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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