Pierre François Verhulst

Pierre François Verhulst

Pierre François Verhulst (28 October 1804, Brussels – 15 February 1849, Brussels) was a mathematician and a doctor in number theory from the University of Ghent in 1825.

Logistic equation

Verhulst published in 1838 the equation:

${\frac {dN}{dt}}=rN-\alpha N^{2}$

where N(t) represents number of individuals at time t, r the intrinsic growth rate, and $\alpha$ is the coefficient of density-dependent growth (also known as intraspecific competition). In this equation, the population equilibrium (sometimes referred to as the carrying capacity, K), $N^{*}$, is

$N^{*}={\frac {r}{\alpha }}$.

Later, Raymond Pearl and Lowell Reed popularized the equation, but with a presumed equilibrium, K, as

${\frac {dN}{dt}}=rN\left(1-{\frac {N}{K}}\right)$

In this equation, K most often represents the maximum number of individuals that the environment can support. The Pearl-Reed logistic equation can be integrated exactly, and has solution

$N(t)={\frac {K}{1+CKe^{-rt}}}$

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,

${\frac {1}{N(t)}}={\frac {1-e^{-rt}}{K}}+{\frac {e^{-rt}}{N(0)}}.$

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.

The concept of R/K selection theory derives its name from the competing dynamics of exponential growth and carrying capacity introduced by the equations above.

See also

• Population dynamics
• Logistic map
• Logistic distribution

Works

• 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". Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles. 18: 1–42. Retrieved 18 February 2013. {{cite journal}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
• 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.

External links

• O'Connor, John J.; Robertson, Edmund F., "Pierre François Verhulst", MacTutor History of Mathematics Archive, University of St Andrews