Environmental Engineering Reference
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exceptions (
Zaikin and Schimansky-Geier
,
1998
;
Dutta et al.
,
2005
). Because these
models use complicated nonlinear terms to represent both the local deterministic
dynamics and the multiplicative-noise terms, their process-based interpretation is
often not straightforward. The example presented in this section shows how additive
noise may cause morphogenesis even in linear systems and with spatial coupling
expressed by (short-range) diffusion.
6.5 Patterns induced by multiplicative white shot noise
In this section we provide an example of patterns induced by multiplicative shot
noise in a nonlinear system. We consider the case of mesic and subhumid savannas
where the persistence of mixed tree-grass plant communities is maintained by fires
(
Sankaran et al.
,
2004
;
Higgins et al.
,
2000
). In the absence of fires these ecosystems
tend to be completely dominated by woody vegetation (e.g.,
Scholes and Archer
,
1997
;
Higgins et al.
,
2000
;
van Wijk and Rodriguez-Iturbe
,
2002
). In these savanna
ecosystems, woody-plant encroachment is limited by disturbances such as fires rather
than by resource availability (see also Subsection
4.3.1
). Moreover, because of the
competitive advantage of trees and shrubs, the dynamics of woody vegetation are
not affected by grasses. Thus only the dynamics of woody biomass
v
are modeled,
whereas grass biomass is assumed to be proportional to
max
is the
ecosystem carrying capacity. In this case vegetation dynamics can be modeled with
only one equation.
D'Odorico et al.
(
2007b
) expressed the temporal variability of
v
v
−
v
,where
v
max
as a growth-death process with tree encroachment modeled as a (deterministic)
diffusion process:
∂v
(
r
,
t
)
=
α
v
(
r
,
t
)
+
v
max
−
v
(
r
,
t
)
]
(6.8)
[
][
∂
t
2
+
D
∇
v
(
r
,
t
)
−
ξ
sn
[
t
,v
(
r
,
t
)]
,
where
r
y
) is the coordinate vector in a 2D domain,
D
is the diffusion coefficient
associated with tree encroachment, and
=
(
x
,
2
is the Laplace operator. The growth rate
is modeled as a deterministic (logistic) function, i.e., with a rate proportional to
the existing woody biomass
∇
v
and the available resources,
v
max
−
v
. The parameter
α
measures the reproduction rate of the logistic growth, and
1 prevents the dynamics
from remaining locked at
0 after all the tree biomass at a site is destroyed by
intense fires. Diffusion would tend to induce a loss of woody biomass from locations
with relatively high vegetation biomass surrounded by areas with lower vegetation
densities, whereas the logistic growth compensates for this loss. Thus the model
mimics a system in which woody vegetation tends (locally) to carrying capacity and
spreads (laterally) into the areas with lower woody biomass. The stochastic component
of the process is due to fire-induced tree death associated with random and intermittent
v
=
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