Environmental Engineering Reference

In-Depth Information

the interested reader to this topic and to
Ludwig et al.
(
1978
) for more details on this

process.

4.4.3 Noise-induced extinction

The effect of random fluctuations in the carrying capacity (
Levins
,
1969
;
May
,
1973
)

may induce interesting phenomena in the logistic process.
May
(
1973
) treated the car-

rying capacity in Eq. (
4.27
) as a normally distributed random variable,
A
c
=
A
0
+
ξ
gn
,

with mean
A
0
>

0 and intensity
s
gn
, and determined the steady-state probability

distribution of
A
. When the noise intensity (i.e.,
s
gn
) is relatively small, the ran-

dom environment induces stochastic fluctuations of
A
about its long-term mean

is smaller than
A
0
and decreases with increasing values of
s
gn
,

suggesting that noise has the effect of reducing the mean value of
A
with respect

to the carrying capacity
A
0
of the underlying deterministic dynamics. As
s
gn
ex-

ceeds a critical-threshold value, the random fluctuations become so intense that no

steady-state pdf of
A
exists, and the system tends to extinction. The occurrence of

extinction as the noise intensity exceeds a critical value is clearly a noise-induced

phenomenon.

When
A
c
=

A

. Moreover,

A

A
0
+
ξ
gn
,Eq.(
4.27
) can be rewritten as (
4.26
) with rescaled time

t
=

at
,and

f
(
A
)

=

A
(
A
0
−

A
)

,

g
(
A
)

=

A

.

(4.35)

Notice that, because for
A

=

0
g
(
A
)

=

0, the dynamics have an intrinsic boundary at

0; it can be shown (
Horsthemke and Lefever
,
1984
) that this boundary is also a

natural boundary (i.e., it cannot be accessed by
A
)if
s
gn

A

=

A
0
.

Conditions conducive to noise-induced extinction were determined by
May
(
1973
),

who used Ito's interpretation of the stochastic equation (see Chapter 2), leading to the

steady-state probability distribution of
A
:

<

C
−
1
A

s
gn

A
0

s
gn
−

2

A

s
gn

e
−

p
(
A
)

=

,

(4.36)

with normalization constant

A
0

s
gn
−

1

C

=

s
gn

(4.37)

and where

(

·

) is the gamma function. Equation (
4.37
) is obtained from the integration

of
p
(
A
)in[0

1. When
s
gn
exceeds
A
0

there is no finite value of
C
and no equilibrium solution exists. In this case,
A

,
+∞

). Notice that
C
is finite only if
A
0
/

s
gn
>

0

is no longer a natural boundary, in that the whole probability mass is concentrated at

this intrinsic boundary; the pdf of
A
becomes
p
(
A
)

=

) is the Dirac

delta function (
Horsthemke and Lefever
,
1984
). Thus when the intensity
s
gn
of the

=
δ

(
A
), where

δ

(

·

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