更改

之后该模型得到进一步扩展，加入了包括依赖于种群密度的猎物增长机制和由C. S. Holling发展出的功能响应机制，被称为'''Rosenzweig–MacArthur模型'''<ref name=":10">{{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> 。Lotka–Volterra和Rosenzweig–MacArthur模型一直被用于解释捕猎双方自然种群的动态变化，例如哈德逊湾<ref name=":11">{{cite journal|last=Gilpin|first=M. E.|year=1973|title=Do hares eat lynx?|journal=American Naturalist|issue=957|pages=727–730|doi=10.1086/282870|volume=107|s2cid=84794121}}</ref>的山猫和雪兔的种群数据，以及罗亚尔岛国家公园的麋鹿和狼的种群数据。<ref name=":12">{{cite journal|last1=Jost|first1=C.|last2=Devulder|first2=G.|last3=Vucetich|first3=J.A.|last4=Peterson|first4=R.|last5=Arditi|first5=R.|doi=10.1111/j.1365-2656.2005.00977.x|title=The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose|journal=J. Anim. Ecol.|volume=74|issue=5|pages=809–816|year=2005}}</ref>

之后该模型得到进一步扩展，加入了包括依赖于种群密度的猎物增长机制和由C. S. Holling发展出的功能响应机制，被称为'''Rosenzweig–MacArthur模型'''<ref name=":10">{{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}}</ref> 。Lotka–Volterra和Rosenzweig–MacArthur模型一直被用于解释捕猎双方自然种群的动态变化，例如哈德逊湾<ref name=":11">{{cite journal|last=Gilpin|first=M. E.|year=1973|title=Do hares eat lynx?|journal=American Naturalist|issue=957|pages=727–730|doi=10.1086/282870|volume=107}}</ref>的山猫和雪兔的种群数据，以及罗亚尔岛国家公园的麋鹿和狼的种群数据。<ref name=":12">{{cite journal|last1=Jost|first1=C.|last2=Devulder|first2=G.|last3=Vucetich|first3=J.A.|last4=Peterson|first4=R.|last5=Arditi|first5=R.|doi=10.1111/j.1365-2656.2005.00977.x|title=The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose|journal=J. Anim. Ecol.|volume=74|issue=5|pages=809–816|year=2005}}</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|s2cid=140537163}}</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>

== 方程求解 ==

== 方程求解 ==

对于通常的三角方程，虽然很容易处理并得到其周期解，但是其解的表达式并不简洁。<ref name=":20">{{cite journal|last1=Steiner|first1=Antonio|last2=Gander|first2=Martin Jakob|year=1999|title=Parametrische Lösungen der Räuber-Beute-Gleichungen im Vergleich|journal=Il Volterriano|volume=7|issue=|pages=32–44|url=http://archive-ouverte.unige.ch/unige:6300/ATTACHMENT01}}</ref><ref name=":21">{{cite journal|last1=Evans|first1=C. M.|last2=Findley|first2=G. L.|title=A new transformation for the Lotka-Volterra problem|journal=Journal of Mathematical Chemistry|volume=25|issue=|pages=105–110|year=1999|doi=10.1023/A:1019172114300|s2cid=36980176}}</ref>

对于通常的三角方程，虽然很容易处理并得到其周期解，但是其解的表达式并不简洁。<ref name=":20">{{cite journal|last1=Steiner|first1=Antonio|last2=Gander|first2=Martin Jakob|year=1999|title=Parametrische Lösungen der Räuber-Beute-Gleichungen im Vergleich|journal=Il Volterriano|volume=7|issue=|pages=32–44|url=http://archive-ouverte.unige.ch/unige:6300/ATTACHMENT01}}</ref><ref name=":21">{{cite journal|last1=Evans|first1=C. M.|last2=Findley|first2=G. L.|title=A new transformation for the Lotka-Volterra problem|journal=Journal of Mathematical Chemistry|volume=25|issue=|pages=105–110|year=1999|doi=10.1023/A:1019172114300}}</ref>