David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Acta Biotheoretica 39 (3-4):197-205 (1991)
Observed biological growth curves generally are sigmoid in appearance. It is common practice to fit such data with either a Verhulst logistic or a Gompertz curve. This paper critically considers the conceptual bases underlying these descriptive models.The logistic model was developed by Verhulst to accommodate the common sense observation that populations cannot keep growing indefinitely. A justification for using the same equation to describe the growth of individuals, based on considerations from chemical kinetics (autocatalysis of a monomolecular reaction), was put forward by Richardson, but met with heavy criticism as a result of his erroneous derivation of the basic equation (Snell, 1929). It errs on the side of over-simplicity (Priestley & Pearsall, 1922). Von Bertalanffy (1957) subsequently based a justification on the assumption that, as a first approximation, the rates of catabolism and anabolism may be assumed to be proportional to weight and power (still vacant places, between the maximal possible and the already accumulated population sizes. This point of view was fiercely challenged by Nicholson (1933), Milne (1962), Smith (1954) and Rubinov (1973). And indeed, what is meant by vacant places has never become entirely clear. Finally Lotka (1925) devised a third leading approach by just truncating a Taylor expansion around zero of the differential law for autonomous growth after the second degree term.
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