Environmental Engineering Reference
In-Depth Information
IF Yn
1 THEN
Y
n
þ
1
¼
<
Y
n
þ
C
1
ELSE
Y
n
þ
1
¼
C
2
*Y
n
To learn about the properties of this system we investigate its behaviour for
C
2
¼
Y
n
0.5. The first 1,000 iterations
are discarded to eliminate transient behaviour. We then plot the results with C on the
x
-axis and the corresponding simulation results on the
y
-axis. To survey the model
outcome across a parameter range we start with C
1
¼
0.5 and start each iteration process with y
¼
0.0 and increase it in small
steps up to 0.7. The result is a highly structured pattern with interesting changes in
periodicities. In the covered range, periods between
þ
2 and infinity approximation
occur (Fig.
1.3
).
The example shows that simplicity and complexity can be in quite a close
relationship: A radically simplified ecosystem model can exhibit aspects of a fractal
dynamic (see also Chaps. 3 and 11). If we find such a situation where we have
captured complexity in a simple approach, then we have found an interesting
underlying mechanism. This is something that we are looking for in various
Fig. 1.3 Iterating the model
IF Y
n
<
1 THEN Y
n
þ
1
¼
Y
n
þ
C
1
ELSE Y
n
þ
1
¼
Y
n
-C
2
*Y
n
,
C
2
0.5,
C
1
is drawn on the
x
-axis. Drawn over the according C
1
value, the
y
-axis shows 500 successive
calculation results which exhibit different periodicities. The first 1,000 iterations of the simulation
for each value of C
1
are not drawn to eliminate transient behaviour
¼