Pierre François Verhulst

From Wikipedia, the free encyclopedia

Pierre Francois Verhulst

Pierre François Verhulst (October 28, 1804, Brussels, Belgium - February 15, 1849, Brussels, Belgium) was a mathematician and a doctor in number theory from the University of Ghent in 1825. Verhulst published in 1838 the logistic equation:

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

when $N (t)$ represents number of individuals at time t, r the intrinsic growth rate and K is the carrying capacity, or the maximum number of individuals that the environment can support. This model was rediscovered in 1920 by Raymond Pearl and Lowell Reed, who promoted its wide and indiscriminate use. The logistic equation can be integrated exactly, and has solution

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

where $C = 1 / N (0) - 1 / K$ is determined by the initial condition $N (0)$ . It's interesting to note that 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 environmental limitation introduced here.

[edit] See also

[edit] References

Verhulst, P. F., (1838). 'Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique 10:113-121.

Verhulst, P. F., Recherches Mathematiques sur La Loi D'Accroissement de la Population, Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18, Art. 1, 1-45, 1845 (Mathematical Researches into the Law of Population Growth Increase)