Environmental Engineering Reference

In-Depth Information

fire occurrences. These occurrences are modeled as a Poisson process in time,

ξ

sn
(
t

,v

),

with each fire killing an exponentially distributed random amount

ω

of tree biomass

(with mean

t
), whichever is less. To account

for the positive feedback between fire occurrences at a point and the local vegetation,

the rate

ω

0
) or the whole existing biomass

v

(
r

,

of the Poisson process is expressed as a state-dependent function (
van

Wilgen et al.
,
2003
;
D'Odorico et al.
,
2006b
),

λ

λ
=
λ

+

v

≤

b

, with
b

0 (i.e., higher

0

tree densities are associated with less-frequent fire occurrences).

D'Odorico et al.
(
2007b
) investigated the conditions associated with the emergence

of phase transitions and spatial patterns in this system. Although the deterministic

counterpart of this process (i.e., the Fisher's equation) is not capable of generating

patterns (see Appendix B), self-organized patches of trees may emerge in the presence

of the stochastic forcing (i.e., random fire dynamics). To investigate the properties of

stochastic process (
6.8
), we assume that the state variable

v

is normalized with respect

to the ecosystem carrying capacity (i.e., 0

≤
v
≤

1) and set

v
max
=

1inEq.(
6.8
). We

calculate the pdf of

by applying the framework presented in Chapter 5 for the case

of dynamics driven by WSN. Thus we use the mean-field approximation (Box 5.4) in

a finite-difference representation of Eq. (
6.8
) and obtain the steady-state probability

distribution of

v

v

(
Porporato and D'Odorico
,
2004
):

)
exp
v

(
u
)
du

C

ρ

λ

(
u
)

p
(

v

;

v

)

=

0
−

,

(6.9)

v

ω

ρ

(

v

where

v

is the mean of

v

across the whole field,

ρ

(

v

)

=
α

(

v
+

)(1

−
v

)

+

D
(

v
−

v

), and
C
is the normalization constant.
p
(

v

;

v

) is defined within the domain [0

,v
lim
],

where

v
lim
≤

1 is the positive root of

ρ

(

v

)

=

0. The probability distribution
p
(

v

;

v

)

of

v

depends on the unknown value of

v

. Therefore the self-consistency equation is

typically used to determine

v

:

v
lim

v
=

v

p
(

v

;

v

)d

v.

(6.10)

0

v
st
of an average tree biomass obtained

through the mean-field approximation of Eq. (
6.8
). For low values of
D
, the system

has only one steady state, which is either at

Figure
6.3
shows the steady-state values

0, depending on the

rates of tree growth and fire occurrence. Multiple steady states exist when
D
exceeds

a critical value (e.g., the bifurcation point at
D

v
=
v
lim
or

v
=

14 in Fig.
6.3
), indicating the

occurrence of a (noise-induced) phase transition when the spatial coupling is relatively

strong. Thus, because of the stronger spatial coupling, for relatively high values of

D
, the system converges toward one out of two mutually exclusive steady states. The

dependence on the initial condition can be investigated through numerical simulations

(Fig.
6.4
). Obtained from Eq. (
6.8
) without invoking the mean-field approximation,

the numerical simulations support the approximated analytical results presented in

=

0

.

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