Belgian mathematician and demographer Pierre-François Verhulst, best known for the conceptualization and specification of the logistic curve, was born in Brussels to wealthy parents. Adolphe Quetelet (1796–1874), the Belgian mathematician and demographer, was first Verhulst's mathematics teacher at the Royal Atheneum, then his professor at the University of Ghent, where Verhulst also earned a doctoral degree in mathematics after just three years of study. Verhulst was a highly versatile man who wrote Latin poetry and even drafted a constitution for the Papal States when he was in Italy, a country that eventually expelled him for that reason. Back in Brussels, Verhulst was invited to join the Academie Royale, of which Quetelet was also a member. Verhulst occupied his chair at the academy in 1848, only one year before his death at age 45.
Verhulst's work in demography was essentially completed by 1833. Through Quetelet, he had been invited to present a mathematical formulation of T.R. Malthus's theories. However, Verhulst was convinced that the geometric or exponential growth of population would be curtailed by constraining factors before Malthus's "positive checks" (emigration, excess mortality due to famine or declining living standards) could limit them. That, in itself, was a departure from Malthus's work. Verhulst also believed that the strength of the curtailing factors would increase in a proportional way to the population expansion itself. To elucidate this, Verhulst needed to introduce a hitherto unknown negative function into the overall formula. The result of the work was a demonstration that any population growth rate would essentially follow a bell-shaped curve, starting from zero, steadily increasing to a maximum, and declining once again to zero in a fashion symmetrical to the positive growth phase. The population stock then evolves according to the elongated Scurve, which has a point of inflection at the maximal value of the growth rate, and then levels off at a new but higher plateau, at which point the growth rate declines to zero. Verhulst checked his theory empirically against population data for France, Belgium, Essex, England, and Russia. Quetelet, however, was not convinced by his student since he knew of no counterpart in physics. After the publication of Verhulst's theory, the logistic curve was forgotten until its rediscovery by the American biometrician Raymond Pearl and demographer Lowell J. Reed in 1921, and British statistician G. Udny Yule's 1925 acknowledgment of the significance of Verhulst's finding of almost a century earlier.
From the 1920s onward, many applications for the theory were found in a wide variety of fields. The logistic curve became one of the essential cornerstones of world systems modeling. It also proved to provide good descriptions of certain diffusion processes, especially of those based on the principles of contagion. Diseases, technical novelties, new ideas and rumors would all grow within a virgin population and reach a maximum, but each would eventually encounter resistance and burn out, or be challenged by a better invention or concept. In the field of mathematics, Verhulst's logistic curve was rediscovered in 1975 by two German physicists who determined that it was one of the essential formulas in the mathematics of fractals.
In the early twenty-first century, Quetelet's contributions to demography have largely faded, while those of Verhulst have steadily increased in importance. However, he is still rarely cited by demographers as the inventor of the logistic curve or the contagion model of diffusion.
selected works by pierre-franÇois verhulst.
Verhulst, Pierre-François. 1838. "Notice sur la loi que la population suit dans son accroissement." Correspondances mathématiques et physiques 10:113.
——. 1845. "Recherches mathématique sur la loi d'accroissement de la population." Mémoires de l'Académie Royale des Sciences et des Lettres de Bruxelles 18: 1–38.
——. 1847. "Deuxième mémoire sur la loi d'accroissement de la population." Mémoire de l'Académie Royale des Sciences, des Lettres et des Beaux Arts de Belgique. 20: 1–38.
selected works about pierre-franÇois verhulst.
Kint, Jos. 1990. "Verhulst, Pierre-François: wiskundige en demograaf." Nationaal Biografisch Woordenboek 13: 822–827.
Yule, G. Udny. 1925. "Verhulst." Journal of the Royal Statistical Society 88: 1.