Environmental Engineering Reference
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0.75
IV
p A
p A
A
1 A
V
0
0.5
p A
p A
a
1 A
1 A
0
0
III
b
0.25
I
p A
c
II
1 A
0
0
0
1
2
3
a
Figure 4.11. Dependence of the shape of the probability distribution p ( A ) of woody
biomass A on parameters of the fire-vegetation dynamics in the presence of positive
feedback between fire frequency and grass biomass ( b
=
0 with b
<
0). Calculated
with the same parameters as in Fig. 4.9 but with b
/
a
=−
0
.
7. With respect to the case
b
=
0 (i.e., Fig. 4.9 ) line (c) shifts to the right, enabling a wider bistable domain, IV.
fire-vegetation feedback destabilizes conditions favorable to tree-grass coexistence
(i.e., the existence of a mode of A between 0 and 1), whereas grassland and woodland
become alternative stable states of the system, as shown in Fig. 4.12 . Notice that the
state dependence
λ ( A ) only enhances this bistable behavior, and its emergence is
induced by the multiplicative noise (see previous section). 3
4.3.3 Stochastic soil-mass balance
In this section we provide an example of systems forced by multiplicative Poisson
noise, with a bound imposed on the distribution p ( h ) of the “jumps” h of the shot
noise [see Eq. ( 4.9 )]. It is shown that the state dependency induced by this bound is
capable of leading to noise-induced transitions. This example is based on the dynamics
of soil production and erosion in landslide-prone landscapes. These dynamics were
traditionally investigated ( Kirkby , 1971 ; Ahnert , 1988 ; Dietrich et al. , 1995 ; Roering
et al. , 1999 ) through a soil-mass-balance equation in which the variability in time of
3
In fact, the deterministic counterpart of the process with state-dependent rate of fire occurrence has only one unstable
state ( A = 0) and one stable state [ A = (1 λα/ a ) / (1 + α b / a )] that can be calculated with ( 4.13 ), replacing λ
with
λ = λ + bA .
 
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