Stimulated by Malthus' "Essay on the Principle of Population", Verhulst (1838) published what he called a "logistique" equation to describe the sigmoidal growth of population density to carrying capacity. The equation was rediscovered by Pearl and Reed (1920). Then, Lotka (1925) derived the same equation mathematically, calling it "the law of population growth", and Gause (1934) demonstrated its validity in laboratory experiments. For an excellent review of these historical events see Kingsland (1985). The discrete form of the logistic equation used in this book was first proposed by Cook (1965) but is, in fact identical to the Ricker (1954) equation (see e.g., Berryman 1978).

Royama (1992) derived the logistic equation by considering a system in which randomly distributed organisms draw resources from a circular area of influence of radius r around themselves, and compete with neighboring organisms when their areas of influence overlap; i.e., when the distance to the neighbor is less than 2r (see figure).

Let G_i be the finite per-capita rate of change of an individual organism competing with i neighbors and Pr(i) be the expected proportion of the population having i overlapping competitors, then the mean finite rate of increase of the population is given by the weighted sum

If the individuals in the consumer population are distributed randomly, then they can be described by a Poisson distribution with mean P, the density of the population (really P_t-1 but we omit the time subscript for convenience). Under these conditions the number of competitors is also Poisson distributed with mean sP, where s = 4πr². From this the expected proportions are given by the Poisson formula

which, can be substituted into the previous equation to yield

Now suppose that the addition of a competitor reduces the average rate of change of an individual by the proportion k (0<k<1), so that the finite rate of change in competition with i neighbors is G_i = G₀kⁱ, and substituting kⁱ for G_i/G₀ in the previous equation, we get

This equation can then be reduced by Taylor's theorem to

which is the well-known Ricker (1954) equation familiar to fisheries ecologists. Continuing after Berryman et al. (1995), we let R_P = lnG and A_P = lnG₀, so that

Noting that the parameter b can be expanded to b = V/H, where V is a measure of the intensity of competition, and H is the density of resources, we obtain the logistic R-function

Notice that V is the same as the demand for prey by a predator, the greater the demand, therefore, the greater the degree of intraspecific competition.

References:

• Berryman, A. A. 1978. Population cycles of the Douglas-fir tussock moth (Lepidoptera: Lymantriidae): the time-delay hypothesis. Canadian Entomologist 110: 513-518.
• Berryman, A. A., A. P. Gutierrez and R. Arditi. 1995. Credible, parsimonious and useful predator-prey models -- a reply to Abrams, Gleeson, and Sarnelle. Ecology 76: 1980-1985.
• Cook, L. M. 1965. Oscillation in the simple logistic growth model. Nature 207: 316.
• Gause, G. 1934. The struggle for existence. Williams and Wilkins, Baltimore.
• Kingsland, S. E. 1985. Modeling nature. University of Chicago Press, Chicago.
• Ricker, W. E. 1954. Stock and recruitment. Journal of the Fisheries Research Board of Canada 11: 559-623.
• Royama, T. 1992. Analytical Population Dynamics. Chapman and Hall, London.
• Verhulst, P. F. 1838. Recherches mathematiques sur la loi d'accrossement de la population. Memoirs de l'Academie Royal Bruxelles 18: 1-38.

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