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| [[Image:Pierre Francois Verhulst.jpg|thumb|250px|right|Pierre Francois Verhulst]]
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| '''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. Verhulst published in 1838 the equation: | |
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| :<math> \frac{dN}{dt} = r N \left(1 - \frac {N}{K} \right)</math>
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| 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. In a paper published in 1845 he called the solution to this the [[logistic function]], and the equation is now called the logistic equation. This model was rediscovered in 1920 by [[Raymond Pearl]] and [[Lowell Reed]], who promoted its wide and indiscriminate use.
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| The logistic equation can be integrated exactly, and has solution
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| :<math> N(t) = \frac{K}{1+ C K e^{-rt}} </math>
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| 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,
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| :<math> \frac{1}{N(t)} = \frac{1-e^{-rt}}{K}+ \frac{e^{-rt}}{N(0)}. </math> | |
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| 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.
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| 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.
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| ==See also==
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| *[[Population dynamics]]
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| *[[Logistic map]]
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| *[[Logistic function]]
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| *[[Logistic distribution]]
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| ==Works==
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| * {{cite journal|first= Pierre-François |last=Verhulst |year= 1838| title = Notice sur la loi que la population poursuit dans son accroissement | journal = Correspondance mathématique et physique |volume = 10| pages = 113–121 |
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| url = http://books.google.com/?id=8GsEAAAAYAAJ&q=
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| | accessdate = 2013-02-18}}
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| * {{cite book
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| | title = Traité élémentaire des fonctions elliptiques : ouvrage destiné à faire suite aux traités élémentaires de calcul intégral
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| | publisher = Hayez
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| | first = Pierre-François
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| | last = Verhulst
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| | year = 1841
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| | place = Bruxelles
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| | url= http://books.google.com/?id=WS8LAAAAYAAJ&printsec=frontcover
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| | isbn =
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| | accessdate = 2013-02-18
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| }}
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| * {{cite journal|first= Pierre-François |last=Verhulst |year= 1845| title = Recherches mathématiques sur la loi d'accroissement de la population | journal = Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles |volume = 18| pages = 1–42 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323640_0018&DMDID=dmdlog7| accessdate = 2013-02-18|trans_title= Mathematical Researches into the Law of Population Growth Increase}}
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| * {{cite journal|first= Pierre-François |last=Verhulst |year= 1847| title = Deuxième mémoire sur la loi d'accroissement de la population | journal = Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique |volume = 20| pages = 1–32 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323659_0020&DMDID=dmdlog29| accessdate = 2013-02-18}}
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| ==External links==
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| * {{MacTutor Biography|id=Verhulst}}
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| {{Authority control|VIAF=66616324}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME =Verhulst, Pierre Francois
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = mathematician
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| | DATE OF BIRTH = 28 October 1804
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| | PLACE OF BIRTH = Brussels, Belgium
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| | DATE OF DEATH = 15 February 1849
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| | PLACE OF DEATH = Brussels, Belgium
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| }}{{dmy|date=January 2011}}
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| {{DEFAULTSORT:Verhulst, Pierre Francois}}
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| [[Category:1804 births]]
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| [[Category:1849 deaths]]
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| [[Category:Belgian mathematicians]]
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| [[Category:19th-century writers]]
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