Author(s):

David A. Smith and Lawrence C. Moore

The von Foerster paper argues that the differential equation modeling growth of world population **P** as a function of time **t** might have the form

**dP/dt = k P**^{1+r},

where **r** and **k** are positive constants. Before attempting to solve this differential equation, we explore whether it can reasonably represent the historical data.

The model asserts that the *rate of change* (derivative) of P should be *proportional* to a *power* of **P**, that is, the rate of change should be a *power function* of **P**. We can test that assertion by looking at a log-log plot of **dP/dt** versus **P**. But first we have to estimate the rate of change from the data. We do this by calculating symmetric difference quotients.

- Explain why
**(P**_{i+1} - P_{i-1}) / (t_{i+1} - t_{i-1}) is a good estimate of **dP/dt** at **t = t**_{i}.
- Construct the symmetric difference quotients (SDQ) approximating
**dP/dt** from the historical data.
- Construct a log-log plot of SDQ versus population. Decide whether you think it is possible that
**dP/dt** is a power function of **P**. Keep in mind that we have only very crude approximations to values of **dP/dt**, and many of them are constructed on intervals that are *not* symmetric about the corresponding year.
- Whatever you think about the linearity of the log-log plot, use your helper application's least squares procedure to find the best fitting line. (The necessary commands are provided in your worksheet. Don't worry if "least squares" is a new idea -- just think of it as "best fitting line.") From the slope and intercept of the best-fitting line, calculate values of the parameters
**r** and **k**. (Keep in mind that logarithmic plots use base 10 logarithms, not base *e*.)
- Now construct a slope field for the model differential equation, and add a sample solution passing through one of the data points. Experiment with the selected data point to see if it makes any difference in the shape of the solution.
- Add a plot of the data points to your slope field plot. Now what do you think about the Coalition Model as a description of the historical data?

David A. Smith and Lawrence C. Moore, "World Population Growth - The Coalition Model," *Convergence* (December 2004)