Pierre François Verhulst (28 October 1804,

Brussels Brussels (french: Bruxelles or ; nl, Brussel ), officially the Brussels-Capital Region (All text and all but one graphic show the English name as Brussels-Capital Region.) (french: link=no, Région de Bruxelles-Capitale; nl, link=no, Bruss ...

– 15 February 1849,

) was a Belgian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

and a doctor in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

from the

University of Ghent Ghent University ( nl, Universiteit Gent, abbreviated as UGent) is a public research university located in Ghent, Belgium. Established before the state of Belgium itself, the university was founded by the Dutch King William I in 1817, when the ...

in 1825. He is best known for the

logistic growth A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...

model.

Logistic equation

Verhulst developed the logistic function in a series of three papers between 1838 and 1847, based on research on modeling

population growth Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...

that he conducted in the mid 1830s, under the guidance of

Adolphe Quetelet Lambert Adolphe Jacques Quetelet FRSF or FRSE (; 22 February 1796 – 17 February 1874) was a Belgian astronomer, mathematician, statistician and sociologist who founded and directed the Brussels Observatory and was influential in introduc ...

; see for details. Verhulst published in the equation: :

\frac = rN - \alpha N^2

where ''N''(''t'') represents number of individuals at time ''t'', ''r'' the intrinsic growth rate, and ''

\alpha

'' is the density-dependent crowding effect (also known as intraspecific competition). In this equation, the population equilibrium (sometimes referred to as the

carrying capacity The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as t ...

, ''K''),

N^*

, is :

N^* = \frac

. In he named the solution the

logistic curve A logistic function or logistic curve is a common S-shaped curve (sigmoid function, sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is ...

. Later,

Raymond Pearl Raymond Pearl (June 3, 1879 – November 17, 1940) was an American biologist, regarded as one of the founders of biogerontology. He spent most of his career at Johns Hopkins University in Baltimore. Pearl was a prolific writer of academic books, ...

and

Lowell Reed Lowell Jacob Reed (January 8, 1886 – April 29, 1966) was 7th president of the Johns Hopkins University in Baltimore, Maryland. He was born in Berlin, New Hampshire, the son of Jason Reed, a millwright and farmer, and Louella Coffin Reed. Edu ...

popularized the equation, but with a presumed equilibrium, ''K'', as :

\frac = r N \left(1 - \frac  \right)

where ''K'' sometimes represents the maximum number of individuals that the environment can support. In relation to the density-dependent crowding effect,

\alpha = \frac

. The Pearl-Reed logistic equation can be integrated exactly, and has solution :

N(t) = \frac

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 In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...

of the initial condition and the carrying capacity, :

\frac = \frac+ \frac.

Although the continuous-time logistic equation is often compared to the

logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...

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 In ecology, ''r''/''K'' selection theory relates to the selection of combinations of traits in an organism that trade off between quantity and quality of offspring. The focus on either an increased quantity of offspring at the expense of individ ...

derives its name from the competing dynamics of

exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a q ...

and

introduced by the equations above.

Works

* * * *

References

* ** Published as:

External links

* {{DEFAULTSORT:Verhulst, Pierre Francois 1804 births 1849 deaths Belgian mathematicians 19th-century male writers

Logistic equation

See also

Works

References

External links