World Population Growth

Part 2: The Coalition Model

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

where r and k are positive constants. Before attempting to solve this differential equation, we explore whether it can reasonably represent the historical data. Our approach will be similar to that used in the module Warming, Cooling, and Urban Ozone Pollution.

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. As in the Ozone module, we do this by calculating symmetric difference quotients.

1. 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. (If necessary, review your work in Part 1 of the Ozone module.)
2. Construct the symmetric difference quotients (SDQ) approximating dP/dt from the historical data.
3. 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.
4. 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. From the slope and intercept of the best-fitting line, calculate values of the r and k.
5. Now construct a slope field for the model differential equation (as in the Slope Fields module), 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.
6. 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?

Last modified: December 2, 1997