Environmental Engineering Reference
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B.3 Kernel-based models of spatial interactions
We classify as kernel-based models those modeling frameworks in which spatial
interactions are expressed through a kernel function, accounting both for short-range
and long-range coupling (see Section 6.2 ). In most of these models self-organized
patterns arise as a result of short-range cooperation (or activation ) and long-range
inhibition. These spatial interactions cause symmetry-breaking instability and the
system converges to an asymmetric state, which exhibits patterns. As in the case of
Turing models, the convergence to this state is due to suitable nonlinear terms, which
prevent the initial (linear) instability from growing indefinitely.
We first consider a particular type of kernel-based models, whereby the nonlinearity
is not in the spatial coupling but in an additive term. These models are often known
as neural models because of their applications to neural systems. Some of the most
fascinating and complex pattern-forming processes existing in nature are associated
with neural systems. Typical examples include the process of pattern recognition, the
transmission of visual information to the brain, and stripe formation in the visual
cortex ( Murray , 2002 ). The framework of a neural model is often used to represent
other systems, including the case of vegetation dynamics in spatially extended systems
( D'Odorico et al. , 2006c ).
Neural models can in general be developed for systems with more than one state
variable. However, unlike Turing models, pattern-forming symmetry-breaking insta-
bility can emerge even when the dynamics have only one state variable. Thus we
concentrate on the case of neural models that are mathematically described by only
one state variable,
φ
, in a 2D domain ( x
,
y ). At any point r
=
( x
,
y ) of the domain, the
variable
) that is indepen-
dent of spatial interactions. The local dynamics exhibit a steady state at
φ
( r ) undergoes local dynamics expressed by a function f (
φ
φ = φ 0 [i.e.,
0]. We express the effect of spatial interactions by using kernel functions, as
explained in detail in Section 6.2 . The impact that other points r ( x
f (
φ 0 )
=
,
y )haveonthe
t ) depends on the relative position of the two points r and r and is
expressed through a weighting (or kernel ) function,
dynamics of
φ
( r
,
r ). We integrate r over the
ω
( r
,
t ) with all points r in
whole domain
to account for the interactions of
φ
( r
,
:
∂φ
r )[
( r ,
φ 0 ]d r .
t =
f (
φ
)
+
ω
( r
,
φ
t )
(B.12)
The terms in Eq. ( B.12 ) are explained in detail in Section 6.2 . In neural models of
pattern formation the interactions between cells are typically represented by short-
range activation and long-range inhibition ( Oster and Murray , 1989 ). In this case the
kernel is positive at small distances, z
r |
, and becomes negative at greater
distances (Fig. B.3 ). This type of framework has been proposed as a model for spatial
interactions within plant communities (e.g., Lefever and Lejeune , 1997 ; Yo ko z awa
et al. , 1999 ; Couteron and Lejeune , 2001 ). A kernel with the shape illustrated in
=|
r
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