Environmental Engineering Reference
In-Depth Information
of interest under the functional interpretation of the DMN, is obtained as a function
of
f
(
φ
)and
g
(
φ
) by use of Eqs. (
2.14
):
)exp
φ
)
d
φ
−
φ
φ
)
φ
)
k
1
(
k
2
(
Cg
(
φ
+
f
(
φ
)
+
1
g
(
φ
)
f
(
φ
)
+
2
g
(
p
(
φ
)
=
.
(2.45)
[
f
(
φ
)
+
1
g
(
φ
)][
f
(
φ
)
+
2
g
(
φ
)]
To demonstrate the possible impact of the feedback between noise and the dynam-
ical system, we consider the simple case in which the alternating processes of growth
and decay are expressed by two linear functions:
f
1
(
φ
)
=
α
(1
−
φ
)
,
f
2
(
φ
)
=−
αφ,
(2.46)
α>
φ
=
where
0 determines the rates of growth and decay. The stationary states,
1
st
,
1
and
0, are also the boundaries of the dynamics. We also assume a linear
dependence of
φ
st
,
2
=
, and a logistic distribution to represent the
variability of the resource
q
, with cumulative density function
θ
on
φ
,
θ
(
φ
)
=
θ
0
+
b
φ
1
σ
−
1
q
−
q
e
−
∗
P
Q
(
q
)
=
+
,
(2.47)
where
is a scale parameter. The mean and the standard deviation of the distribution
are
q
∗
and
√
3
σ
σ
and the use
of the logistic distribution are aimed at simplifying the mathematical treatment of the
problem, but other choices [i.e., other monotonic forms of the
, respectively. The choice of a linear dependence of
θ
on
φ
) function or other
probability distributions] could be equally adopted. Under the preceding assumptions,
the transition rates are found as
θ
(
φ
1
−
1
e
−
θ
0
−
q
∗
+
b
·
φ
k
1
(
φ
)
=
1
−
k
2
(
φ
)
=
+
,
(2.48)
σ
and the corresponding steady-state pdf from Eq. (
2.44
)is
⎡
⎤
1
α
d
y
1
α
−
⎣
−
⎦
,
1
(1
)
−
1
exp
p
(
φ
)
=
C
φ
−
φ
y
)
1
(2.49)
2
e
−
θ
0
−
q
∗
+
b
·
y
y
(1
−
+
φ
σ
where
C
is the normalization constant we calculate by imposing the condition that
the integral of
p
(
) in the domain [0,1] be equal to 1.
Figure 2.8 shows an example of pdf (
2.49
) along with the one corresponding to
no feedback (i.e.,
b
φ
0).
It is obvious that the state dependency of the transition rates induces a remarkable
change in
p
(
=
0). In particular, the feedback is chosen to be positive (
b
<
). This change is not only quantitative but also qualitative. In Chapter
3 we discuss thoroughly the ability of feedbacks to induce structural changes to the
shape of the pdf's.
φ
Search WWH ::
Custom Search