Environmental Engineering Reference
In-Depth Information
(a)
(b)
Figure 4.26. The pdf of average soil moisture during the growing season. The pa-
rameters for soil and vegetation are as follows: n
=
0
.
43, Z r =
1
.
40 m, K s =
9
.
5
10 6 m/s, s fc =
8, s =
0
.
0
.
36, E max =
3
.
2 mm/day. The rainfall is characterized by
21 d 1 , with CVs as follows: (a) CV[
α =
12
.
4 mm/storm and
λ =
0
.
α
]
=
0
.
45,
CV[
λ
]
=
0
.
23; (b) CV[
α
]
=
0
.
22, CV[
λ
]
=
0
.
11 (after D'Odorico et al. , 2000 ).
parameters. The two preferential states correspond to “dry” and “wet” seasonal soil-
moisture conditions. The implications of the emergence of bimodal behavior is of
foremost importance for ecohydrologic dynamics because it implies that the system
is more likely to be found in two states that are far from the long-term average,
whereas the long-term average conditions occur with low probability ( D'Odorico
et al. , 2000 ). Moreover, the bimodal behavior enhances the likelihood of occurrence
of dry conditions and the effect of disturbance of climate fluctuations on terrestrial
ecosystems ( Ridolfi et al. , 2000 ).
4.7 Noise-induced phenomena in multivariate systems
The study of the effect of noise on the dynamics of multivariate environmental sys-
tems is limited by the lack of a general analytical framework for the solution of
sets of nonlinear stochastic differential equations. In most cases, these systems are
investigated through numerical simulations, though some approximate methods were
also developed (e.g., May , 1973 ; Horsthemke and Lefever , 1984 ). In this section we
present the case of multispecies population dynamics, investigated through either a
linearization of the stochastic equations ( May , 1973 ) or with methods based on the
mean-field theory 5 ( Mankin et al. , 2004 ). Other multivariate systems are presented
in this chapter in the context of populations with random interspecies interactions
(Subsection 4.7.3 ) or with coherence-resonance behavior (Subsection 4.8.4 ).
5 A comprehensive description of the mean-field approach is given in Chapter 5 (in particular, see Box 5.4), in which
this technique is used to solve systems of stochastic ordinary differential equations obtained by spatial discretization
of stochastic partial differential equations.
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