Environmental Engineering Reference
In-Depth Information
Now, for a population that is inhomogeneously distributed across its habitat, the
population density
Y
Y
(
t
, x) of a single species changes not only over time
t
, but
also with spatial location x. In biological scenarios, the domain or habitat will
usually be some bounded subset of three-dimensional space. In this case, x usually
denotes the Cartesian coordinates of a point in that domain. Let us first consider
only one-dimensional spatial domains, in which case the spatial location is simply
denoted by the real number x.
If it is assumed that the motion of individuals on this domain can be approxi-
mated by a random walk, the rate of change is given by the reaction-diffusion
equation:
¼
2
Y
@
Y
@
D
@
t
¼
fY
ðÞþ
;
x
(7.2)
@
x
2
As in the non-spatial case, the growth term
f
describes the species growth, which
now may additionally depend on the spatial location
x
. To shorten notation, the
variable
x
is also usually omitted in the following. The second term now describes the
species dispersal, usually down its own spatial density gradient. The diffusion
coefficient
D
reflects how motile the individuals of the population are. The fact
that the population density now depends on two independent variables is reflected in
the partial derivatives in (
7.2
), one with respect to time
t
and the other of second order
with respect to the spatial variable
x
. A solution to this equation is a real-valued
function
Y
, whose partial derivatives satisfy (
7.2
) and which has a given initial
population distribution
Y
(0,
x
)
Y
0
(
x
). If the spatial domain is bounded with
boundary
d
, the solution additionally needs to fulfil suitable boundary conditions.
An important special case is referred to as the no-flux boundary conditions, given by:
¼
@
Yt
ðÞ
@
;
x
n
¼
0
for all points
x
2 d
. Here, with respect to the outward pointing normal vector, the
partial derivative is perpendicular to the boundary. This boundary condition simply
reflects the assumption that no individual leaves or enters the domain through the
boundary; i.e., because the population is physically confined to a certain habitat or the
habitat is surrounded by a hostile environment. In the next sections we will see how
the form of the growth term
f
determines which spatiotemporal patterns these solu-
tions may exhibit. Note, that for all examples we assume no-flux boundary conditions.
7.2.1 Exponential Growth
For the Malthusian assumption (Malthus 1798) of exponential growth of a single
species, the growth term takes the form:
f
ð
Y
Þ¼
rY
(7.3)
