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(These examples do not exhaust the ability of the ‘logistic’ equation to describe populations: both oscillatory and chaotic variants exist, but they have growth rates r in excess of human experience.)

Until 1940 it was possible to believe that human population growth followed that of fruitflies, because Verhulst’s ‘logistic equation’ of 1845, with a constant carrying capacity K, gives a nearly perfect fit to the US population from 1640 to 1940. Raymond Pearl and L.J. Reed of the US Census Bureau independently reinvented Verhulst’s model and applied it to Census data {Pear20, Pear30, Pear40}, explaining with much ingenuity how US might support the improbable maximum population of 186 million it predicted.

The upslope of the Gaussian in Fig. 20.4 (the ‘baby boom’) apparently discredited simple models in the minds of demographers. Although some later claimed to have predicted the ‘boom’, Scientific American (1951) could find no evidence that they had, and only Cox observed—30 years later—that they should have been able to {CoxP70: 444}.

Replacements were multi-state component-cohort models which follow the births, deaths, and migration of social cohorts, such as Lotka’s ‘birth trajectory of the renewal equation’ {Lotk24}, coupled with the dogma that simple equations were passé. The new approach required integral equations (with the unknown on both sides of the equation, once under an integral), much data (or many guesses), and considerable mathematical skill, and were limited to projections of a decade or less {Monr93} because the behavior of cohorts changes quickly. It would be 2 generations before Ahlburg succeeded in publishing the obvious fact that simple and complex approaches were equally valid and answered different questions [Ahlb95].

Meanwhile, W.E. Howland, an engineer at Purdue (memorialized there by a web server bearing his name), combined the insights of Malthus and Verhulst. The resulting model, with K rising linearly, required the transcendental exponential integral, which in precomputer days was painful to work with. Howland devoted much effort to nomographs on which the solution could be read from a ruler going through 2 chosen parameters.

Shortly before his death he sent me typescripts of his unpublished papers and a set of pencilled nomographs (which accurately predicted the 1990 Census). His only attempt to publish that was not rejected by demographic journals was a short letter to Science {Howl61}. One of Howland’s asides, which he did not pursue, was that he obtained a better fit to the US data if he increased the slope of the limit midway through the period. The obvious approach was to try an exponential limit for K—which both improved the fit and greatly simplified the arithmetic, making the nomographs unnecessary. The resulting equation has 4 parameters, the initial population and its biological growth rate (well constrained), and the initial value and growth rate of the population limit K. The rate is constrained by the technological growth rate—which turns out to be about 1/3 of the government’s estimates of ‘growth’ as measured from inflated economic figures. The initial value is the free ‘fitting parameter’.

See also: Logistic curve Growth curves Gompertz curve

References: Ahlburg, DA (1995) ‘Simple versus complex models: evaluation, accuracy and combining’ Mathematical Population Studies 5(#3): 281-290. MacIntyre, F (2005) ‘The Maltho-Marxian Hypothesis “Economics Controls Population”: A Test and a Projection’ Population Review 44(#2): 24-49. Malthus, TR (1890) Malthus on Population, reproduced from the last (6th) edition of An Essay on the Principle of Population ... (1798-1826), with an introduction by G. T. Bettany (Ward, Lock & Co., London). Howland, WE (1961) Letter. Science 133: 939.

Pearl, R & LJ Reed (1920) ‘On the rate of growth of the population of the United States since 1870 and its mathematical representation’ Proc. Nat. Acad. Sci. 6: 275-288. Pearl, R & LJ Reed (1930) ‘The Logistic Curve and the Census Count of 1930’ Science 72: 399–401. Pearl, R, LJ Reed & JF Kish (1940) ‘The logistic curve and the Census count of 1940’ Science 92: 486-488. Verhulst, PF (1845) ‘Recherches mathematiques sur la loi d’accroissement de la population’ Nouv. Mem. Acad. Roy. Soc. Belle-lettr. Bruxelles 18: 1-38.