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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