This equation was meant to model population. Rather than count individual animals, this equation represents the population as a fraction, p, of the maximum population that is supported by the habitat. The g represents the population growth ratio of the species. The formula is:

p'=pg(1-p)

To use this formula, pick a value between 0 and 1 for p (which represents the first generation), and pick a growth rate for g. Use the formula to calculate the population of the second generation. You can then use the population of the second generation to calculate the population of the third generation, and so on.

For growth rates less than 1, the population decreases and dies.

Population graph for g = .9:

At low values of g, (from 1 to 3) the population increases for a few generations, then becomes stable.

Population graph for g = 2:

At higher growth rates (3 to 3.45), the graphs start to get interesting.

The population will settle into a pattern where it alternates between two populations.

Population graph for g = 3.2:

At even higher growth rates (3.45 to 3.54), it will alternate between four populations.

Population graph for g = 3.53:

Then 8, then 16, then 32, etc. Then it turns into chaos.

What do I mean by chaos? In the graph above, the first generation was 10% of the maximum. In the graph below, the growth rate is the same, but the first generation is 90%. Notice that the end result is the same.

Another graph for g = 3.53:

The next two graphs are chaotic. The first graph starts at 20%, the second at 21%.

Population graphs for g = 3.96:

The two graphs turn out radically different. One of the principle ideas of chaos, known as the butterfly effect, is that small changes may have large consequences over time. These graphs show what a difference 1% can make.

For values of g greater than four, the population will grow above its limit and die.

To make this equation easier to study, May and Feigenbaum created one graph that shows the results for many different values of g. The logistic map (below) is a graph for all values of g from 0 to 4 that shows the end result(s) of the equation. The population growth (g) is on the x (horizontal) axis, and the final populations are shown on the y (vertical) axis.

You can now see where the population dies, where it becomes stable, where it alternates between two populations, etc.

For the most part, complexity increases as you go from left to right, but it is interesting that there are a few small windows in the chaotic section of the graph where there is order again. The largest one is around 3.8, where the population starts alternating between three values.

To display a graph, enter the growth and initial population or click a place on the logistic map and hit Graph.

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