Environmental Engineering Reference
In-Depth Information
A
m
1
0.8
0.6
I
IV
III
0.4
Stable
Unstable
IV
0.2
0
a
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 4.10. Bifurcation diagram showing the stable and unstable states (thick solid
and dashed lines) of the dynamics as functions of
λ/
a
and with
α
=
0
.
75. It is
observed that with increasing values of
λ/
a
the preferential state of the system shifts
from
A
m
=
0 (zone III). This transition occurs through
a bistability domain (zone IV, between the two vertical dotted lines), which exists
only for
1 (zone I of Fig.
4.9
)to
A
m
=
5, as shown in Fig.
4.9
. The dynamics exhibit a hysteresis in that the
transitions from
A
m
=
α>
0
.
1to
A
m
=
0 and from
A
m
=
0to
A
m
=
1 occur for different
values of
λ/
a
capable of destabilizing the states
A
m
=
1 and
A
m
=
0, respectively.
Landscapes with denser tree or shrub cover (i.e., higher
A
) exhibit lower grass biomass
and a consequently lower probability of fire occurrence (e.g.,
van Wilgen et al.
,
2003
;
Goldammer and de Ronde
,
2004
;
D'Odorico et al.
,
2006b
). Thus the rate
λ
of fire occurrence is a decreasing function of
A
, which we assume to be linear,
λ
(
A
)
=
λ
+
bA
(with
b
≤
0,
λ>
−
b
). When the rate of (deterministic) growth is
expressed by a logistic function,
f
(
A
)
=
aA
(1
−
A
), and the effect of fire occurrences
=−
is again taken to be proportional to
g
(
A
)
A
;usingEq.(
2.73
) we obtain that the
steady-state pdf of
A
is again a beta distribution (
D'Odorico et al.
,
2006b
),
1
α
−
a
−
A
)
a
+
b
a
−
1
(1
1
p
(
A
)
=
CA
−
,
(4.14)
with normalization constant
1
a
b
α
+
C
=
1
α
−
a
a
+
a
.
(4.15)
b
0) is able to increase the size of
the portion of the parameter space in which noise-induced bistable dynamics are
observed (zone IV). In fact, a comparison with Fig.
4.9
(case with no feedback)
shows that curve (c) shifts to the right, thereby allowing for the emergence of bista-
bility, even with lower values of
Figure
4.11
shows that the feedback (i.e.,
b
<
α
. By enhancing this bistable behavior, the positive
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