Environmental Engineering Reference
In-Depth 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 climate-driven 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 V-shape 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 linear-stability 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 water-stressed conditions)
increases along a rainfall gradient and can be considered as a surrogate variable
 
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