Environmental Engineering Reference
In-Depth Information
4.2.4 State-dependent dichotomous Markov noise in environmental systems
We consider a process in which feedback exists between random environmental
drivers and the state of the system. In a number of systems this feedback is mediated
either by the dynamics of resource availability or by the disturbance regime. For
example, dryland vegetation is typically limited by soil moisture; thus random rainfall
inputs affect vegetation through the dynamics of soil moisture. Vegetation growth is
sustainable only when soil moisture exceeds a certain threshold; otherwise a mortality-
induced decrease in (live) plant biomass occurs. As a result, rainfall inputs determine
the switching between stressed and unstressed conditions. However, these dynamics
are often more complex because of the existence of a positive feedback between
soil moisture and vegetation: In arid and semiarid environments moister near-surface
soils are found beneath vegetation canopies rather than in the surrounding bare-soil
areas ( Greene et al. , 1994 ; Breman and Kessler , 1995 ; D'Odorico et al. , 2007a ). Thus
growth requires less rainwater on well-vegetated soils than on soils with only a thin,
sparse canopy cover. This fact translates into a state dependency in the threshold
θ
of
the random driver q (precipitation), as indicated in Chapter 3.
Another example is represented by the dynamics of woody vegetation in semiarid,
fire-prone environments. In this case the encroachment of woody plants has been
found to be limited by fires (e.g., Anderies et al. , 2002 ; van Wilgen et al. , 2003 ),
which, in turn, depend both on ignition and on the presence of grass fuel (see also
Section 3.2). In the study of the dynamics of woody vegetation in savannas, fire
ignition is the random external forcing. Ignition does not act directly on woody
plants, in that its effect is mediated by grass-fuel availability, which, in turn, is
inversely related to woody-plant biomass B . Relatively high values of B correspond
to a system dominated by woody vegetation, in which herbaceous vegetation (i.e.,
fuel load) is present only in limited amounts. In this system, with the potential for
random ignition (e.g., lightning) being the same, a woodland savanna is less prone
to fires than an open savanna with relatively low tree density. Thus positive feedback
exists between vegetation and fire pressure. The dynamics of woody-plant biomass
can be modeled by Eqs. ( 4.2 ) with state-dependent threshold
( B ). Thus, in this
case, state dependency exists in the probabilities P 1 of being in state 1 and in the
transition rates of the DMN, k 1 ( B )
θ = θ
P 1 ( B ) (see Chapter 2).
This state dependency implies that the dichotomous noise acts on the dynamics as a
multiplicative noise, in that its parameters are state dependent. As noted in Chapter
3, this multiplicative noise cannot be factorized, i.e., it cannot be expressed as the
product of a state-independent dichotomous noise with a suitable function of B ( Laio
et al. , 2008 ).
We now consider the particular case in which the functions f 1 ( B )and f 2 ( B )
are linear as in Eqs. ( 4.2 ). Moreover, we use a linear dependence of k 1 and k 2
on B : k 2 ( B )
=
1
P 1 ( B )and k 2 ( B )
=
=
P 0 +
bB , with b being positive in the case of positive feedback
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