Population Growth Models

Part 3.1: Natural and Coalition Models

In this part we explore whether the population of the world might be growing exponentially, i.e., whether the natural growth model is appropriate for our data. We repeat the data below.

Recall from Part 2 that if a population is growing exponentially, then the doubling times are constant.

1. How long did it take to double the world 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?

   Does this imply that world population is growing exponentially?

2. The historical data is already in your worksheet. Plot it, and decide whether you think this looks like exponential growth. You may want to think about what you said in answer to question 1.

3. Test the data for exponential growth by plotting the natural logarithm of the population versus time. Does it confirm or refute your answer to the preceding question? Explain.

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. To emphasize the productivity rate, we may rewrite the differential equation in the form

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^h, where the power h is positive and presumably small. (If h were 0, this would reduce to the natural model -- which we now know does not fit.) The differential equation for this model is

or

In Part 3.2 we consider the question of whether such a model can fit the historical data.