Pierre François Verhulst
Encyclopedia
Brussels
Brussels , officially the Brussels Region or Brussels-Capital Region , is the capital of Belgium and the de facto capital of the European Union...
, Belgium
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...
– 15 February 1849, Brussels
Brussels
Brussels , officially the Brussels Region or Brussels-Capital Region , is the capital of Belgium and the de facto capital of the European Union...
, Belgium
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...
) was a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and a doctor in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
from the University of Ghent in 1825. Verhulst published in 1838 the equation:
when N(t) represents number of individuals at time t, r the intrinsic growth rate and K is the carrying capacity
Carrying capacity
The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment...
, or the maximum number of individuals that the environment can support. In a paper published in 1845 he called the solution to this the logistic function
Logistic function
A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...
, and the equation is now called the logistic equation. This model was rediscovered in 1920 by Raymond Pearl
Raymond Pearl
Raymond Pearl was an American biologist, regarded as one of the founders of biogerontology. He spent most of his career at Johns Hopkins University in Baltimore....
and Lowell Reed
Lowell Reed
Lowell Reed was 7th president of the Johns Hopkins University in Baltimore, Maryland.He had a long career as a research scientist in biostatistics and public health administration at Hopkins, where he was previously dean and director of the School of Public Health and later as vice president in...
, who promoted its wide and indiscriminate use.
The 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
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
of the initial condition and the carrying capacity,
Although the continuous-time logistic equation is often compared to the logistic map
Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
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
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 or quality of offspring...
derives its name from the competing dynamics of exponential growth
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
and environmental limitation introduced here.
See also
- Population dynamicsPopulation dynamicsPopulation dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...
- Logistic mapLogistic mapThe logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
- Logistic functionLogistic functionA logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...
- Logistic distribution