Population Growth Models

Part 3.4: When is Doomsday?

The parameter T in the model function

is obviously important. We could try to determine a value for T by substituting particular values for P and t in the formula above and then solving for T. However, this approach has several drawbacks. First, this is dependent on the particular population estimate chosen. Even more important is the fact that it depends on the rather shaky estimates that we have for h and k. For these reasons, we will take a different approach -- one that does not depend on our choices of h and k and does not overemphasize the importance of any particular population estimate.

When we take the natural logarithms of both sides of the equation,

we obtain the equivalent form

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

1. Commands are provided in your worksheet to construct a plot of ln P versus ln (T - t) for your choice of 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?

2. You now have values for all the parameters -- k, h, T -- in your model function. Plot the model function, and superimpose your plot of the historical data. Does this model describe the data adequately?

3. Recall that we computed k and h 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 h 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 h.)

4. The last date represented in our historical data was 1985. With your best estimates of the parameters k, h, and T, what does your model function "predict" for populations that have already occurred in 1990 and 1995?

5. What does your model function predict for world population in 2000, 2010, 2020?

6. 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?