Environmental Engineering Reference
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underlying (nonlinear) deterministic dynamics. We now consider a different type of
stochastic system: In this case the deterministic counterpart of the process is bistable,
and we investigate the effect of noise on these bistable dynamics.
We refer to the case of dryland plant ecosystems, which have been shown to exhibit
bistable behavior, with two stable states corresponding to unvegetated- and vegetated-
land surface conditions ( Walker et al. , 1981 ; Zeng and Neelin , 2000 ). The existence
of these two stable states is usually ascribed to positive feedbacks between vegetation
and its most limiting resource: water (e.g., Walker et al. , 1981 ; Rietkerk and van de
Koppel , 1997 ; Zeng et al. , 2004 ; D'Odorico et al. , 2007a ).
Two different mechanisms are often invoked to explain these feedbacks at different
scales. At the regional or subcontinental scales, vegetation may affect the rainfall
regime ( Zeng et al. , 1999 ) as suggested by global- and regional-climate models
( Charney , 1975 ; Xue and Shukla , 1993 ; Xue , 1997 ; Brovkin et al. , 1998 ; Zeng et al. ,
1999 ; Wang and Eltahir , 2000 ). At smaller (patch-to-landscape) scales, soils beneath
vegetation canopies are generally moister than adjacent bare-soil plots (e.g., Greene
et al. , 1994 ). This feedback has often been attributed to the larger infiltration capac-
ity of vegetated soils ( Walker et al. , 1981 ), which are less exposed to rain-splash
compaction and exhibit higher hydraulic conductivity resulting from the presence of
roots. This mechanism is invoked by a number of competition-facilitation models of
pattern formation ( von Hardenberg et al. , 2001 ; Rietkerk et al. , 2002 ). Other authors
( Zeng and Zeng , 1996 ; Scholes and Archer , 1997 ; Zeng et al. , 2004 ; D'Odorico et al. ,
2007a ) explain soil-moisture-vegetation feedbacks as an effect of the lower evapo-
transpirational losses from subcanopy soils, compared with bare-soil evaporation. As
a result of this feedback in some regions, seedling establishment can occur only in
vegetated-soil plots.
Natural and anthropogenic disturbances acting on bistable dynamics may in-
duce abrupt transitions from the stable vegetated state to the alternative unvegetated
(“desert”) state (e.g., Scheffer et al. , 2001 ). After this transition has taken place, a
significant increase in resource availability (i.e., rainfall) is required to destabilize the
desert state and reestablish a vegetation cover. This view of drylands as deterministic
bistable systems appears to contrast with the existence of a middle ground between
desert and completely vegetated landscapes. It has been shown ( von Hardenberg et al. ,
2001 ; Rietkerk et al. , 2002 ) that spatial heterogeneities and lateral redistribution of
resources can explain the emergence of patchy distributions of vegetation and the
consequent existence of an intermediate stable condition of vegetation between the
two stable states of the zero-dimensional (i.e., nonspatially explicit) system ( van de
Koppel and Rietkerk , 2004 ). 1 In this section we show that a similar result can be in-
duced by temporal fluctuations in environmental conditions rather than by the spatial
1 These studies refer to systems in which vegetation bistability is induced by positive feedbacks between vegetation
and soil moisture at the patch scale, rather than by regional-scale feedbacks between vegetation and the rainfall
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