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The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

'''<font color="#ff8000"> Lotka–Volterra方程式</font>'''（又称为'''捕猎方程'''）是一对一阶非线性微分方程，它经常被用来描述两个物种间因相互作用，从而产生的生物系统动力学反应。其中一个物种被视作捕食者，而另一个物种被视作猎物，它们共同组成的系统称为'''捕猎系统'''。其个体数量会随时间变化遵循如下一对方程组：

'''<font color="#ff8000"> Lotka–Volterra方程式</font>'''（又称为'''捕食者-猎物方程'''，以下简称为'''捕猎方程'''）是一对一阶非线性常微分方程组，我们经常用它来描述两个物种间因捕食和被捕食关系而形成的动力学系统，这样的系统可称为'''捕猎系统'''。该方程组反映了此二者物种的种群数量会随时间变化并遵循如下规律：

其中：

其中：

:{{mvar|x}}是猎物（例如兔子）的数量；

:{{mvar|x}}是猎物（例如兔子）的数量；

:{{mvar|y}}是捕猎者者（例如狐狸）的数量；

:{{mvar|y}}是捕食者（例如狐狸）的数量；

:<math>\tfrac{dy}{dt}</math> 和 <math>\tfrac{dx}{dt}</math>代表个体的瞬时增长率；

:<math>\tfrac{dy}{dt}</math> 和 <math>\tfrac{dx}{dt}</math>代表个体的瞬时增长率；

:{{mvar|t}}代表时间；

:{{mvar|t}}代表时间；

The Lotka–Volterra system of equations is an example of a [[Kolmogorov equations|Kolmogorov model]],<ref>{{cite book |last=Freedman |first=H. I. |title=Deterministic Mathematical Models in Population Ecology |publisher=[[Marcel Dekker]] |year=1980}}</ref><ref>{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chavez |first2=C. |title=Mathematical Models in Population Biology and Epidemiology |publisher=[[Springer-Verlag]] |year=2000}}</ref><ref name="scholarpedia">{{cite journal |last=Hoppensteadt |first=F. |title=Predator-prey model |journal=[[Scholarpedia]] |volume=1 |issue=10 |page=1563 |year=2006|doi=10.4249/scholarpedia.1563 |bibcode=2006SchpJ...1.1563H |doi-access=free }}</ref> which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, [[Competition (biology)|competition]], disease, and [[mutualism (biology)|mutualism]].

The Lotka–Volterra system of equations is an example of a [[Kolmogorov equations|Kolmogorov model]],<ref name=":0">{{cite book |last=Freedman |first=H. I. |title=Deterministic Mathematical Models in Population Ecology |publisher=[[Marcel Dekker]] |year=1980}}</ref><ref name=":1">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chavez |first2=C. |title=Mathematical Models in Population Biology and Epidemiology |publisher=[[Springer-Verlag]] |year=2000}}</ref><ref name="scholarpedia">{{cite journal |last=Hoppensteadt |first=F. |title=Predator-prey model |journal=[[Scholarpedia]] |volume=1 |issue=10 |page=1563 |year=2006|doi=10.4249/scholarpedia.1563 |bibcode=2006SchpJ...1.1563H |doi-access=free }}</ref> which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, [[Competition (biology)|competition]], disease, and [[mutualism (biology)|mutualism]].

The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.

The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.

 

 

 

== History 历史 ==

== History 历史 ==

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|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> 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|df=|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|issue=|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}}</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|location=|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|issue=|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.|location=|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. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.<ref>{{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>

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 name=":2">{{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|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> This was effectively the [[Logistic function#In ecology: modeling population growth|logistic equation]],<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)|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|df=|doi=10.2307/1940005|jstor=1940005}}</ref> originally derived by [[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> 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 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> and in 1925 he used the equations to analyse predator–prey interactions in his book on [[biomathematics]].<ref name=":6">{{cite book|last=Lotka|first=A. J.|title=Elements of Physical Biology|location=|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 name=":7">{{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|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> 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. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.<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>

The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. This was effectively the logistic equation, originally derived by Pierre François Verhulst. In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example and in 1925 he used the equations to analyse predator–prey interactions in his book on biomathematics. The same set of equations was published in 1926 by Vito Volterra, a mathematician and physicist, who had become interested in mathematical biology. 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. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.

The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. This was effectively the logistic equation, originally derived by Pierre François Verhulst. In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example and in 1925 he used the equations to analyse predator–prey interactions in his book on biomathematics. The same set of equations was published in 1926 by Vito Volterra, a mathematician and physicist, who had become interested in mathematical biology. 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. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.

