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
)
[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 )
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
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
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