Pierre François Verhulst
Pierre François Verhulst (28 October 1804, Brussels – 15 February 1849, Brussels) was a Belgian mathematician and a doctor in number theory from the University of Ghent in 1825. He is best-known for the logistic growth model.
Verhulst developed the logistic function in a series of three papers between 1838 and 1847, based on research on modeling population growth that he conducted in the mid 1830s, under the guidance of Adolphe Quetelet; see Logistic function § History for details.
Verhulst published in Verhulst (1838) the equation:
where N(t) represents number of individuals at time t, r the intrinsic growth rate, and is the density-dependent crowding effect (also known as intraspecific competition). In this equation, the population equilibrium (sometimes referred to as the carrying capacity, K), , is
where K sometimes represents the maximum number of individuals that the environment can support. In relation to the density-dependent crowding effect, . The Pearl-Reed logistic equation can be integrated exactly, and has solution
where C = 1/N(0) − 1/K is determined by the initial condition N(0). The solution can also be written as a weighted harmonic mean of the initial condition and the carrying capacity,
Although the continuous-time logistic equation is often compared to the logistic map because of similarity of form, it is actually more closely related to the Beverton–Holt model of fisheries recruitment.
- Verhulst, Pierre-François (1838). "Notice sur la loi que la population suit dans son accroissement". Correspondance mathématique et physique. 10: 113–121. Retrieved 18 February 2013.
- Verhulst, Pierre-François (1841). Traité élémentaire des fonctions elliptiques : ouvrage destiné à faire suite aux traités élémentaires de calcul intégral. Bruxelles: Hayez. Retrieved 18 February 2013.
- Verhulst, Pierre-François (1845). "Recherches mathématiques sur la loi d'accroissement de la population" [Mathematical Researches into the Law of Population Growth Increase]. Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles. 18: 1–42. Retrieved 18 February 2013.
- Verhulst, Pierre-François (1847). "Deuxième mémoire sur la loi d'accroissement de la population". Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. 20: 1–32. Retrieved 18 February 2013.
- Cramer 2002, pp. 3–5.
- Cramer, J. S. (2002). The origins of logistic regression (PDF) (Technical report). 119. Tinbergen Institute. pp. 167–178. doi:10.2139/ssrn.360300.
- O'Connor, John J.; Robertson, Edmund F., "Pierre François Verhulst", MacTutor History of Mathematics archive, University of St Andrews.