Environmental Engineering Reference

In-Depth Information

The normalization constant
C
is finite for
A
0

>

0 (independent of
s
gn
). The modes of

(
4.39
) can be easily obtained with Eq. (
3.48
):

A
m
=
A
0
−
s
gn

for

s
gn
<
A
0
/

2

,

(4.40)

whereas if
s
gn
exceeds
A
0
/

2, the mode is zero. Thus, in the Stratonovich interpretation,

this second transition in the shape of the distribution still exists, is controlled by the

noise intensity, and is induced by the random fluctuations (i.e., it is a noise-induced

transition). However, for
A
0
>

0 (i.e., in the typical ecological applications of this

logistic-Verhulst process to population dynamics) no noise-induced extinction (i.e.,

the noise-induced attainment of
A

1) occurs. Thus in this

case the interpretation rule seems to play a key role in the noise-induced behaviors of

these dynamics (e.g.,
Turelli
,
1978
;
Braumann
,
1983
).

Similar results can be obtained with a (stochastic) Malthusian growth model

d
A

d
t
=

=

0 with probability

→

(
r

+
ξ
gn
)
A

,

(4.41)

where
r
is the average growth rate. Using Ito's calculus, we can observe that, in the

presence of relatively strong fluctuations in the growth rate (i.e.,
s
gn
>

r
), extinction

occurs with probability tending to one even with positive values of the mean growth

rate
r
, i.e., when in model (
4.41
) the expected value of
A
grows indefinitely with time

(
Lewontin and Cohen
,
1969
;
Braumann
,
1983
). However, these dynamical properties

do not appear when Stratonovich's calculus is used. In this case the probability of

extinction tends to one when
r

0, i.e., with negative values of the average growth

rate. This condition is independent of
s
gn
, i.e., of the intensity of random environmental

fluctuations. Thus, with Stratonovich's interpretation of (
4.41
), extinction does not

occur as a noise-induced effect.

Overall, this discussion stresses how the choice of either Ito's or Stratonovich's

interpretation may lead to dramatically different results. It is therefore crucial to

choose the correct interpretation rule while solving stochastic differential equations.

As noted in Chapter 2, Stratonovich's calculus prevents the emergence of biases.

However, these biases appear only in the case of systems driven by multiplicative

white noise. Therefore, in the case of systems forced by multiplicative white noise,

Stratonovich's interpretation should preferably be used (
van Kampen
,
1981
). In fact,

in Chapter 2 we showed that Stratonovich's rule is consistent with the interpretation

of white Gaussian noise and WSN as limit cases of DMN. Alternatively,
Braumann

(
1983
,
2007
), showed that the two methods would lead to the same results when the

growth-rate parameter
r
is adequately reinterpreted.

The model of noise-induced extinction presented in the previous section treats the

carrying capacity as a random variable to account for stochastic fluctuations inherent

in the variability of the limiting resources. In particular, in that case the carrying

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