Environmental Engineering Reference

In-Depth Information

a
0
and intensity
s
gn
; i.e.,
a

=

a
0

+
ξ

gn
. Thus the stochastic dynamics can be expressed

as (
4.26
) with

f
(
A
)

=

a
0
A
(
A
c
−

A
)

−

kA

,

g
(
A
)

=

A
(
A
c
−

A
)

.

(4.29)

This model is well suited to investigate the possible emergence of noise-induced

transitions as an effect of random environmental fluctuations in the growth rate

of the dynamics. In what follows we normalize
A
with respect to
A
c
and use the

state variable
A
=

A

/

A
c
. Moreover, to simplify the notation we drop the prime

superscript.

The steady-state pdf of
A
can be calculated with Eq. (
2.83
) from Chapter 2:

1

s
gn

k

1

s
gn
(
a
0

1

s
gn
(
a
0

Ce
−

−

k
)

−

1

A
)
−

−

k
)

−

1

p
(
A
)

=

A

(1

−

,

(4.30)

1

−

A

where
C
is the normalization constant. It can be shown that when
k

=

0 the normal-

ization constant
C
is not finite and
p
(
A
)

0),

as the noise intensity (i.e.,
s
gn
) increases, the dynamics undergo a noise-induced tran-

sition from a system with only one stable state (for the deterministic system or for

low values of
s
gn
) to bistable dynamics. Figure
4.21
(a) shows the modes of
A
calcu-

lated with Eq. (
3.48
): As
s
gn
exceeds a critical value
s
c
=

=
δ

(
A

−

1). In all the other cases (
k

=

k
), a noise-induced

transition occurs, in that multiple equilibria emerge and the probability distribution

of
A
becomes bimodal [Fig.
4.21
(b)]. Thus the multiplicative noise is able to in-

duce a new stable state
A

(
a
0
−

=

0 that did not exist in the underlying deterministic

system.

4.4.2 Stochastic genetic model

Logistic harvest model (
4.29
) can be generalized to account for possible inputs of
A

occurring at constant rate

β

.Knownasa
genetic model
(
Horsthemke and Lefever
,

1984
), this process finds a number of applications to population biology and dy-

namic ecology. For example, the genetic model could describe the dynamics of a

population affected by logistic growth, state-dependent emigration or harvest, and

state-independent immigration (

). If, for the sake of simplicity,
A
is normalized with

respect to
A
c
in a way that it ranges between 0 and 1, these dynamics are expressed

by

β

d
A

d
t
=

aA
(1

−

A
)

+
β
−

kA

.

(4.31)

Dividing both sides of (
4.31
)by
k
, normalizing the model's parameters
a
=

a

/

k
,

β
=
β/

k
, rescaling the time variable (
t
=

kt
), and dropping the prime superscripts,

we obtain

d
A

d
t
=

aA
(1

−

A
)

+
β
−

A

.

(4.32)

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