Chaos. Write a program to study the following simple model for population growth, which might be applied to study fish in a pond, bacteria in a test tube, or any of a host of similar situations. We suppose that the population ranges from 0 (extinct) to 1 (maximum population that can be sustained). If the population at time t is x, then we suppose the population at time t + 1 to be rx (1 − x), where the argument r, known as the fecundity parameter, controls the rate of growth. Start with a small population—say, x = 0.01—and study the result of iterating the model, for various values of r. For which values of r does the population stabilize at x = 1 − 1/r ? Can you say anything about the population when r is 3.5? 3.8? 5?
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