Environmental Engineering Reference
In-Depth Information
The constructive role of noise autocorrelation is evident also in another example
that allows us to discuss some issues related to the use of a functional approach in the
stochastic modeling of systems forced by DMN. In this case we start with the study
of the deterministic dynamics d
), and we use DMN to introduce auto-
correlated stochastic fluctuations in these dynamics. Consider the classical Verhulst
(or logistic) model (Section
2.2.3
, Example 2.4):
φ/
d
t
=
f
(
φ
d
d
t
=
f
(
φ
)
=
φ
(
β
−
φ
)
,φ
∈
[0
,
∞
[
,
(3.9)
where
0 is a parameter, called the Malthusian growth parameter or carrying
capacity. This nonlinear model was originally proposed to describe the growth of
a population and then adopted in many other fields. In dynamical system (
3.9
), the
growth rate of the variable
β>
itself but is limited by a
saturation term that stands for the fact that resources are limited. Despite its apparent
simplicity, the model may exhibit a remarkable variety of behaviors with different
types of noise. For this reason it is often used to introduce a classical example of
noise-induced transitions (e.g.,
Horsthemke and Lefever
,
1984
).
For any given positive and constant value of
φ
increases with the value of
φ
β
, deterministic dynamical system (
3.9
)
has two steady states:
φ
=
0 (unstable) and
φ
=
β
(stable). The deterministic
st
,
1
st
,
2
dynamics of
φ
can be analytically obtained as a solution of Eq. (
3.9
):
(0)
e
β
t
φ
φ
(
t
)
=
1)
,
(3.10)
+
φ
(0)
β
1
(
e
β
t
−
where
φ
(0)
>
0 is the initial condition. The (asymptotic) deterministic steady state is
φ
=
β
(0).
We can now assume that an additive dichotomic noise forces the dynamics:
, regardless of the initial condition
φ
d
d
t
=
φ
(
β
−
φ
)
+
ξ
,
(3.11)
dn
where, for the sake of simplicity,
ξ
dn
is assumed to be a symmetric noise (i.e.,
1
=
−
=
and
k
1
=
k
2
=
k
). In this case, the pdf of
φ
is obtained by use of Eq. (
2.32
)
2
and reads
k
p
k
q
−
−
C
−
2
φ
+
β
−
p
−
2
φ
+
β
−
q
p
(
φ
)
=
,
(3.12)
φ
2
(
β
−
φ
)
2
−
2
−
2
φ
+
β
+
p
−
2
φ
+
β
+
q
=
β
=
β
where
C
is the normalization constant,
p
2
+
4
,
q
2
−
4
,andthe
domain of
p
(
φ
)is
β
+
q
,
β
+
p
φ
∈
,
(3.13)
2
2
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