Environmental Engineering Reference
InDepth Information
1
ζ
_____
Analytical
Numerical
0.8
0.6
Unstable
0.4
0.2
Stable
Stable
P
P
1
P
2
0.2
0.4
0.6
0.8
1
ζ
=
ζ
∗
as a function
Figure 6.10. Analytical and numerical marginal stability curve
of the probability
P
of being in unstressed conditions (with
χ
=
2
.
0;
=
0
.
25; and
ζ>ζ
∗
. Figure taken from
D'Odorico et al.
(
2006c
).
η
=
1
.
454). Patterns emerge for
=
w
)
e
i
k
·
r
d
r
. Figure
6.9
(b) shows that, for given values
with
k
=
k

and
W
(
k
)
(

r

of
1, indicating that the
corresponding homogeneous solutions are stable. Thus no patterns emerge in the
absence of climate fluctuations (i.e., with no switching) because the system evolves
toward a homogeneous state.
ζ
,
,and
η
,
γ
(
k
) is negative both for
P
=
0andfor
P
=
(
k
) may be positive for intermediate values of
P
,
indicating how climatedriven random switching between deterministic dynamics
may indeed trigger instability.
The marginal stability conditions are shown in Fig.
6.10
: The solid line [Eq. (
6.26
)]
separates stable from unstable states in the parameter space. The Vshape is due to the
discontinuous dependence of
γ
v
ζ
(Fig.
6.10
,
dotted curve) there are two values,
P
1
and
P
2
,of
P
(Fig.
6.10
) marking the transition
between stable and unstable states. For
P
<
P
1
the unvegetated state is stable; for
P
1
<
P
<
P
2
the system tends to a spatially heterogeneous stable state with organized
vegetated patches bordered by bare ground. For
P
0
on
P
[Eq. (
6.25
)]. For a given value of
>
P
2
the homogeneous state
v
0
is
stable. The linearstability analysis of the state
v
=
v
0
does not account for the exis
tence of a bound for
0. The numerical simulations carried out to investigate
the effect of this bound show that conditions of instability and pattern formation are
reached for values of
P
1
(Fig.
6.10
, dashed curve) that are slightly different from those
predicted by the analytical methods. However, the existence of a bound at
v
at
v
0
=
0 does
not qualitatively change the stability of the unvegetated state. Interestingly, Fig.
6.10
(solid line) also shows the emergence of a completely nondeterministic limit behavior
in the local dynamics for
v
=
ζ
=
0 (i.e., with no deterministic spatial interactions) and
P
≈
0
.
4, though this behavior disappears when the bound of the dynamics at
v
=
0
is accounted for (Fig.
6.10
).
The parameter
P
(i.e., the probability of not being in waterstressed conditions)
increases along a rainfall gradient and can be considered as a surrogate variable
Search WWH ::
Custom Search