Environmental Engineering Reference
In-Depth Information
Fig. 7.1
Left
: Spirals in an amoeba population (
Dictyostelium discoideum
). The base line of the
photo is about 28.9 mm (Courtesy of Christiane Hilgardt and Stefan C. M
uller, University of
Magdeburg).
Right
: Satellite image of tiger bush in Niger, the darker lines of woodland are on
average about 20-40 m wide and 50-100 m apart (Courtesy of the US Geological Survey)
€
in models of biological dynamics, and as such they are related to coupled map lattices
and cellular automata. Often, PDEs take the form of reaction-diffusion equations,
especially if individuals are assumed to perform a random walk, similar to small
particles in a fluid whose molecules are in constant thermal motion. PDEs can also be
used to model directed motion in the form of advection and even random environ-
mental fluctuations. The next few pages give an overview of some basic PDE models
and the interesting range of patterns they may generate.
7.2 Single Population Models
In the case of a single species homogeneously distributed in space, the rate of
change of the species population density
Y
Y
(
t
), that is, its temporal behaviour, is
described by the ordinary differential equation (ODE):
¼
dY
dt
¼
fY
ð
;
t
;
c
Þ
(7.1)
Here,
f
describes all processes relevant for the species growth, i.e. reproduction,
competition and predation. In general,
f
will depend on a set
c
R
k
of biological
parameters, like birth and mortality rates. Additionally, the growth rate parameters
of the species may also explicitly depend on the time
t
, i.e. reflecting seasonality of
reproduction or increased mortality in harsh winter conditions. In the following, we
always assume that
f
does not explicitly depend on time, and the variables
t
and
c
are dropped from the notation. A real-valued function
Y
is a solution of this
equation if its temporal derivative satisfies (
7.1
). In order to uniquely identify a
particular solution, it is also necessary to specify the initial population density
condition in the form
Y
(0)
2
¼
Y
0
.
