Environmental Engineering Reference
In-Depth Information
sections we review some examples of noise-induced patterns in self-organized en-
vironmental dynamics.
6.2 Models of spatial interactions
Spatial interactions in environmental systems typically result from the dependence of
a state variable
φ
=
,
(e.g., plant biomass, population density, etc.) at any point r
( x
y )
( x , y ), in the neighborhood of r . The nature of this
dependence may in general change as a function of the direction and of the distance
between r and r . In this chapter we concentrate mainly on the case of systems that are
mathematically described by only one state variable,
at points r =
on the value of
φ
φ
, in a 2D domain ( x
,
y ). At any
point r
=
( x
,
y ), the state variable
φ
( r
,
t ) undergoes local dynamics, expressed by a
function f (
0]. For the sake of simplicity
we assume that the local dynamics exhibit only one stable state. To express the effect
of spatial interactions on the dynamics of
φ
) with steady state
φ = φ 0 [i.e., f (
φ 0 )
=
φ
( r
,
t ), we account for the impact that
other points r =
( x ,
y ) of the domain have on
t ). It is sensible to assume that
this impact depends on the relative position of the two points r and r . Because the
strength of the interactions with other sites is likely to vary with distance and direction,
a weighting function
φ
( r
,
r ) is introduced to model spatial interactions as a function
of r and r . We integrate r over the whole domain
ω
( r
,
to account for the interactions
at any point r in the domain
of
φ
( r
,
t ) with the values of
φ
and obtain
ω
∂φ
( r )
r )[
( r ,
0 ]d r .
=
f (
φ
( r ))
+
( r
,
φ
t )
φ
(6.1)
t
Known as the neural model ,Eq.( 6.1 ) has been used to simulate pattern recognition
by the brain and morphogenesis in a variety of natural systems ( Murray , 2002 ). The
right-hand side of ( 6.1 ) consists of two terms: f (
φ
) describes the local dynamics, i.e.,
the dynamics of
that would take place in the absence of spatial interactions with
other points of the domain. The second term expresses the spatial interactions and
depends both on the shape of the weighting function (or kernel ) and on the values
of
φ
r )
φ
in the rest of the domain
.If
ω
( r
,
>
0, the spatial interactions affect the
( r ) is greater or
dynamics of
φ
( r ) positively or negatively, depending on whether
φ
r )
smaller than
0.
When the processes underlying the spatial interactions are homogeneous (i.e., they
do not change from point to point) and isotropic (i.e., they are independent of the
direction), the kernel function is independent of r and exhibits axial symmetry. In this
case
φ 0 , respectively. The opposite happens when
ω
( r
,
<
r
ω
is a function only of the distance, z
=|
r
|
, between the two interacting
r )
points [i.e.,
( z )]. It will be shown that even though the underlying mecha-
nisms are homogeneous, they can lead to pattern formation, i.e., to nonhomogeneous
distributions of the state variable.
ω
( r
,
= ω
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