Chaotic Growth with the Logistic Model of P.-F. Verhulst

  • Hugo Pastijn
Part of the Understanding Complex Systems book series (UCS)


Pierre-François Verhulst was born 200 years ago. After a short biography 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.


Logistic Model Chaotic Behaviour Discrete Version Chaos Theory Continuous Time Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    A. De Palma, F. Droesbeke, Cl. Lefevre, C. Rosinski: Modèles mathématiques de base pour la diffusion des innovations, Jorbel, Vol 26, No 2, 1986, pp 37–69Google Scholar
  2. 2.
    D. Gulick: Encounters with chaos (McGraw-Hill, New York 1992)Google Scholar
  3. 3.
    T. Kinnunen, H. Pastijn: Chaotic Growth — attractors for the logistic model of P.-F. Verhulst. In: Revue X (RMA Brussels 1986), 4, pp 1–17, 1986Google Scholar
  4. 4.
    T. Kinnunen, H. Pastijn: The chaotic behaviour of growth processes, ICOTA proceedings, Singapore, 1987Google Scholar
  5. 5.
    V.A. Kostitzin: Biologie mathématique (Armand Colin, Paris 1937)Google Scholar
  6. 6.
    J.D. Lebreton, C. Millier: Modèles dynamiques déterministes en biologie (Masson, Paris 1982)Google Scholar
  7. 7.
    T.-Y. Li, J. Yorke: Period three implies chaos, American mathematical monthly (82), pp 985–992, 1975MathSciNetzbMATHGoogle Scholar
  8. 8.
    A.J. Lotka: Elements of physical Biology (Williams & Wilkins, Baltimore 1925)zbMATHGoogle Scholar
  9. 9.
    R. May: Nature 261, 459 (1976)CrossRefADSGoogle Scholar
  10. 10.
    R. Pearl: Introduction to Medical Biometry and Statistics (W.B. Saunders, Philadelphia London 1923)Google Scholar
  11. 11.
    R. Pearl, L.J. Reed: Metron 5, 6 (1923)Google Scholar
  12. 12.
    A. Quetelet: Notice sur Pierre-François Verhulst. In: Annuaire de l’Académie royale des Sciences, des Lettres et des Beaux-Arts de Belgique (Impr. Hayez, Brussels 1850) pp 97–124Google Scholar
  13. 13.
    P.-F. Verhulst: Traité élémentaire des fonctions elliptiques (Impr. Hayez, Brussels 1841)Google Scholar
  14. 14.
    P.-F. Verhulst: Recherches mathématiques sur la loi d’accroissement de la population. In: Mem. Acad. Royale Belg., vol 18 (1845) pp 1–38Google Scholar
  15. 15.
    P.-F. Verhulst: Deuxième mémoire sur la loi d’accroissement de la population. In Mem. Acad. Royale Belg., vol 20 (1847) pp 1–32Google Scholar
  16. 16.
    V. Volterra: Leçons sur la théorie mathématique de la lutte pour la vie (Gauthier-Villars, Paris 1931)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hugo Pastijn
    • 1
  1. 1.Department of MathematicsRoyal Military AcademyBrusselsBelgium

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