Environmental Engineering Reference
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B.2 Turing-like instability
In the study of nonlinear chemical systems, Turing ( 1952 ) found that the diffusion
of two species (reagents) may lead to pattern formation when they have different
diffusivities. In the absence of diffusion both species reach a stable and spatially
uniform steady state, whereas diffusion may be able to destabilize this state ( diffusion-
driven instability ), leading to the formation of spatial patterns. Known as Turing's
instability , this mechanism seems to be counterintuitive. In fact, diffusion is usually
believed to act as a homogenizing process, leading to the dissipation of concentration
gradients of the diffusing species. Conversely, Turing 's model (1952) shows that
diffusion may lead to the emergence of spatial heterogeneity in the coupled nonlinear
dynamics of two diffusing species. In the literature on symmetry-breaking instability
in chemistry and biology, the two diffusive species are often called activator and
inhibitor and pattern emergence requires (i) nonlinear local dynamics and (ii) a faster
diffusion for the inhibitor than for the activator (e.g., Murray and Maini , 1989 ).
Patterns emerging from Turing's instability are self-organized, in that they origi-
nate from the internal dynamics of the system and are not imposed by heterogeneities
in the external drivers. Thus this mechanism is often invoked to explain the emer-
gence of self-organized patterns also in fields other than chemistry, such as physics
and biology, in systems with two or more diffusing species. Notable examples in-
clude convection in fluid mixtures ( Platten and Legros , 1984 ), the formation of shell
patterns from pigment diffusion ( Murray , 2002 ), and vegetation pattern formation
from diffusion-induced instability in arid landscapes (e.g., HilleRisLambers et al. ,
2001 ). The emergence of natural patterns from Turing's instability was experimen-
tally demonstrated in a chemical system ( Castets et al. , 1990 ) and in nonlinear optics
(e.g., Staliunas and Sanchez-Morchillo , 2000 ). We are not aware of any similar ex-
periment for the case of environmental patterns. Thus, although models based on
Turing's instability are capable of generating patterns resembling those observed in
nature, there is no conclusive experimental evidence suggesting that these patterns do
emerge from Turing's dynamics. One of the major challenges in the application of
Turing's activator-inhibition model to environmental systems arises from the need to
recognize two or more leading state variables and to assess whether they do diffuse in
space. The diffusive character of the spatial dynamics of both activator and inhibitor
is fundamental to the development of a sound Turing-like model of pattern forma-
tion, in that diffusion is crucially important to the emergence of symmetry-breaking
instability in a Turing system.
We present the mathematical framework of Turing's models for the case of two
species, u and
, diffusing across a 2D infinite domain
. The dynamics of u
are modeled by two differential equations involving both diffusive terms and
functions of the local values of the state variables (e.g., Murray and Maini , 1989 ;
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