Lotka–Volterra猎捕模型最初是由阿尔弗雷德·J·洛特卡Alfred J. Lotka于1910年在自催化化学反应理论中提出的。这个模型实际上是源于皮埃尔·弗朗索瓦·韦吕勒Pierre François Verhulst得出的逻辑方程。1920年，洛特卡以植物和草食性动物为例，通过Andrey Kolmogorov将该模型扩展到了“有机系统”，并于1925年在他的生物数学书中使用这些方程式分析了捕食者与猎物之间的相互作用。1926年，数学家和物理学家维托·沃尔泰拉Vito Volterra发表了同样的方程组。沃尔泰拉对数学生物学非常感兴趣。他对此的探索是受到他与海洋生物学家翁贝托·德安科纳Umberto D'Ancona互动的启发，后者当时正向他的女儿求婚，不久便成了他的女婿。德安科纳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），捕捞的掠食性鱼类其百分比有所增加。这使他感到困惑，因为在战争年代，捕鱼工作已大大减少。沃尔泰拉后来独立于洛特卡开发了他的模型，并用它来解释德安科纳的观察结果。

Lotka–Volterra捕猎模型最初是由阿尔弗雷德·J·洛特卡（Alfred J. Lotka）于1910年在自催化化学反应理论中提出的<ref name=":2" /><ref name="Goelmany" /> 。这个模型实际上是一类逻辑方程<ref name=":3" /> ，源自于皮埃尔·弗朗索瓦·韦吕勒（Pierre François Verhulst）<ref name=":4" /> 。1920年，洛特卡以植食和草食性动物为例<ref name=":5" />，在安德雷·柯尔莫哥洛夫（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" /> 。1926年，数学和物理学家维托·沃尔泰拉（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" /><ref name=":8" /> ，他对该领域的研究受到了与海洋生物学家翁贝托·德安科纳（Umberto D'Ancona）交流的启发，当时德安科纳正向他的女儿求婚，不久后便成了他的女婿。德安科纳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，沃尔泰拉独立于洛特卡发展了自己的模型，并用它来解释德安科纳的观察结果<ref name=":9" />。

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, such as the [[lynx]] and [[snowshoe hare]] data of the [[Hudson's Bay Company]]<ref>{{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> and the moose and wolf populations in [[Isle Royale National Park]].<ref>{{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>

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 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> Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey, such as the [[lynx]] and [[snowshoe hare]] data of the [[Hudson's Bay Company]]<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> and the moose and wolf populations in [[Isle Royale National Park]].<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>

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. Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey, such as the lynx and snowshoe hare data of the Hudson's Bay Company and the moose and wolf populations in Isle Royale National Park.

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. Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey, such as the lynx and snowshoe hare data of the Hudson's Bay Company and the moose and wolf populations in Isle Royale National Park.

该模型后来得到进一步扩展，衍生出了包括基于密度依赖的猎物生长机制和由C. S. Holling开发的功能响应机制。该模型后被称为'''<font color="#ff8000"> Rosenzweig–MacArthur模型</font>'''。Lotka–Volterra和Rosenzweig–MacArthur模型现均用于解释捕猎双方自然种群的动态，例如哈德逊湾公司的山猫和雪鞋野兔数据，以及皇家岛国家公园的麋鹿和狼种群。

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霍林（C. S. Holling）发展出的功能响应机制，被称为'''<font color="#ff8000"> Rosenzweig–MacArthur模型<ref name=":10" /></font>''' 。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" />的山猫和雪兔的种群数据，以及罗亚尔岛国家公园<ref name=":12" />的麋鹿和狼的种群数据。

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}}</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>

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 name=":13">{{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}}</ref> The validity of prey- or ratio-dependent models has been much debated.<ref name=":14">{{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>

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 model. The validity of prey- or ratio-dependent models has been much debated.

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 model. The validity of prey- or ratio-dependent models has been much debated.

在1980年代后期，出现了Lotka–Volterra捕猎模型（泛指常规食饵依赖模型）的替代模型，即比率依赖模型或'''<font color="#ff8000"> Arditi–Ginzburg模型</font>'''。但时至今日，食饵依赖和比率依赖模型的有效性一直存在争议。

在1980年代末，出现了Lotka–Volterra捕猎模型（泛指常规猎物依赖模型）的替代模型，即比率依赖模型或'''<font color="#ff8000"> Arditi–Ginzburg模型<ref name=":13" /></font>'''。但时至今日，猎物依赖和比率依赖模型的有效性一直存在争议<ref name=":14" />。

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.|authorlink=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>

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 name=":15">{{cite journal|last=Gandolfo|first=G.|authorlink=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 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>

The Lotka–Volterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in 1965 or 1967.

