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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