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Pierre-François Verhulst’s final triumph. (English) Zbl 1103.01011

Ausloos, Marcel (ed.) et al., The logistic map and the route to chaos. From the beginnings to modern applications. Berlin: Springer (ISBN 3-540-28366-8/hbk). Understanding Complex Systems, 13-28 (2006).
The so-called Logistic function of Verhulst has lived through a long and difficult history before it was finally and generally recognized as a universal milestone marking the road to unexpected fields of research. During the past three decades it has been claimed as the prototype of a chaotic oscillation and as a model of a fractal figure.
In the first part of the paper the life of Verhulst (October 28, 1804–February 15, 1849) is described. He studied very successfully at the Brussels Atheneum, where Adolphe Quetelet (1796–1876) was his mathematics teacher, and at the University of Ghent, where his highest achievement was the defence of his PhD dissertation in mathematics in 1825, being scarcely 21 years old. After returning to Brussels he took a keen interest in the calculus of probability and in political economy, an interest which he shared with Quetelet. His poor health caused his brief stay in Italy. During his stay in Rome in September 1830, the Belgian Revolution broke out in Brussels. Back in Brussels, in 1831, he wrote a document on behalf of the recently established Congress – the present Belgian parliament – in which he deplores the situation at the university and formulates a way to resolve it. In 1835 he was appointed professor at the Royal Military Academy, and also Professor at the University Libre of Brussels. In 1837 he married a miss Debiefve, who would bear him a daughter about a year later.
Verhulst and Quetelet were closely associated in their life and work. They were both professors at the Military School, they were both members of the Academic royale des Sciences et des Belles Lettres de Bruxelles and they were both interested in mathematical statistics which could be the key to revealing the “natural laws” of human society. However, the debate which must have been going on between Verhulst and Quetelet for several years on the problem, is there some analogy between physical laws and social phenomena, came to a sudden end with Verhulst’s untimely death.
In the remaining part of the paper Verhulst’s scientific work and its later appropriation are analyzed. His first research in the field of population growth dates from shortly after the independence of Belgium. He did not accept Malthusianism and considered an alternative stating that \[ {dp\over dt}= mp- np^2 \] with \(p\) for the population figure. The first results were published in 1838, a much more elaborate study in 1845 and 1847. This work of Verhulst was completely ignored during the whole nineteenth century. In 1920 two American demographers Pearl and Reed rediscovered Verhulst’s work. A first sign of real recognition of Verhulst’s merits came in 1925 by the English statistician Udny Yule. But Verhulst’s formula got its final victory only after 1965: his publications are now cited about 15 times a year. The authors of the paper see that there are at least five reasons for this. The most important is their implication in chaos theory. To obtain deterministic chaos from Verhulst’s formula one has to replace the continuous logistic differential equation by its discrete form \[ p_{n+1}- p_n= rp_n(1- p_n). \] This gives the discrete Verhulst iteration. If we allow \(p\) and \(r\) to be complex then one obtains the logistic fractal of Verhulst, illustrated by a figure in the paper. Also predictability and chaos alternating in Verhulst’s formula are illustrated. The paper is concluded with the words: “Some scientific ideas have to wait for a long period before they come to their final triumph. Verhulst’s logistic function is certainly one among them”.
For the entire collection see [Zbl 1085.37001].

MSC:

01A55 History of mathematics in the 19th century

Biographic References:

Verhulst, Pierre-François
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