Environmental Engineering Reference
In-Depth Information
θ
θ
a
b
θ
2
θ
1
θ
θ
1
2
φ
φ
φ
φ
φ
φ
st,2
st,1
st,2
st,1
Figure 3.8. Qualitative sketch of the dependency of the threshold
θ
on the state
variable
φ
in the case of (a) positive feedback and (b) negative feedback. The values
θ
1
and
θ
2
are also shown.
where
f
1
(
0 express the state-dependent growth and decay
rates, respectively. In the presence of feedback,
φ
)
>
0and
f
2
(
φ
)
<
. This depen-
dence translates into a state dependence of the switching rates of DMN,
k
1
(
θ
is a function of
φ
φ
)and
k
2
(
).
As in the case with no feedback (Section 2.1), if we decrease the variance of the
driving force
q
while maintaining constant its mean
q
∗
φ
in the zero-variance limit,
q
tends to a constant deterministic value,
q
q
∗
. We can now analyze two cases of
systems with (i) positive feedback (i.e., the threshold
=
θ
decreases as
φ
increases), and
(ii) negative feedback (i.e.,
increases).
(i) In the case of positive feedback, the threshold
θ
increases as
φ
φ
[see Fig.
3.8
(a)]. In these conditions the deterministic counterpart of the stochastic
dynamics depends on the relation between
θ
is a decreasing function of
θ
(
φ
)and
q
∗
. We first define the maximum
and minimum possible values of
θ
by setting in the relation
θ
(
φ
) the boundaries of
φ
φ
φ
the domain of
p
(
). As noted, these boundaries,
2
(minimum) and
1
(maxi-
st
,
st
,
φ/
=
φ
mum), are the steady states of the two deterministic dynamics, d
d
t
f
1
,
2
(
). Thus
two values
φ
st
,
2
) (maximum) are obtained [see
Fig.
3.8
(a)]. If
q
∗
<θ
1
the deterministic dynamics monotonically decrease, converg-
ing to the stable state
θ
1
=
θ
(
φ
st
,
1
) (minimum) and
θ
2
=
θ
(
φ
st
,
2
. Analogously, if
q
∗
>θ
2
the system persists in state 1
(growth) regardless of the value of
φ
. Thus
φ
(
t
) converges to the deterministic stable
state
q
∗
<θ
2
. In this case the deter-
ministic system is bistable: If the threshold value associated with the initial condition
φ
0
is smaller than
q
∗
φ
st
,
1
. The most interesting situation is with
θ
1
<
[i.e.,
θ
(
φ
0
)
<
q
∗
], the dynamics of
φ
exhibit a deterministic
growth with rate determined by
f
1
(
) decreases and the system
persists in the growth conditions, thereby converging to the steady state,
φ
). As
φ
grows,
θ
(
φ
φ
st
,
1
. Con-
versely, if the system is initially in decay (or “stressed”) conditions [i.e.,
θ
(
φ
0
)
>
q
∗
],
φ
decreases with rate
f
2
(
φ
). As
φ
decreases,
θ
(
φ
) increases. Thus
φ
tends to the
steady state
φ
st
,
2
in that the system persists in state 2 (decay).
(ii) In the presence of negative feedback,
θ
(
φ
) increases with
φ
[see Fig.
3.8
(b)].
The two limiting values of
θ
are again defined as
θ
=
θ
(
φ
1
)and
θ
=
θ
(
φ
2
).
1
st
,
2
st
,
However,
θ
1
is now the maximum and
θ
2
the minimum value of
θ
[see Fig.
3.8
(b)].
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