## Chaos Algorithm

This algorithm uses a population growth model developed by biologist Robert May in 1976. The model represents a logistic difference equation used by ecologists for calculating demographics of populations over time. The model is a modification of the Malthusian scenerio of unrestrained growth (x next = rx) a linear growth function developed by Thomas Malthus in Essay on the Principle of Population (1798). The growth model is an expression of how populations increase geometrically to the power x (e.g. 1,3,9,27,81...) while food supplies increase at an arithmetic rate ( e.g. 0, 10, 20, 30 40 ...). When left unchecked, populations will catastrophically outpace food supplies. Ecologists added an extra term x next = rx(1-x) to reflect the realities of population dynamics and survival rates. X is a number between 0 and 1 representing a population at a given time. r is a positive number representing the rate of reproduction and starvation combined.

Please note that the return values from the chaos algorithm are represented by numbers with several decimal places. In order to normalize these values the algorithm must use integers; therefore, the algorithm uses a multiplier to adjust the results.

Users can anticipate the return values from the following information:

When r is between 0 and 1 the population will die off, because the rate of reproduction is not high enough to sustain a population.

When r is between 1 and 2 the population will increase and quickly stabilize to a steady state.

When r is between 2 and 3 the population will take longer to increase and stabilize to a steady state than values between 1 and 2.

When r is between 3 and 3.45 the population values will increase and bifurcate. The population size alternate between two values forever. This phenomenon is called "period 2."

When r is between 3.45 and 3.54 the population values will increase until it reaches period 4, a continuos repetition of four values.

When r is between 3.54 and 3.56 the values will reach several different states: period 8, period 16, period 32 etc... depending on the exact value of r.

When r is equal to or greater than 3.57 the population values will demonstrate chaotic behavior.

For more information see "Chaos: Making a New Science" by James Gleick
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