This module provides a mathematical model for the study of world population growth. It compares the  "natural" and "coalition" differential equation models as possible descriptions of the growth pattern.

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Copyright 1998-2000, CCP and the authors

Published December, 2001

Only in the 20th century has it become possible to make reasonable estimates of the entire human population of the world, current or past. The following table lists some of those estimates, based in part on data considered "most reliable" in a 1970 paper and in part on both overlapping and more recent data from the U. S. Census Bureau. Of course, the earliest entries are at best educated guesses. The later entries are more likely to be correct -- at least to have the right order of magnitude -- but you should be aware that there is no "world census" like the decennial U. S. census, in which an attempt is made to count every individual in this country.

1. How long did it take to double the population from a half billion to one billion? How long to double again from one billion to two billion? How long to double from two billion to four billion? What do you conclude about doubling times?

The natural growth model for biological populations suggests that the growth rate is proportional to the population, that is,

where k is the productivity rate, the (constant) ratio of growth rate to population. We know that the solutions of this differential equation are exponential functions of the form

where P₀ is the population at whatever time is considered to be t = 0.

2. The historical data are already in your worksheet. Plot the data points, and decide whether you think the graph looks like exponential growth. You may want to think about what you said in answer to question 1.
3. We know how to test data for exponential growth by using a semilog plot. Make such a plot. Does it confirm or refute your answer to the preceding question? Explain.

In 1960 Heinz von Foerster, Patricia Mora, and Larry Amiot published a now-famous paper in Science (vol. 132, pp. 1291-1295). The authors argued that the growth pattern in the historical data can be explained by improvements in technology and communication that have molded the human population into an effective coalition in a vast game against Nature -- reducing the effect of environmental hazards, improving living conditions, and extending the average life span. They proposed a coalition growth model for which the productivity rate is not constant, but rather is an increasing function of P, namely, a function of the form kP^r, where the power r is positive and presumably small. (If r were 0, this would reduce to the natural model -- which we know does not fit.) Since the productivity rate is the ratio of dP/dt to P, the model differential equation is

In Part 3 we consider the question of whether such a model can fit the historical data. Note that there is no particular scientific reason to expect that "coalition" will be expressed in a growth rate that is proportional to a power of the population. This is merely the simplest model for expressing the observation that the historical growth has been faster than the exponential model would suggest.

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.

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 (P_i+1 - P_i-1) / (t_i+1 - t_i-1) is a good estimate of dP/dt at t = t_i.

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?

We now take up the solutions of the coalition model and consider the implications of faster-than-exponential growth. We may separate the variables in the differential equation

to write it in the form

Then we may integrate both sides to get an equation relating P and t.

1. Carry out the integrations, and then solve for P as a function of t. You may do the calculations on paper or in your worksheet, as you choose. (Don't forget the constant of integration. The solution makes no sense without it.)

2. Show that your solution can be written in the form
   
   P = 1/[r k (T - t)]^1/r
   for some constant T. Specifically, how is T related to your constant of integration C?

This calculation shows that there is a finite time T at which the population P becomes infinite -- or would if the growth pattern continues to follow the coalition model. The von Foerster paper calls this time Doomsday.

It's clear that Doomsday hasn't happened yet. To assess the significance of the population problem, it's important to know whether the historical data predict a Doomsday in the distant future or in the near future. We take up that question in the next Part.

The parameter T in the model function

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

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).

1. 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?

2. 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?

3. 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.)

4. 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?

5. What does your model function predict for world population in 2010; in 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?

1. How has historical human population growth compared with the "natural" (or exponential) growth model for biological populations? What does this imply about percentage growth rates?

2. Historically, the population growth rate has been proportional to what power of the population? Recall that the assumption underlying the coalition model was that r would turn out to be "small." Was it small?

3. The title of the 1960 paper by von Foerster, Mora, and Amiot was "Doomsday: Friday, 13 November A. D. 2026." How close was your prediction of Doomsday to theirs? (Keep in mind that you were working with more "reliable" data before 1960, plus later data not accessible to them, so there was no reason for your conclusion to be the same.)

4. What would be the social and political implications -- as predicted by the coalition model -- of human beings continuing to behave indefinitely as they always have in the past?

5. Recent evidence suggests that human behavior might in fact be changing. According to the
   U. S. Census Bureau, in what year did the average percentage growth rate peak? In what year did the annual growth in population peak?

6. Given the predictions of population growth through your lifetime, how serious a problem do you see for your generation in coping with the pressures of increased population?

While the Doomsday authors wrote with their tongues firmly in their respective cheeks, it's rather remarkable that they also constructed the most accurate predictor of real population growth for almost two generations. Historical data is a good predictor when the behavior that produced it does not change. Only now are we beginning to see any substantial change in this behavior on a global scale.

Here are some additional links for further study: 

• Visit our sister site at Montana State for a completely different take on the Census Bureau data.
• Visit the Population Reference Bureau for much more information about human population growth.
• Visit Zero Population Growth for information about the problems of overpopulation.
  Editor's note, 12/04: ZPG has changed its name to Population Connection. The link to ZPG now goes to a transitional page that explains the name change and links to the new site.