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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 
Year 
Population 

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 
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 = P_{0} e^{kt},
where P_{0} is the population at whatever time is considered to be t = 0.
In 1960 Heinz von Foerster, Patricia Mora, and Larry Amiot published a nowfamous paper in Science (vol. 132, pp. 12911295). 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
dP/dt = k P^{r+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," Convergence (December 2004)
Journal of Online Mathematics and its Applications