Editing Lotka–Volterra equations (section)

==History==

The Lotka–Volterra predator–prey [[mathematical model|model]] was initially proposed by [[Alfred J. Lotka]] in the theory of autocatalytic chemical reactions in 1910.<ref>{{cite journal|last=Lotka|first=A. J.|title=Contribution to the Theory of Periodic Reaction|journal=[[Journal of Physical Chemistry A|J. Phys. Chem.]]|volume=14|issue=3|pages=271–274|year=1910| doi=10.1021/j150111a004|url=https://zenodo.org/record/1428768}}</ref><ref name="Goelmany">{{cite book |last1=Goel |first1=N. S. |first2=S. C. |last2=Maitra |first3=E. W. |last3=Montroll |display-authors=1 |title=On the Volterra and Other Nonlinear Models of Interacting Populations |publisher=Academic Press|year=1971 |isbn=0-12-287450-1 }}</ref> This was effectively the [[Logistic function#In ecology: modeling population growth|logistic equation]],<ref>{{cite journal|last=Berryman|first=A. A.| url=http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf|title=The Origins and Evolution of Predator-Prey Theory|journal=[[Ecology (journal)|Ecology]]|volume=73|issue=5|pages=1530–1535|year=1992|url-status=dead|archive-url=https://web.archive.org/web/20100531204042/http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf|archive-date=2010-05-31|doi=10.2307/1940005|jstor=1940005}}</ref> originally derived by [[Pierre François Verhulst]].<ref>{{cite journal|last=Verhulst|first=P. H.|url=https://books.google.com/books?id=8GsEAAAAYAAJ|title=Notice sur la loi que la population poursuit dans son accroissement|journal=Corresp. Mathématique et Physique|volume=10|pages=113–121|year=1838}}</ref> In 1920 Lotka extended the model, via [[Andrey Kolmogorov]], to "organic systems" using a plant species and a herbivorous animal species as an example<ref>{{cite journal|last=Lotka|first=A. J.|pmc=1084562|title=Analytical Note on Certain Rhythmic Relations in Organic Systems|journal=[[Proc. Natl. Acad. Sci. U.S.A.]]|volume=6|issue=7|pages=410–415|year=1920|doi=10.1073/pnas.6.7.410|pmid=16576509|bibcode=1920PNAS....6..410L|doi-access=free}}</ref> and in 1925 he used the equations to analyse predator–prey interactions in his book on [[biomathematics]].<ref>{{cite book|last=Lotka|first=A. J.|title=Elements of Physical Biology|publisher=[[Williams and Wilkins]]|year=1925}}</ref> The same set of equations was published in 1926 by [[Vito Volterra]], a mathematician and physicist, who had become interested in [[mathematical biology]].<ref name="Goelmany"/><ref>{{cite journal|last=Volterra|first=V.|title=Variazioni e fluttuazioni del numero d'individui in specie animali conviventi|journal=[[Accademia dei Lincei|Mem. Acad. Lincei Roma]]|volume=2|pages=31–113|year=1926}}</ref><ref>{{cite book|last=Volterra|first=V.|chapter=Variations and fluctuations of the number of individuals in animal species living together|title=Animal Ecology|editor-last=Chapman|editor-first=R. N.|publisher=[[McGraw–Hill]]|year=1931}}</ref> Volterra's enquiry was inspired through his interactions with the marine biologist [[Umberto D'Ancona]], who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the [[Adriatic Sea]] and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18). This puzzled him, as the fishing effort had been very much reduced during the war years and, as prey fish the preferred catch, one would intuitively expect this to increase of prey fish percentage. Volterra developed his model to explain D'Ancona's observation and did this independently from Alfred Lotka. He did credit Lotka's earlier work in his publication, after which the model has become known as the "Lotka-Volterra model".<ref>{{cite book|last=Kingsland|first=S.| title=Modeling Nature: Episodes in the History of Population Ecology|publisher=University of Chicago Press|year=1995| isbn=978-0-226-43728-6}}</ref>

The model was later extended to include density-dependent prey growth and a [[functional response]] of the form developed by [[C. S. Holling]]; a model that has become known as the Rosenzweig–MacArthur model.<ref>{{cite journal|last1=Rosenzweig|first1=M. L.|last2=MacArthur|first2=R.H.|year=1963|title=Graphical representation and stability conditions of predator-prey interactions|journal=American Naturalist|issue=895|pages=209–223|doi=10.1086/282272|volume=97|s2cid=84883526}}</ref> Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey.

In the late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or [[Arditi–Ginzburg equations|Arditi–Ginzburg model]].<ref>{{cite journal|last1=Arditi|first1=R.|last2=Ginzburg|first2=L. R.|year=1989|url=http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf|title=Coupling in predator-prey dynamics: ratio dependence|journal=Journal of Theoretical Biology|volume=139|issue=3|pages=311–326|doi=10.1016/s0022-5193(89)80211-5|bibcode=1989JThBi.139..311A|access-date=2013-06-26|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304053545/http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf|url-status=dead}}</ref> The validity of prey- or ratio-dependent models has been much debated.<ref>{{cite journal|last1=Abrams|first1=P. A.|last2=Ginzburg|first2=L. R.| year=2000|title=The nature of predation: prey dependent, ratio dependent or neither?|journal=Trends in Ecology & Evolution| volume=15| issue=8|pages=337–341|doi=10.1016/s0169-5347(00)01908-x|pmid=10884706}}</ref>

The Lotka–Volterra equations have a long history of use in [[Economics|economic theory]]; their initial application is commonly credited to [[Richard M. Goodwin|Richard Goodwin]] in 1965<ref>{{cite journal|last=Gandolfo|first=G.|author-link=Giancarlo Gandolfo|title=Giuseppe Palomba and the Lotka–Volterra equations|journal=Rendiconti Lincei|volume=19|issue=4| pages=347–357| year=2008|doi=10.1007/s12210-008-0023-7|s2cid=140537163}}</ref> or 1967.<ref>{{cite book|last=Goodwin|first=R. M.|chapter=A Growth Cycle|title=Socialism, Capitalism and Economic Growth|chapter-url=https://archive.org/details/socialismcapital0000fein | chapter-url-access=registration|editor-last=Feinstein|editor-first=C. H.|publisher=[[Cambridge University Press]]|year=1967}}</ref><ref>{{cite journal|last1=Desai|first1=M.|last2=Ormerod|first2=P.|url=http://www.paulormerod.com/pdf/economicjournal1998.pdf |title=Richard Goodwin: A Short Appreciation|journal=[[The Economic Journal]]|volume=108 |issue=450 |pages=1431–1435|year=1998|doi=10.1111/1468-0297.00350|citeseerx=10.1.1.423.1705|access-date=2010-03-22|archive-url=https://web.archive.org/web/20110927154044/http://www.paulormerod.com/pdf/economicjournal1998.pdf|archive-date=2011-09-27|url-status=dead}}</ref>