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World Population Growth - The Coalition Model

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 P1+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.

  1. Explain why (Pi+1 - Pi-1) / (ti+1 - ti-1) is a good estimate of dP/dt at t = ti.

  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. (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.)

  5. 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.

  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?

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