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

. We choose a value for the parameter

muand a startingxvalue, then repeatedly substitute the results of each mapping intoxon 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

mube 2, and letxstart 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

mube 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

mubetween 1 and about 3, the iteration settles down to a single value.If we let

mube 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

mubetween 3 and about 3.4494.If we choose

mu= 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

muis calledbifurcation.For greater values of

mu, the list of iterates is chaotic; the iteration does not settle down to a fixed point or cycle, but rather apparently behaves randomly.Bifurcation diagram:

Here we visualize the iterations of the mapping for values of

muranging from 2.4 to 3.8 in increments of 0.2. We see that for small values ofmuthe iteration settles down to a fixed point or a cycle, but that for larger values the behavior is chaotic.Designed and rendered using

Mathematica3.0 for the Apple Macintosh, with more than a little help from Peitgenet al, Chaos and Fractals, Springer-Verlag, 1992.

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