The Logistic Map

In: Business and Management

Submitted By yansa
Words 138629
Pages 555
Chaotic Growth with the Logistic Model of P.-F. Verhulst
Hugo Pastijn
Department of Mathematics, Royal Military Academy B-1000 Brussels, Belgium
Hugo.Pastijn@rma.ac.be

Summary. Pierre-Fran¸ois Verhulst was born 200 years ago. After a short biograc phy of P.-F. Verhulst in which the link with the Royal Military Academy in Brussels is emphasized, the early history of the so-called “Logistic Model” is described. The relationship with older growth models is discussed, and the motivation of Verhulst to introduce different kinds of limited growth models is presented. The (re-)discovery of the chaotic behaviour of the discrete version of this logistic model in the late previous century is reminded. We conclude by referring to some generalizations of the logistic model, which were used to describe growth and diffusion processes in the context of technological innovation, and for which the author studied the chaotic behaviour by means of a series of computer experiments, performed in the eighties of last century by means of the then emerging “micro-computer” technology.

1 P.-F. Verhulst and the Royal Military Academy in Brussels
In the year 1844, at the age of 40, when Pierre-Fran¸ois Verhulst on November c 30 presented his contribution to the “M´moires de l’Acad´mie” of the young e e
Belgian nation, a paper which was published the next year in “tome XVIII” with the title: “Recherches math´matiques sur la loi d’accroissement de la e population” (mathematical investigations of the law of population growth), he did certainly not know that his work would be the starting point for further research by Raymond Pearl and Lowell J. Reed [10, 11], by the famous
A.-J. Lotka [8] and independently by Volterra [16] and later by V.A. Kostitzin [5], in the fields of mathematical biology, biometry, and demography.
It was in these M´moires that he introduced a growth model for a…...

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