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
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