Continuing 33, a 1920 study by Pearl and Reed (appearing

Chapter 1, Problem 1.34

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Continuing 33, a 1920 study by Pearl and Reed (appearing in the Proceedings of the National Academy of Sciences) suggested the values a = 0.03134, b = (1.5887)1010 for the population of the United States. Table 1.1 gives the census data for the United States in ten year increments from 1790 through 1980. Taking 1790 as year zero to determine p0, show that the logistic model for the United States population is P(t) = 123, 141.5668 0.03072 + 000062e0.03134t e0.03134t . Calculate P(t) in ten year increments from 1790 to fill in the P(t) column in the table. Remember that (with 1790 as the base year) 1800 is year t = 10 in the model, 1810 is t = 20, and so on. Also, calculate the percentage error in the model and fill in this column. Plot the census figures and the numbers predicted by the logistic model on the same set of axes. You should observe that the model is fairly accurate for a long period of time, then diverges from the actual census numbers. Show that the limit of the population in this model is about 197, 300, 000, which the United States actually exceeded in 1970. Sometimes an exponential model Q (t) = k Q(t) is used for population growth. Use the census data (again with 1790 as year zero) to solve for Q(t). Compute Q(t) for the years of the census data and the percentage error in this exponential prediction of population. Plot the census data and the exponential model predicted data on the same set of axes. It should be clear that Q(t) diverges rapidly from the actual census figures. Exponential models are useful for very simple populations (such as bacteria in a dish) but are not sophisticated enough for human or (in general) animal populations, despite occasional claims by experts that the population of the world is increasing exponentially.

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