R DT
ð Þ ¼ 4:57=DT
ð
ÞÀ 1 þ 2 þ 4 þ 8 þ Á Á Á1=DT
½
Š
ð 4:5Þ
or
R DT
ð Þ ¼ R 1
ð Þ þ 1
ð
Þ =DT À 1 þ 2 þ 4 þ 8 þ Á Á Á1=DT
ð
Þ ,
ð4:6Þ
the boundary between chaos and finitely numbered solutions for the spectrum of
discrete steps.
This model shows you that the R for DT ¼ 0 is infinity. This result is correct
since chaos is typically not noticed on the continuous level. Chaos occurs on the
continuous level only if you are stuck with a specific DT in your particular problem,
and the parameters lie within the critical range. For a full discussion of this and
other versions of chaos, see Jenson [1].
Compare the chaotic time paths of your model to a truly random number.
We defined such a number in the model as RAND and calculated its delayed
value (Fig. 4.6).
The random number, plotted against its delayed value, is shown in Fig. 4.7. Re-run
the model several times and see how the graph changes. The difference between
chaotic—but deterministic—behavior and random behavior should become apparent.
At this point we should ask if chaos occurs in nature. We find that indeed it does.
Water drips from a faucet chaotically, heart beats and brain wave variations show
chaos. Both living and non-living systems seem to show chaos. Why? To what
Fig. 4.6
Fig. 4.7
4.1 The Emergence of Chaos in Population Models
51