Environmental Engineering Reference

In-Depth Information

p A

p A

p A

p A

A

A
m

A
0

A

A

A

1

0.8

______
Stable

Unstable

0.6

0.4

0.2

0

s
gn

A
0

0.25

0.5

0.75

1

1.25

1.5

1.75

Figure 4.23. Transition points in the stochastic
Verhulst model
(Ito's interpretation),

i.e., of a logistic process with random normally distributed carrying capacity.

environmental fluctuations exceeds
A
0
,
A
tends to extinction with a probability of

one. Another transition point can be found with respect to the shape of
p
(
A
). In fact,

the mode (i.e., preferential “stable” state)
A
m
of
A
[which we obtain by setting equal

to zero the first-order derivative of the right-hand-side of (
4.36
)] is

A
m

=

A
0

−

2
s
gn

for

s
gn

<

A
0

/

2

,

(4.38)

and
A

=

0 is an unstable state of the system. When
s
gn
exceeds
A
0

/

2, the stable

state is
A

0. Figure
4.23
shows the stable and unstable states of the system as a

function of the intensity of the environmental fluctuations: It can be observed that

the qualitative properties of the dynamics change as an effect of intensification of

random fluctuations with two transition points (
Horsthemke and Lefever
,
1984
). As

s
gn
exceeds
A
0

=

/

=

2,
A

0 becomes the preferential state of the dynamics, whereas for

>

=

s
gn

A
0
the steady state of the system
A

0 (extinction) occurs with a probability

that tends to one.

To be consistent with the other analytical results presented in this chapter, we look at

the results obtained when Eqs. (
4.26
)-(
4.35
) are integrated by using the Stratonovich

interpretation (see Chapter 2). In this case, the pdf of
A
differs from the one obtained

by May (1973):

C
−
1
A

s
gn

A
0

s
gn

A
0

s
gn

−

1

A

s
gn

e
−

p
(
A
)

=

with
C

=

s
gn

.

(4.39)

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