Environmental Engineering Reference
In-Depth Information
3.2.1.2 Noise-induced transitions for processes driven by multiplicative DMN
We now increase the complexity of the system by considering the case of a Verhulst
model forced by multiplicative noise (see also
Horsthemke and Lefever
,
1984
). We
concentrate on the case in which
ξ
dn
is symmetric and the noise term is a linear
function of
φ
, i.e.,
d
d
t
=
φ
β
−
φ
+
φξ
=
φ
β
+
ξ
−
φ
.
(
)
[(
dn
)
]
(3.19)
dn
In this system the noise randomly modifies the carrying capacity. The corresponding
pdf,
2
k
β
k
+
β
(
k
β
−
∝−
φ
+
β
−
φ
φ
+
−
β
(
)
)
2
2
−
β
p
(
φ
)
,
(3.20)
φ
[(
φ
−
β
)
2
−
2
]
is defined in the domain [max(0
].
In this case the second term in Eq. (
3.3
) does not vanish. Equation (
3.3
) becomes
,β
−
)
,β
+
2
]
φ
m
[(
φ
m
−
β
)(2
k
−
3
φ
m
+
β
)
−
=
,
0
(3.21)
2
k
and the modes and antimodes of
φ
obtained as solutions of (
3.21
)are
3
1
3
[
k
φ
m
,
1
=
0
,
m
,
2
,
3
=
±
2
+
(
k
−
β
)
2
+
2
β
]
.
(3.22)
) it is again useful to investigate the behavior
of the process close to the boundaries of the domain. Using approximation (
2.41
), we
find that, at the upper boundary,
To sketch the possible shapes of
p
(
φ
φ
=
β
+
, the possible behaviors are
p
(
β
+
)
→∞
,
if
β
+
>
k
;
φ
=
β
+
→∞
,
d
p
(
φ
)
k
2
p
(
β
+
)
=
0
,
if
<β
+
<
k
;
d
φ
φ
=
β
+
=
φ
d
p
(
)
k
2
β
+
=
,
,
β
+
<
.
p
(
)
0
0
if
d
φ
If the lower boundary is
φ
=
β
−
, the possible behaviors at this boundary are
p
(
β
−
)
→∞
,
if
β
−
>
k
;
φ
=
β
−
→∞
,
d
p
(
φ
)
k
2
p
(
β
−
)
=
0
,
if
<β
−
<
k
;
d
φ
φ
=
β
−
=
d
p
(
φ
)
k
2
p
(
β
−
)
=
0
,
0
,
if
β
−
<
.
d
φ
In contrast, when the lower boundary is
φ
=
0, it is not possible to use limit
expression (
2.41
) because
φ
=
0 is a deterministic steady state. However, studying
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