A simple iterative model of population growth (and decline) over time is

*x* → *μ* *x* (1 − *x*)

We choose a value for the parameter *μ* and a starting *x* value, then
repeatedly substitute the results of each mapping into *x* on the right-hand
side of the mapping; the list of partial results can be viewed as a time series
of the population levels over a discrete sequence of times.

For example, if we let *μ* be 2, and let *x* start with 0.25, the
first few iterates are

{0.25, 0.375, 0.46875, 0.498047, 0.499992, 0.5, 0.5, 0.5, ...}.

Similarly, if we let *μ* be 2.5, the first few iterates are

{0.25, 0.46875, 0.622559, 0.587448, 0.605882, 0.596973, 0.601491, 0.599249, 0.600374, 0.599813, 0.600094, 0.599953, 0.600023, 0.599988, 0.600006, 0.599997, 0.600001, 0.599999, 0.6, 0.6, 0.6, ...}.

For values of *μ* between 1 and about 3, the iteration settles down to a
single value.

If we let *μ* be 3.2, the list of iterates is

{0.25, 0.6, 0.768, 0.570163, 0.784247, ..., 0.799455, 0.513045, 0.799455, 0.513045, ...};

the iteration eventually oscillates forever between the two values 0.799455
and 0.513045. The iteration has period two for values of *mu* between 3
and about 3.4494.

If we choose *μ* = 3.46, the iterates are

{0.25, 0.64875, 0.788442, 0.577132, ..., 0.838952, 0.467486, 0.861342, 0.413234, 0.838952, ...},

where the last four values repeat forever, or fall into a period-4 cycle. The
phenomenon of period-doubling for greater values of *μ* is called
*bifurcation*.

For greater values of *μ*, the list of iterates is chaotic; the iteration
does not settle down to a fixed point or cycle, but rather apparently behaves
randomly.

Here we visualize the iterations of the mapping for values of *μ*
ranging from 2.4 to 3.8 in increments of 0.2. We see that for small values
of *μ* the iteration settles down to a fixed point or a cycle,
but that for larger values the behavior is chaotic.

Designed and rendered using *Mathematica* 3.0 for the Apple Macintosh,
with more than a little help from Peitgen et al, Chaos and Fractals,
Springer-Verlag, 1992.

Copyright © 1997–2017 by Robert Dickau

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