Environmental Engineering Reference
In-Depth Information
the equation needs to be extended itself. The basic formulation for the development
of a biological species is the
logistic growth
equation:
¼:
@c
@t
¼ rc
1
c
k
f ðcÞ
(19.1)
with growth rate
r
and carrying capacity
. For populations we maintain to use the
symbol
c
, which was introduced in previous chapters (with an eye on concentrations
to be described). The term 'logistic growth' was introduced by Verhulst
2
, who
studied the equation already in the first half of the nineteenth century. For small
populations the first (linear)
rc
term is dominant, describing first order growth
r >
k
0.
For high concentrations the population approaches the carrying capacity
k
,
while the temporary growth rate
r
1
approaches zero. Equation (
19.1
)is
a nonlinear differential equation. It has an analytical solution:
ð
c=k
Þ
c
0
k
exp
ðrtÞ
cðtÞ¼
(19.2)
k þ c
0
exp
ð
ðrtÞ
1
Þ
which can easily be implemented using MATLAB
. Here an example command
sequence for parameter input, execution and graphical output:
®
%---------------------- input ---------------------------------
T = 10;
% maximum time
r = 1;
% rate
kappa = 1;
% capacity
c0 = 0.01;
% initial value
%---------------------- execution ---------------------------------
t = linspace (0,T,100);
e = exp(r*t);
c = c0*kappa*e./(kappa+c0*(e-1));
%---------------------- graphical output -------------------------
plot (t,c); grid;
xlabel ('time'); legend ('population');
title ('logistic growth');
The complete code is included in the accompanying software under the name
“logistic.m”.
The graphical output of the M-file is given in Fig.
19.1
, showing the increase of
the population from an initial value towards a maximum value, at which the
carrying capacity of the system is reached.
Also for nonlinear equations it makes sense to examine equilibria, as introduced
in Chap. 18. The logistic equation has two equilibria that are obtained by finding
2
Pierre Fran¸ois Verhulst (1804-1849), Belgian mathematician.
