Environmental Engineering Reference
In-Depth Information
(e.g.,
Murray
,
2002
). For each of the two climate-controlled states (i.e., stressed and
unstressed conditions) vegetation dynamics can be expressed as
∂v
,
(
r
t
)
r
−
(
r
,
t
)d
r
,
=
f
1
,
2
[
v
(
r
,
t
)]
+
w
(
|
r
|
)
v
(6.18)
∂
t
where the subscripts 1 and 2 denote the local dynamics of stressed and unstressed
vegetation, respectively, and
represents vegetation biomass, which is normalized
with respect to the ecosystem carrying capacity in state 2 (i.e., 0
v
≤
v
≤
1). The
local dynamics are expressed by
f
1
,
2
(
v
)asinEqs.(
6.16
)and(
6.17
). The weighting
(or
kernel
) function
(
z
) is modeled here as the difference between two Gaussian
functions (e.g.,
Murray
,
2002
):
w
b
1
exp
2
b
2
exp
2
z
d
1
z
d
2
w
(
z
)
=
−
−
−
,
(6.19)
where
b
1
and
b
2
express the relative importance of facilitation and competition pro-
cesses and
d
1
and
d
2
are related to the radii of canopy and root footprints, respectively.
The kernel function
w
(
z
) qualitatively has the shape shown in Fig.
6.9
when
d
1
<
d
2
and
b
1
b
2
.
We use DMN as a mechanism to drive the switching between the two local dynamics
[Eqs. (
6.16
)and(
6.17
)]: With probability (1
>
−
P
) vegetation is water stressed and
its dynamics are modeled by Eqs. (
6.16
)and(
6.18
). With probability
P
plants are
unstressed and vegetation growth is modeled by Eqs. (
6.17
)and(
6.18
). Thus the effect
of (large-scale) random, interannual rainfall fluctuations cause the switching between
these dynamics. This alternation simultaneously occurs at all points in the 2D domain.
When DMN is used to model the random switching, vegetation dynamics can be
expressed by the stochastic integral-differential equation
∂v
(
r
,
t
)
=
f
+
[
v
(
r
,
t
)]
+
f
−
[
v
(
r
,
t
)]
ξ
(6.20)
dn
∂
t
b
1
exp
b
2
exp
r
−
2
r
−
2
−
|
r
|
−
|
r
|
(
r
,
t
)d
r
,
+
−
v
d
1
d
2
where
ξ
dn
is a DMN assuming values of
−
1 and 1 with probability
P
and (1
−
P
),
1
2
respectively, and
f
±
[
.
We focus on the case in which neither one of the dynamics of stressed and unstressed
vegetation [Eqs. (
6.15
)-(
6.17
)] is - separately - able to generate patterns, whereas the
random switching induces pattern formation (
D'Odorico et al.
,
2006c
). The response
of (woody) vegetation to water-stress conditions is relatively slow (a few decades)
compared with the year-to-year climate variability considered in this study (e.g.,
Archer et al.
,
1988
;
Barbier et al.
,
2006
). Thus we can investigate pattern emergence
v
(
r
,
t
)]
=
{
f
1
[
v
(
r
,
t
)]
±
f
2
[
v
(
r
,
t
)]
}
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