Les modeles sigmoides en biologie vegetale

Resume

Revue des fondements conceptuels de modèles sigmoïdes très souvent employés. Examen de leur applicabilité aux modes de croissance multiplicatif et accrétionnaire.

Abstract

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 (< 1) of weight respectively. Gause (1934) rederived the Verhulst equation in a population context by assuming the per capita growth rate to be proportional to the difference, interpreted as the number of “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.

The empirically based Gompertz model avowedly describing human mortality was applied to the growth of organisms by Davidson (1928). The closely related model by Gray (1929) also has a pure phenomenological basis only. A theoretical interpretation has recently been propounded by Makany (1991). This author introduces the following biological system: an inexhaustible generator which produces essentially inert, i.e. non-destructible and non-reproducing elements, which accommodate in a space where they can be counted. It is a probabilistic problem of occupancy and an urn model can be used. The asymptotic law of the obtained discrete distribution is a Gompertz one.

The comprehensive growth functions, more flexible, of Richards (1959) and Nelder (1961) are based on sheer mathematical generalizations. There is no other theoretical way to explain them.

The conceptual bases of these models and their applicability are discussed in cases of multiplicative or accretionary growth (Richards, 1969). In practice it is advisable to use the Von Bertalanffy model in case of a multiplicative growth, the Makany model in accretionary growth and the logistic function in both cases, but only as an approximation at the second degree term and as a purely descriptive formula of empirical data. Thus without any reference to an autocatalytic theory or “vacant places”.