

φ
st
are the zeros of
f
(

φ

), i.e.,
f
(

φ
st
)

=

d
t
=

d
t
=

d

d

f
1
(

φ

)

>

0

,

f
2
(

φ

)

<

0

,

(3.2)

depending on whether the value of a stochastic external driver
q
is greater or smaller

than a given threshold

[see Eqs. (
2.12
) in Chapter 2]. In this case noise is used as a

mechanism to switch between these two processes. The expression of the steady-state

pdf is given by Eq. (
2.31
). Also in this case the underlying deterministic dynamics

are obtained by turning the noise variance to zero. If we decrease the variance of the

driving force
q
while maintaining its mean as constant,
q
∗
, in the zero-variance limit

q
becomes a constant deterministic value,
q

θ

=

q
∗
. The deterministic stationary state

is determined by the position of
q
∗
relative to

, the dynamics are expressed

by the first of Eqs. (
3.2
). In this case the (constant) resources
q
are abundant enough to

sustain the growth of

θ

:If
q
∗
>θ

φ

at a rate
f
1
(

φ

). The deterministic steady state

φ
st
,
1
is obtained

asasolutionof
f
1
(

φ
st
,
1
)

=

0. Conversely, if
q
∗
<θ

, the available resources are scarce

and

decays at a rate expressed by the second equation in (
3.2
). In this case we find

the deterministic steady state

φ

0.

Once the deterministic counterparts of the dynamics have been identified, it is

possible to investigate how noise modifies the stable states of the system. To this end,

we analyze the modes and antimodes

φ

2
by setting
f
2
(

φ

2
)

=

st

,

st

,

(
t
) forced by

dichotomous noise. We can obtain these modes by setting equal to zero the first-order

derivative of (
2.32
)or(
2.31
), depending on the interpretation adopted for the DMN. In

the functional interpretation, using Eq. (
2.32
), we find that the modes and antimodes

are solutions of

φ

m
of the pdf of the process

φ

φ
m
)
g
(

1
+
2
)
f
(

f
(

φ
m
)

+
τ
c
1
2
g
(

φ
m
)

+
τ
c
(

φ
m
)
g
(

φ
m
)

+
τ
c
2
f
(

f
2
(

m
)
g
(

φ

φ

m
)

φ
m
)
f
(

φ
m
)

−

=

0

,

(3.3)

g
(

φ
m
)

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