Environmental Engineering Reference

In-Depth Information

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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