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γ
Re( )
k 1
k 2
0
k max
k
Figure B.1. Example of dispersion relation for a two-diffusive-species monodimen-
sional system. k 1 and k 2 are the extremes of the range of unstable Fourier modes, and
k max represents the most unstable Fourier mode.
0 ), we consider full sys-
tem ( B.1 ) and use a Taylor expansion to linearize this set of equations around the
homogeneous steady state, ( u 0
To study the effect of diffusion on the stability of ( u 0
,v
,v
0 );
10
w
.
2 w
t =
J w
+
D
,
D
=
(B.6)
0 d
The solution of Eqs. ( B.6 ) can be written in the form of a sum of Fourier modes:
W k e γ t + i k · r
w ( r
,
t )
=
,
(B.7)
where k
=
( k x ,
k y ) is the wave-number vector, r
=
( x
,
y ) is the coordinate vector, and
W k are the Fourier coefficients ( Murray , 2002 ).
We can obtain the relation between eigenvalues and wave numbers (known as the
dispersion relation, see Fig. B.1 ) b y insert ing Eq. ( B.7 )into( B.6 ) and searching for
nontrivial solutions. Setting k
k x +
=
k y , we obtain
Dk 2
| γ
I
J
+
|=
0
.
(B.8)
For the state ( u 0 ,v 0 ) to be unstable with respect to small perturbations, the solution
of dispersion relation ( B.8 ) should exhibit positive values of Re[
γ
( k )] for some wave
number k
=
0. Using Eq. ( B.8 ), we find that this condition is met when
d f
2
4 d f
d f
u + g
u + g
u g
∂v f
g
∂v >
0
,
>
0
.
(B.9)
∂v
∂v
u
The first of conditions ( B.5 ) combined with the first of conditions ( B.9 ) implies that
d
=
1, indicating that the systemcannot be unstable with respect to small perturbations    Search WWH ::

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