Environmental Engineering Reference

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A
m

a

1

0.8

0.6

0.4

______
Stable

Unstable

s
c

0.2

s
gn

1

2

3

4

5

6

7

p
A

2.5

b

2

1.5

1

______

s
gn

0.5

s
c

0.5

s
gn

8.0

s
c

A

0.2

0.4

0.6

0.8

1

Figure 4.22. Stochastic genetic model: (a) stable and unstable states of the system as

a function of the noise intensity (
s
gn
), (b) pdf of

φ

with
a
0
=

0,

β
=

0

.

5 (in this case

s
c
=

2).

becomes unstable and two new stable states emerge. The steady-state probability

distribution of
A
can be determined from Eq. (
2.83
):

s
gn
1
−
β

A

1

1

s
gn
(
a
0

1

s
gn
(
a
0

Ce
−

+

+

2

β
−

1)

−

1

A
)
−

+

2

β
−

1)

−

1

p
(
A
)

=

A

(1

−

,

(4.34)

1

−

A

where
C
is the normalization constant. Figure
4.22
(b) shows some examples of

probability distributions of
A
, which become bimodal when
s
gn
exceeds
s
c
, indicating

the occurrence of noise-induced bistability.

We note that in a number of systems the harvest rate is a nonlinear function of
A
.

For example, the consumption by predators is likely to reach saturation for high values

of
A
as the predator needs are met (
Ludwig et al.
,
1978
). To account for this saturation

effect the harvest rate can be expressed as
kA
b

A
b
), as suggested for the case of

insect outbreak dynamics (
Ludwig et al.
,
1978
). Interesting noise-induced transitions

may also emerge in these systems when in Eq. (
4.28
) the coefficient
k
(instead of
a
)

is treated as a (Gaussian) random variable to account for the effect of rapidly varying

(random) environmental conditions. These dynamics were investigated in detail by

Horsthemke and Lefever
(
1984
) in the context of predator-prey systems. We refer

/

(1

+

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