Environmental Engineering Reference

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fluctuations of the state variable, which can be either spatially correlated or uncor-

related. In another major class of stochastic models, noise-induced patterns result

from the random switching between dynamics that simultaneously occurs at all points

of the spatial domain (
Buceta and Lindenberg
,
2002a
,
2002b
;
Buceta et al.
,
2002a
,

2002b
). This mechanism is based on the idea that if the random switching between

the dynamics is global (i.e., it simultaneously occurs across the domain) and rapid, the

system behaves as if it was undergoing the average dynamics obtained as a weighted

mean of the two states. In this context, by rapid switching we mean that the average

residence time of the dynamics in each of its two dynamical states is much shorter

than the time needed by the system to reach the equilibrium state.

Thus if we take as an example two Turing models
5
and we randomly and rapidly

switch between them, the system experiences only the average Turing dynamics. We

can envision cases in which, separately, neither one of the two dynamics is able to

lead to pattern formation, and their average exhibits diffusion-induced symmetry-

breaking instability. In these conditions patterns emerge from the nonequilibrium

random (global) alternation between the dynamics. Similar models can be constructed

with two suitable biharmonic (or neural) models (
Buceta and Lindenberg
,
2002a
)

that by themselves are unable to exhibit symmetry-breaking instability. The random

switching between these two models may lead to mean dynamics that are capable of

generating patterns.

We consider as an example a system in which Turing's instability is induced by

noise. To this end, consider a two-variable dynamical system described by reaction-

diffusion equations, namely

∂φ

1

2

t
=

f
1
,
2
(

φ
1
,φ
2
)

+∇

φ
1
,

(5

.

95a)

∂

∂φ

2

2

t
=

g
1
,
2
(

φ
1
,φ
2
)

+

d
1
,
2
∇

φ
2
,

(5

.

94b)

∂

where
f
1
,
2
(

2
) are the two pairs of functions describing the local

dynamics of states 1 and 2. The system switches between state 1, in which the local

kinetics is expressed by
f
1
and
g
1
and the diffusion ratio is
d
1
(these three quantities

are hereafter simply indicated as
A
1
), and state 2, in which the kinetics is modeled

by the functions
f
2
and
g
2
and the diffusion ratio is
d
2
(indicated as
A
2
). Neither of

Eqs. (
5.95
) for state 1 or 2 meets all the conditions for pattern formation in the Turing

models [see Appendix B, Eqs. (
B.5
)and(
B.9
)]. Thus, separately, the two dynamics

do not lead to pattern formation. Each control parameter (
f
,
g
,or
d
) alternates

φ

,φ

2
)and
g
1
,
2
(

φ

,φ

1

1

5
Turing models (
Turing
,
1952
) involve a system of two or more coupled deterministic reaction-diffusion equations

in which a homogeneous equilibrium state is destabilized by the spatial coupling (i.e., by diffusion). This instability

leads to pattern formation when the most unstable mode has a wave number different from zero. More details on

the Turing model can be found in Appendix B.

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