Environmental Engineering Reference
In-Depth Information
ω
Z
r
r
Z
Figure B.3. Typical kernel that exhibits local activation-long-range inhibition.
Fig. B.3 can be obtained for example as the difference between two exponential
functions of the form
= b 1 exp
2
b 2 exp
2
z
q 1
z
q 2
ω
( z )
,
(B.13)
where 0
< q 1 < q 2 and b 1 and b 2 are two coefficients expressing the relative im-
portance of the facilitation and competition components of the kernel (see also
Chapter 6).
The dynamics expressed by Eq. ( B.12 ) may lead to pattern formation through
mechanisms that resemble those of Turing's instability. In fact, patterns emerge as a
result of the spatial interactions, which destabilize the uniform stable state
φ 0 of the
local dynamics. To study the stability of the state
φ = φ 0 with respect to infinitesimal
perturbations, we linearize Eq. ( B.12 ) around the steady state
φ = φ 0 . Indicating by
ˆ
φ = φ φ 0 the amplitude of the (“small”) perturbation, we obtain
ˆ
φ
ˆ
) ˆ
f (
r
( r ,
t )d r ,
=
φ
φ
0 )
+
ω
(
|
r
|
φ
(B.14)
t
where f (
φ = φ 0 .
Solutions of Eq. ( B.14 ) can be expressed in the form of integral sums of the
harmonics ˆ
φ 0 ) is the derivative of the function f (
φ
), calculated for
φ
( r
,
t )
exp[
γ
t
+
i k
·
r ], where each harmonic is a solution of ( B.14 ),
k
=
( k x ,
k y ) is the wave-number vector, and the growth factor
γ
is an eigenvalue of
r
Eq. ( B.14 ). Substituting this solution into Eq. ( B.14 ), setting z
, and cancel-
ing out the exponential function, we obtain the dispersion relation, that is, the relation
between k and
=|
r
|
γ
in solutions of ( B.12 ) obtained as small perturbations of the state
φ = φ 0 :
f (
f (
γ
( k )
=
φ 0 )
+
ω
( z )exp[ i k
·
z ]d z
=
φ 0 )
+
W ( k )
,
(B.15)        Search WWH ::

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