Environmental Engineering Reference
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if both species have the same diffusivity. If all four conditions ( B.5 )and( B.9 ) hold,
at least one eigenfunction is unstable with respect to small perturbations and grows
exponentially with time as a consequence of the destabilizing effect of diffusion.
Dispersion relation ( B.8 ) imposes a specific link between eigenvalues
and wave
numbers k . The wave number k max , corresponding to a maximum positive value
of Re[
], represents the most unstable mode of the system. This implies that if
( k max )]
0, this mode grows faster than the others, and the state of the system
for t →∞
is dominated by k max in the sense that, as t →∞
, only k max dictates the
length scale of the spatial pattern.
Because this linear-stability analysis is developed in the limit
0 (i.e., under
the assumption of small perturbations of the homogeneous, steady state), it cannot
provide any information on the state of the system when the perturbation grows in
amplitude. In the absence of nonlinearities in f ( u
| w |→
), solutions of ( B.7 )
of the linearized model would coincide with the exact solutions of ( B.1 )evenaway
from the state ( u 0 ,v 0 ). In this case Eq. ( B.7 ) clearly shows that if the state ( u 0 ,v 0 )
is unstable the perturbations w grow indefinitely. Thus suitable nonlinear terms are
needed to stabilize the pattern through higher-order terms in the Taylor expansion,
which become important when the amplitude of the perturbation is finite. In other
words, the system reaches a steady configuration when the exponential growth of the
eigenfunction is limited by second-order (or higher) terms that come into play once
the perturbation has finite amplitude. In these conditions the (nonlinear) stability of
the system can be partly studied through a more complex mathematical framework
based on the so-called amplitude equations , which investigate the dynamics of the
system in the neighborhood of the most unstable mode. This appendix does not review
these nonlinear methods, and we refer the interested reader to specific literature on
this topic for further details ( Cross and Hohenberg , 1993 ; Leppanen , 2005 ).
)and g ( u
B.2.1 An example of a Turing model
We consider a simple example of a Turing model able to generate spatial patterns
in a system with two species, u (activator) and
(inhibitor). To this end, we use
Eqs. ( B.1 ) with local kinetic functions:
f ( u
u ( a
e )
cu 2
= v
g ( u
( b
where a , b , c ,and e are dimensionless positive constants.
The first equation describes the growth or the decay of the activator and accounts
for a positive interaction between u and
increases, the growth rate of
species u increases. Moreover, the growth rate of u increases with increasing values
of u . The second equation is a generalized logistic growth (e.g., Murray , 2002 ) with
. In fact, as
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