### World Population Growth

Part 1: Background: Natural and Coalition Models

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.

 Year (CE) Population (millions) Year (CE) Population (millions) 1000 200 1940 2295 1650 545 1950 2517 1750 728 1955 2780 1800 906 1960 3005 1850 1171 1965 3345 1900 1608 1970 3707 1910 1750 1975 4086 1920 1834 1980 4454 1930 2070 1985 4850
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,

dP/dt = k P,

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

P = P0 ekt,

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

1. The historic 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.
2. 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 historic 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 kPr, 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

dP/dt = k Pr+1.

In Part 2 we consider the question of whether such a model can fit the historic data.

Send comments to the authors <modules at math.duke.edu>