Environmental Engineering Reference
In-Depth Information
In what follows we use the dimensionless variables
A
i
K
,
JK
α
A
i
=
t
=
α
J
=
,τ
c
=
ατ
c
,
t
,
(4.60)
and drop the prime superscript to simplify the notation. To study the effect of noise and
interspecies interactions on the dynamics of this system, some analytical expressions
can be obtained for the distribution of the state variables
A
i
by use of the mean-
field assumption (1
N
)
j
=
i
A
j
≈
. Inserting Eq. (
4.57
) into Eq. (
4.55
), we can
express the dynamics of
A
i
as a function of the mean-field conditions
/
A
A
and of the
other parameters of the system:
d
A
i
d
t
=
A
i
1
+
a
ξ
dn
,
i
(
t
)]
+
J
A
−
γ
A
i
[1
.
(4.61)
The solution of Eq. (
4.61
) can be determined with the methods presented in Chapter
2:
A
−
(1
+
β
)
(1
β
+
J
A
)
aB
2
,
)
p
(
A
,
A
)
=
(4.62)
1
γ
2
βτ
c
(1
+
J
A
2
1
−
1
)
2
2
βτ
c
(1
+
J
A
)
(1
+
J
A
γ
1
A
β
×
1
−
)
−
,
γ
2
a
2
(1
+
J
A
where
B
[
] is the beta function. Using the conditions presented in Chapter 2, we find
that the extremes
A
1
and
A
2
of the domain of
A
are
·
1
1
β
+
J
A
A
1
,
2
=
.
(4.63)
γ
∓
a
)
(1
A
,
which in turn depends on the distribution of
A
. Using the Weiss mean-field approach,
Mankin et al.
(
2004
) imposed the self-consistency condition
The steady-state probability distribution of
A
expressed by (
4.62
) depends on
A
2
A
=
Ap
(
A
,
A
)d
A
.
(4.64)
A
1
We can then calculate the probability distribution of
A
by inserting into Eq. (
4.62
)
values of
obtained as solutions of Eq. (
4.64
). Multiple solutions of (
4.64
)cor-
respond to dynamical systems, which exhibit two alternative stable phases, i.e., two
statistical regimes characterized by distinct probability distributions.
To evaluate the effect of noise on these dynamics, we first consider the deterministic
counterpart of the system [i.e., Eq. (
4.61
) with
A
ξ
dn
,
i
(
t
)
=
0]. In the case
β
=
1the
dynamics are stable (see Box 4.3) for
J
<γ
, with the system converging to the state
A
→
1
/
(
γ
−
J
). However, as
J
exceeds
γ
, the dynamics become unstable and
A
diverges to
1 the deterministic system exhibits transitions from
stable to unstable conditions induced by the coupling (expressed by the parameter
J
)
∞
. Thus for
β
=
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