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Populations of organisms often show an S-shaped growth pattern, where population growth accelerates rapidly in an early stage, then slows to little or no growth in a later stage. S-shaped or sigmoidal growth can be caused by a number of factors, however a common interpretation involves an external limit to growth based on environmental factors. Populations might be limited to some value (often referred to as the carrying capacity) because of finite food resources, space, water, etc.

Data on US residents from 1790-1940 show an S-shaped curve of population as a function of time. The data were obtained during the decadal census starting in 1790. One might ask how accurate these data are, and whether there was overcounting, undercounting or both.

Sigmoidal growth curves are often modeled using a modified exponential model, the logistic model, where the growth rate decreases as the population increases. The US population data for this limited time period fit a logistic model very well. To model the data using logistic difference equations, students must choose reasonable values for the initial population and initial time (tricky!), the carrying capacity, and the intitial/uninhibited growth rate in the logistic model; none of these parameters are given in the actual data set.

The US population since 1940 is well known, and not surprisingly, does not fit a logistic model at all. The best fit logistic model of Pearl et al (1940) suggested a carrying capacity of about 180 million US residents; the current population is about 270 million residents. Pearl et al (1940) were very cautious, however, and would not even speculate about the 1950 population, let alone that of the year 2000. One might ask: why did the population appear logisitic over this time period, and why did the US population turn out to be non-logistic?

Reference: Pearl, R., Reed, L. J. and Kish, J. F. (1940), The logistic curve and the census count of 1940; Science, vol 92, no. 2395, pp. 486-488.

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 Pearl, Reed and Kish (1940) US population from 1790-1940 year pop (millions) 1790 3.929 1800 5.308 1810 7.240 1820 9.638 1830 12.866 1840 17.069 1850 23.192 1860 31.443 1870 38.558 1880 50.156 1890 62.948 1900 75.995 1910 91.972 1920 105.711 1930 122.775 1940 131.410

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