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→历史
== 历史 ==
== 历史 ==
Lotka–Volterra捕猎模型最初是由Alfred J. Lotka于1910年在自催化化学反应理论中提出的<ref name=":2">{{cite journal|last=Lotka|first=A. J.|title=Contribution to the Theory of Periodic Reaction|journal=Journal of Physical Chemistry A|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|last=Goel|first=N. S.|display-authors=etal|title=On the Volterra and Other Non-Linear Models of Interacting Populations|location=|publisher=Academic Press|year=1971}}</ref>。这个模型实际上是一类逻辑方程<ref name=":3">{{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)|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|df=|doi=10.2307/1940005|jstor=1940005}}</ref> ，源自于Pierre François Verhulst<ref name=":4">{{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|issue=|pages=113–121|year=1838}}</ref> 。1920年，Lotka以植食和草食性动物为例<ref name=":5">{{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}}</ref>，在Andrey Kolmogorov的帮助下将该模型扩展到了“有机系统”，并于1925年，他在自己编写的生物数学书中使用了这些方程式分析了捕食者与猎物之间的相互关系<ref name=":6">{{cite book|last=Lotka|first=A. J.|title=Elements of Physical Biology|location=|publisher=Williams and Wilkins|year=1925}}</ref>。1926年，数学和物理学家Vito Volterra发表了同样的方程组。沃尔泰拉对数理生物学非常感兴趣<ref name="Goelmany"/><ref name=":7">{{cite journal|last=Volterra|first=V.|title=Variazioni e fluttuazioni del numero d'individui in specie animali conviventi|journal=Accademia dei Lincei|volume=2|issue=|pages=31–113|year=1926}}</ref><ref name=":8">{{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.|location=|publisher=McGraw–Hill|year=1931}}</ref>，他对该领域的研究受到了与海洋生物学家Umberto D'Ancona交流的启发，当时D'Ancona正向他的女儿求婚，不久后便成了他的女婿。D'Ancona研究了亚得里亚海的渔获物，并注意到在第一次世界大战期间（1914-1918年），捕捞的肉食性鱼类的百分比有所增加。因为这种现象恰好发生在捕鱼量已大大减少的战争年代，这使他感到困惑不已。后来，Volterra独立于洛特卡发展了自己的模型，并用它来解释D'Ancona的观察结果<ref name=":9" />。
Lotka–Volterra捕猎模型最初是由Alfred J. Lotka于1910年在自催化化学反应理论中提出的<ref name=":2">{{cite journal|last=Lotka|first=A. J.|title=Contribution to the Theory of Periodic Reaction|journal=Journal of Physical Chemistry A|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|last=Goel|first=N. S.|display-authors=etal|title=On the Volterra and Other Non-Linear Models of Interacting Populations|location=|publisher=Academic Press|year=1971}}</ref>。这个模型实际上是一类逻辑方程<ref name=":3">{{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)|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|df=|doi=10.2307/1940005|jstor=1940005}}</ref> ，源自于Pierre François Verhulst<ref name=":4">{{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|issue=|pages=113–121|year=1838}}</ref> 。1920年，Lotka以植食和草食性动物为例<ref name=":5">{{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}}</ref>，在Andrey Kolmogorov的帮助下将该模型扩展到了“有机系统”，并于1925年，他在自己编写的生物数学书中使用了这些方程式分析了捕食者与猎物之间的相互关系<ref name=":6">{{cite book|last=Lotka|first=A. J.|title=Elements of Physical Biology|location=|publisher=Williams and Wilkins|year=1925}}</ref>。1926年，数学和物理学家Vito Volterra发表了同样的方程组。沃尔泰拉对数理生物学非常感兴趣<ref name="Goelmany"/><ref name=":7">{{cite journal|last=Volterra|first=V.|title=Variazioni e fluttuazioni del numero d'individui in specie animali conviventi|journal=Accademia dei Lincei|volume=2|issue=|pages=31–113|year=1926}}</ref><ref name=":8">{{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.|location=|publisher=McGraw–Hill|year=1931}}</ref>，他对该领域的研究受到了与海洋生物学家Umberto D'Ancona交流的启发，当时D'Ancona正向他的女儿求婚，不久后便成了他的女婿。D'Ancona研究了亚得里亚海的渔获物，并注意到在第一次世界大战期间（1914-1918年），捕捞的肉食性鱼类的百分比有所增加。因为这种现象恰好发生在捕鱼量已大大减少的战争年代，这使他感到困惑不已。后来，Volterra独立于洛特卡发展了自己的模型，并用它来解释D'Ancona的观察结果。<ref name=":9">{{cite book|last=Kingsland|first=S.|title=Modeling Nature: Episodes in the History of Population Ecology|location=|publisher=University of Chicago Press|year=1995|isbn=978-0-226-43728-6}}</ref>
Lotka-Volterra方程在理论经济学中有很长的应用历史，最早由Richard Goodwin应用于1965<ref name=":15">{{cite journal|last=Gandolfo|first=G.|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}}</ref>与1967年。<ref name=":16">{{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 name=":17">{{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>
Lotka-Volterra方程在理论经济学中有很长的应用历史，最早由Richard Goodwin应用于1965<ref name=":15">{{cite journal|last=Gandolfo|first=G.|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}}</ref>与1967年。<ref name=":16">{{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 name=":17">{{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>
== 方程的物理意义 ==
== 方程的物理意义 ==