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by Derek Rush (Submitted: 7/18/2002 )

[screen shot]

Download Popeq
If clicking does not initiate a download, try right clicking or control clicking and choosing "Save" or "Download".

(You can also run this model in your browser, but we don't recommend it; details here.)

Model "Popeq" presents the well known population equation of biology Pn+1=APn(1-Pn)
in a form both attractive and instructive, where the multiple iteration solutions are
displayed in full rather than plotting only the final values.
The NetLogo plot window shows the overall picture and the graphics window (although
somewhat smaller than usual) is used to graph the complete set of iterations for a
single step solution. A "print" of the values from the iterations is also provided.

Verhulst in the 19th century showed a model of growth restricted by the environment
in which the largest population that can be supported is unity and the Malthusian
factor or growth is curtailed as a function of the population size. It is this
original relationship that is modeled here but it has wider associations in the
field of mathematics with fractals and chaos theory.

To use the model press "setup" which clears the plot and the graph. Command Center has
its own clear for the "print" of the values of the iterations.
"Run" starts the full solution plot using 0.001 increments of 'A', a selected initial
value of 'Pn' and a selected number of iterations 5 to 80. "Run" blocks use of "Step"
and run time is a minute or so depending on the selected number of iterations.
"Step" runs a solution for selected single values of both 'A' and 'Pn', with graph and
printed values of Pn+1 for each iteration. Selecting 'Pn' as well as 'A' provides the means
to explore sensitivity to the initial value of 'Pn'. 'A' is limited to a max of 4 so that
the solutions remain not greater than unity; also the min is set at 1.8 to give good
resolution and because the solutions for 'A' below this value are less important.
It is easy to relate the value of the growth factor 'A' to the plotted solution because
the 'A' slider has the same dimensions as the plot.
Fifteen iterations gives a plot that is both attractive and interesting; also
sensitivity to Pn is not great, a setting of 0.35 is suggested for general viewing.

The model makes for easy appreciation of some features, for example,
using 80 iterations and Pn set at 0.35:
for 1 < A <= 2 e.g. 1.8 a simple approach to the final value
for 2 < A < 3 e.g. approx 2.3 or 2.7 an overshoot or oscillatory approach to the final value
for A >= 3 the solution oscillates first between two final values and then
for A = 3.461 it oscillates between four final values and then
for A = 3.560 there are eight final values and
for A = 3.679 there is an interesting result and
for A = 3.840 there are three final values and there is more....including chaos.

This subject is also related to Fractals and a suggested reference for further reading
is FRACTALS Images of Chaos, by Hans Lauwerier, published by Princeton University Press.

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