Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 241

( Dubois-Violette et al. , 1978 ; Chandrasekhar , 1981 ; DiPrima and Swinney , 1981 ;
Cross and Hohenberg , 1993 ; Drazin , 2002 ), chemistry ( Turing , 1952 ), and biology
( Murray , 2002 ). This broad body of literature on patterns in nature inspired a number
of studies proposing a variety of models to explain pattern formation in the environ-
mental sciences, particularly in the case of vegetation.
One of the early examples of process-based models of self-organized vegetation
can be found in Wa t t ( 1947 ), who invoked mechanisms of reallocation of nutrients
and water to explain the emergence of patchy vegetation distributions: This model
showed that, as nutrient and water availabilities decrease, plants tend to grow in
clumps. The emergence of these aggregated structures is motivated by the need to
concentrate scarce resources (e.g., soil moisture, soil nutrients) in smaller areas,
thereby increasing the likelihood of vegetation survival within vegetated patches that
are richer in resources. In subsequent years, the idea that the mechanisms underlying
vegetation pattern formation are intrinsically dynamic and originate from interactions
among plant individuals was better articulated and formalized (e.g., Greig-Smith ,
1979 ; Thiery et al. , 1995 ). These studies paved the way for a new generation of models
( Lefever and Lejeune , 1997 ; Klausmeier , 1999 ; von Hardenberg et al. , 2001 ; Rietkerk
et al. , 2002 ), explaining vegetation patterns as the result of self-organization emerging
from symmetry-breaking instability, i.e., as a process in which the existence of both
cooperative and inhibitory interactions at two slightly different spatial ranges may
induce the appearance of heterogeneous distributions of vegetation with wavelengths
determined by the interactions between the two spatial scales ( Lefever and Lejeune ,
1997 ; Lejeune et al. , 1999 ; Lejeune and Tlidi , 1999 ; Barbier et al. , 2006 ).
These phenomena are often interpreted as the result of deterministic mechanisms
(see Appendix B) of symmetry-breaking instability, whereas stochastic environmental
drivers are usually considered as the source of random perturbations in the ordered
states of the system. Thus random environmental fluctuations have been frequently
associated with disturbances able to destroy patterns formed by deterministic dy-
namics (e.g., Rohani et al. , 1997 ). However, in Chapter 5 we showed that random
fluctuations are also able to play a constructive role in the dynamics of nonlinear
systems, in that they can induce new dynamical behaviors that did not exist in the
deterministic counterpart of the system (e.g., Horsthemke and Lefever , 1984 ). In
particular, stochastic fluctuations have been associated with the emergence of new
ordered states in dynamical systems, both in time (e.g., Horsthemke and Lefever ,
1984 ) and in space ( Garcia-Ojalvo and Sancho , 1999 ). Thus random environmental
drivers are not necessarily in contraposition to pattern formation. Indeed, patterns have
been shown ( van den Broeck et al. , 1994 ; Garcia-Ojalvo and Sancho , 1999 ; Loescher
et al. , 2003 ) to emerge as a result of the randomness inherent in environmental fluc-
tuations. These random drivers may induce a symmetry-breaking instability, which
leads to morphogenesis. Alternatively, noise may stabilize transient patterned features
temporarily emerging in the underlying deterministic dynamics. We refer the reader