Environmental Engineering Reference
In-Depth Information
Prey density
100
10
Time
Space
0
Fig. 7.6 Diffusion-induced chaos in (
7.10
and
7.11
) along a gradient in the prey growth rate.
Parameters:
K ¼
1,
a ¼ b ¼
5,
e ¼
1,
m ¼
0.61,
g ¼ h ¼ Y
3
¼
0,
D
1
¼ D
2
¼
10
4
(a.u.),
no-flux boundary conditions
X
n
@
Y
i
@
t
¼
2
Y
j
rn
i
Y
i
þ
f
i
ð
Y
Þþ
D
ij
r
F
i
Y
ðÞ
;
t
(7.12)
j¼
0
Here, Y
¼
{
Y
i
;
i
¼
0, 1, 2,
...
,
N
} denotes the population densities of the
N
species at time
t
and position x
(
x
,
y
,
z
). The term
f
i
describes the growth,
death and interactions of the
i
th species, which, as we have seen, may depend on the
time
t
, spatial location x and a set of constant parameters. The
D
ij
are the self- and
cross-diffusion coefficients. As in the previous examples, the self-diffusion coeffi-
cients
D
ij
reflect the motility of the species with respect to its own spatial gradient.
Cross-diffusion is the dispersal of a species along the gradient of others, which
facilitates the description of some behavioural strategies like neutrality, attraction
or repulsion (Skellam 1973). The velocity vector v
i
of the
i
th species gives the
speed and direction for both the common passive advection, with the surrounding
transport medium as water or air and the potential individual capacity of active
locomotion. The nabla operator
¼
z
) is simply the vector of the
partial derivatives with respect to the spatial directions, with the dot product
r
r¼
(
/
x
,
/
y
,
/
∂
∂
∂
∂
∂
∂
denoting now the three-dimensional Laplacian. Environmental and/
or demographic variability may be introduced into the model via a density-depen-
dent external stochastic force
F
i
with certain noise characteristics. Note, that for
F
i
2
¼rr
0(
7.1
) has to be interpreted as a stochastic PDE and solutions Y to (
7.12
) then
constitute stochastic processes. This chapter has provided a very small collection
and short description of selected spatiotemporal pattern forming mechanisms,
6¼
