Environmental Engineering Reference
In-Depth Information
where
α>
0 determines the rates of growth and decay. In this system
φ
=
1
st
,
1
and
0. Similar to the example shown in Subsection
2.2.3.5
,herewealso
assume a linear dependence of
φ
=
st
,
2
θ
on
φ
,
θ
(
φ
)
=
θ
+
b
φ
, and a logistic distribution
0
}
−
1
to represent the variability of the resource
q
.The
P
Q
(
q
)
={
1
+
exp[
−
(
q
−
q
∗
)
/σ
]
steady-state pdf of
is given by Eq. (
2.49
).
Figure
3.9
summarizes the behavior of
p
(
φ
φ
σ
) as a function of the parameters
and
α
=
<
for the case of no feedback [
b
0, Fig.
3.9
(a)], positive feedback [
b
0, Fig.
3.9
(b)], or negative feedback [
b
>
0[Fig.
3.9
(a)]
refers to a situation in which
q
∗
<θ
0
, which implies that the deterministic stable
state is
0, Fig.
3.9
(c)]. The case with
b
=
φ
st
=
0. For small noise intensities, i.e., for small values of
σ
, the probability
distribution is L-shaped, i.e., the most probable state is
φ
=
0. For increasing
σ
values, two different kinds of noise transitions occur: If
is small relative to the rate
of switching of DMN - i.e., the system responds slowly to the external forcing -
a new mode appears at
α
is relatively large a bifurcation
occurs, i.e., the distribution becomes U-shaped, with two stable states in
φ
m
=
(
k
1
−
α
)
/
(1
−
α
); if
α
φ
=
0and
φ
=
1. In Fig.
3.9
(a) the separation among these three regimes is marked by two lines
determined through the analysis of the pdf at the boundaries of the domain. It is found
that these lines are
}
−
1
and
α
=
k
1
= {
1
+
exp[
−
(
θ
0
−
q
∗
)
/σ
]
α
=
k
2
=
1
−
k
1
(
Laio
et al.
,
2008
).
Figure
3.9
(b) shows the case with positive feedback (
b
<
0). For simplicity, we
take
b
=
2(
q
∗
−
θ
0
), which implies that the distribution is symmetrical with respect to
φ
=
5 for any value of the parameters [see the expression of pdf (
2.49
)]. The
distribution is U-shaped for low noise intensities, as expected from the bistable
behavior of the deterministic counterpart of the dynamics. With increasing val-
ues of
0
.
σ
α<
.
+
/
σ
), with two other modes ap-
pearing and the distribution assuming an M-shape. If
, a first transition occur for
0
5
b
(8
σ
is further increased, and
}
−
1
(see
Laio et al.
,
2008
), a
second transition occurs and the system exhibits a new stable state in
α<
k
1
(
φ
=
1)
=
k
2
(
φ
=
0)
= {
1
+
exp[(
θ
0
−
q
∗
)
/σ
]
5,
and no other stable state exists. The stochastic forcing therefore stabilizes the system
around a new statistically stable state. This state is clearly noise induced, in that it does
not exist in the deterministic counterpart of the process. The ability of noise to turn
a bistable deterministic system into a stochastic process with only one stable state
(contained between the two stable deterministic states) is known as
noise-induced
stability
(
D'Odorico et al.
,
2005
;
Borgogno et al.
,
2007
;
Ridolfi et al.
,
2007
).
We finally turn to the case with negative feedback [
b
φ
m
=
0
.
>
0, Fig.
3.9
(c)]. We take
again
b
=
2(
q
∗
−
θ
0
) to have symmetrical distributions. For low values of
σ
the
distribution has a single mode in
5, which corresponds to the deterministic
stable state. For increasing noise intensity, the distribution becomes first W-shaped,
for
φ
m
=
0
.
}
−
1
, and then U-shaped, for
α>
k
1
(
φ
=
1)
=
k
2
(
φ
=
0)
= {
1
+
exp[(
θ
−
q
∗
)
/σ
]
0
α>
). This is a clear example of purely noise-induced bistability, because
bistability does not appear in the corresponding deterministic dynamics.
0
.
5
+
b
/
(8
σ
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