Chaos and ProbabilityHere we will investigate the long term behavior of the population dynamics. We will let the population evolve for a long time, and see how often it takes values in certain intervals (For example, how often is it between 0 and .1? How often is it between .1 and .2? etc...). A bar graph containing such information is called a histogram, and we will create histograms for various initial populations and choices of r and see if we can learn anything interesting about the population behavior from a statistical point of view. To try out the histogram set c = 1, r = 3, P_{0} = 0.1, and N = 50000. Click [Iterate] to fill up the lower graph window which shows the population evolution (Note, only the first 200 points will be plotted for better visualization), then click [Draw Histogram] to visualize the population distribution. The numbers on the top of each column give the percentage of the total number of points contained in each range. The population range (minimum population to maximum population) is divided into 20 equal bins (labelled 0 to 19). What does the histogram look like if you set N to 100 ? Why do you think this happens? At what value of N does it "stabilize"? Try a few different values of r in the range 2 < r < 3. You are now ready to do Part Two of the Birth, Growth, Death, and Chaos Problem Set.
