Author(s):

David A. Smith and Lawrence C. Moore

The parameter **T** in the model function

**P = 1/[r k (T - t)]**^{1/r}

is obviously important. It is just as obviously not directly observable from measurements or estimates of population. However, we can take logs of both sides of this equation to find the equivalent form

**log P = (-1/r) [log (rk) + log (T - t)]**

If this model fits the data (and we have seen some evidence that it does), then we should find that **log P** is a *linear* function of **log (T - t)**.

- Commands are provided in your worksheet to construct a log-log plot of
**P** versus **T - t** for your choice of **T**. [That is, **log P** is plotted as a function of **log (T - t)**.] Experiment with **T** until you can make this plot as straight as possible. Is your best estimate of **T** in the *near* future or the *distant* future relative, say, to your lifetime?
- You now have values for all the parameters --
**k**, **r**, **T** -- in your model function. Plot the model function, and superimpose your plot of the historical data. Does this model describe the data adequately?
- Recall that we computed
**k** and **r** from crude approximations to **dP/dt**, so these may not be the best values for fitting a model to the data. Experiment with *small* changes in **k** and/or **r** to see if you can get a better fit with your model function. (These adjustments will not affect **T** because the procedure for finding **T** did not involve **k** or **r**.)
- The last date represented in our historical data was 1985. With your best estimates of the parameters
**k**, **r**, and **T**, what does your model function "predict" for populations that have already occurred in 1990, 1995, and 2000?
- What does your model function predict for world population in 2010; in 2020?
- For the five dates in the two preceding steps, compare the estimates and projections at the

U. S. Census Bureau. What do you conclude about the recent trend in population growth and projections for the near-term future? What does the Census Bureau predict for the longer term?

David A. Smith and Lawrence C. Moore, "World Population Growth - When is Doomsday?," *Convergence* (December 2004)