Environmental Engineering Reference
In-Depth Information
Y(x,
t)
1
Y(
x
, t)
1
0.5
0.5
0
0
1
1
2
2
3
3
Fig. 7.3 Spatial reverse (
left
) or forward (
right
) propagation of a population with Allee effect
for an initial condition less or larger than the critical radius. Parameters:
r ¼
3,
K ¼
1,
K
-
¼
0.4,
D ¼
10
1
(a.u.), no-flux boundary conditions
7.4 Two Population Models
As we have seen, growth and dispersal of a single species in a constant and
homogeneous environment does not support spatial pattern formation, it merely
balances out spatial differences in population density. However, spatial patterns can
appear in models with at least two interacting and moving populations. Since these
patterns are more striking in two spatial dimensions, we will also move on from one
spatial dimension to a two-dimensional model; that is x
(
x
,
y
). The interaction
and dispersal of both populations is then described by the two equations:
¼
@
Y
1
@
t
¼
2
Y
1
f
1
ð
Y
Þþ
D
1
r
(7.8)
@
Y
2
@
t
¼
2
Y
2
f
2
ð
Y
Þþ
D
2
r
(7.9)
The basic structure of each equation is the same as for the single species model of
(
7.2
), but the respective growth terms
f
1,2
now depend on the vector Y
(
Y
1
,
Y
2
)of
both populations. Also, dispersal of the populations is now possible in two dimen-
sions, indicated by the two-dimensional Laplacian:
¼
2
2
¼
@
x
2
þ
@
2
r
@
@
y
2
which is simply the sum of the second order partial derivatives with respect to the
spatial dimensions. The diffusivity or motility of the populations is given by
D
1
and
D
2
, respectively. A selection of stationary and dynamic patterns will be described
below. All of these patterns arise from the following, very important model of a
prey species
Y
1
, a predator
Y
2
, and a constant top predator population
Y
3
: