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World Population Growth - Background: Natural and Coalition Models

Author(s): 
David A. Smith and Lawrence C. Moore

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 
 
Sources: (1) A. L. Austin and J. W. Brewer, "World Population Growth and Related Technical Problems", IEEE Spectrum 7 (Dec. 1970), pp. 43-54. (2) U. S. Census Bureau.
  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 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.
  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 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 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 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.

David A. Smith and Lawrence C. Moore, "World Population Growth - Background: Natural and Coalition Models," Loci (December 2004)

JOMA

Journal of Online Mathematics and its Applications

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