Logistic Functions

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Response to Malthus

Malthus influenced Darwin.

"“In October 1838, that is fifteen months after I had begun my systematic enquiry, I happened to read for amusement Malthus on Population, and being well prepared to appreciate the struggle for existence which everywhere goes on from long-continued observation of the habits of animals and plants, it at once struck me that under these circumstances favorable variations would tend to be preserved, and unfavorable ones to be destroyed. The result of this would be the formation of new species.” (Darwin, 1993, p. 120)."

The Malthusian idea of an uncontrollable population growth only restricted by the bounds of natural resources was first put into a mathematical equation describing human population growth by Pierre F. Verhulst, Professor of Mathematics in Brussels, Belgium, in 1838 (Verhulst, 1838).5 He checked the results of the equation with censuses of population developments in France, Belgium, Russia and in Essex, England, over 20 years in the early 19th century and found confirming results.The equation of logistic growth is: where N is the population, r is the growth rate and K is the carrying capacity.

Critiques of the Logistic Model

From Stephen Strogatz' Nonlinear Dynamics and Chaos

"The algebraic form of the model is not to be taken literally. The model should really be regarded as a metaphor for populations that have a tendency to grow from zero population up to some carrying capacity K.

Originally a much stricter interpretation was proposed, and the model was argued to be a universal law of growth (Pearl 1927). The logistic equation was tested in laboratory experiments in which colonies of bacteria, yeast, or other simple organisms were grown in conditions of constant climate, food supply and absence of predators. For a good review of this literature, see Krebs (1972, pp. 190-200). These experiments often yielded sigmoid growth curves, in some cases with an impressive match to the logistic predictions.

On the other hand, the agreement was much worse for fruit flies, flour beetles and other organisms that have complex life cycles, involving eggs, larvae, pupae and adults. In these organisms, the predicted asymptotic approach to a steady carrying capacity was never observed--instead the populations exhibited large, persistent fluctuations after an initial period of logistic growth. See Kregs (1927) for a discussion of the possible causes of these fluctuations, including age structure and time-delayed effects of overcrowding in the population.

For further reading on population biology, see Pielou (1969) or May (1981). Edelstein-Keshet (1988) and Murray (1989) are excellent textbooks on mathematical biology in general."