Environmental Engineering Reference
In-Depth Information
Figure B.2. Spatial pattern emerging for the variable
u
in the Turing system in
Eqs. (
B.10
). The parameters are
a
2.
The parameters
a
and
g
do not influence the emergence of spatial patterns (see the
end of Section
B.2
); they influence only the shape of spatial patterns. The simulation
is carried out over a domain of 256
=
22,
b
=
84,
c
=
113
.
33,
e
=
18, and
d
=
27
.
×
256 cells, each cell representing a spatial step
x
=
y
=
0
.
2.
carrying capacity
b
and a strong negative influence (inhibition) of species
u
on the
growth rate of
v
: In fact, as
u
increases, the second term of the function
g
decreases
nonlinearly.
The homogeneous steady state of this system is
u
0
=
ce
2
ba
2
.The
ab
/
ce
,
v
0
=
/
derivatives of the two functions of (
B.10
) calculated in (
u
0
,v
0
)are
∂
f
/∂
u
=
e
,
a
3
b
2
c
2
e
2
,
2
g
2
e
3
ba
3
,and
∂
b
, and the four condi-
tions (
B.5
)and(
B.9
) leading to diffusion-driven instability become
e
f
/∂v
=
/
∂
g
/∂
u
=−
/
∂
g
/∂v
=−
−
b
<
0,
eb
>
0,
0, and
d
2
e
2
b
2
de
0, respectively.
Figure
B.2
shows an example in which these conditions are met and patterns emerge
from diffusion-driven instability as a hexagonal arrangement of spots with wavelength
−
b
>
−
6
bde
+
>
1
1
e
+
b
−
√
2
bde
1
+
d
λ
2
π/
,
(B.11)
−
d
d
in agreement with the wavelength of the most unstable mode obtained through dis-
persion relation (
B.8
).
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