The Lotka–Volterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in 1965 or 1967.

Lotka-Volterra方程在经济理论学中存在了很久，最初由Richard Goodwin在1965或1967年应用。

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" /> 与1967年<ref name=":16" /><ref name=":17" />。

== Physical meaning of the equations 方程的物理意义 ==

== Physical meaning of the equations 方程的物理意义 ==

The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:<ref>{{Cite web|url=http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html|title=PREDATOR-PREY DYNAMICS|website=www.tiem.utk.edu|access-date=2018-01-09}}</ref>

The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:<ref name=":18">{{Cite web|url=http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html|title=PREDATOR-PREY DYNAMICS|website=www.tiem.utk.edu|access-date=2018-01-09}}</ref>

The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:

The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:

* 捕食者种群的食物供应充足与否完全取决于猎物种群的大小。

* 捕食者种群的食物供应充足与否完全取决于猎物种群的大小。

* 各种群数量变化率与其规模成正比。

* 各种群数量变化率与其规模成正比。

* 在此过程中，环境不会因一种物种而改变，并且无关于遗传适应性。

* 在此过程中，环境不会因一种物种而改变，并且忽略遗传适应性。

* 食肉动物有无限的食欲。

* 捕食者有无限的食欲。

In this case the solution of the differential equations is [[deterministic system|deterministic]] and [[Continuous function|continuous]]. This, in turn, implies that the generations of both the predator and prey are continually overlapping.<ref>{{cite book|last1=Cooke|first1=D.|last2=Hiorns|first2=R. W.|display-authors=etal|title=The Mathematical Theory of the Dynamics of Biological Populations|volume=II|publisher=Academic Press|year=1981}}</ref>

In this case the solution of the differential equations is [[deterministic system|deterministic]] and [[Continuous function|continuous]]. This, in turn, implies that the generations of both the predator and prey are continually overlapping.<ref name=":19">{{cite book|last1=Cooke|first1=D.|last2=Hiorns|first2=R. W.|display-authors=etal|title=The Mathematical Theory of the Dynamics of Biological Populations|volume=II|publisher=Academic Press|year=1981}}</ref>

In this case the solution of the differential equations is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.

In this case the solution of the differential equations is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.

{{see also|Competitive Lotka–Volterra equations}}

{{see also|Competitive Lotka–Volterra equations}}

When multiplied out, the prey equation becomes

When multiplied out, the prey equation becomes

当猎物数量成倍增趋势时，猎物方程变为:

当猎物数量成倍增趋势时，猎物方程可表示为:

The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by βxy. If either  or  is zero, then there can be no predation.

The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by βxy. If either  or  is zero, then there can be no predation.

假定猎物具有无限的食物供应，且除非受到捕食，否则可以成倍繁殖，那么其指数增长由上式中的''αx''来表示。假设掠食者的掠食率，与掠食者和猎物的相遇率成正比，则用上式中的''βxy''表示。注意如果{{mvar|x}}或{{mvar|y}}为零，则表示没有捕猎。

假定猎物具有无限的食物供应，且除非受到捕食，否则可以成倍繁殖，那么其指数增长可表示为上式中的''αx''。假设捕食者的捕食率，与捕食者和猎物的相遇率成正比，那么即可表示为上式中的''βxy''。注意如果{{mvar|x}}或{{mvar|y}}为零，则表示没有发生捕猎。

With these two terms the equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus the rate at which it is preyed upon.

With these two terms the equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus the rate at which it is preyed upon.

The predator equation becomes

The predator equation becomes

捕食者方程可以表示为:

捕食者方程可表示为:

In this equation,  represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term  represents the loss rate of the predators due to either natural death or emigration, it leads to an exponential decay in the absence of prey.

In this equation,  represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term  represents the loss rate of the predators due to either natural death or emigration, it leads to an exponential decay in the absence of prey.

在此等式中，{{math|''δxy''}}代表捕食者种群的增长。（请注意此处与捕食率表达式虽然相似；但是使用了一个不同的常数，因为捕食者的生长速度不一定等于其消耗猎物的速度）。另外{{math|''γy''}}表示由于自然死亡或迁徙造成的捕食者数量减少率，它在没有猎物的情况下是指数型衰减的。

在此等式中，{{math|''δxy''}}代表捕食者种群的增长。（请注意此处与捕食率表达式虽然相似；但是使用了一个不同的常数，因为捕食者的增长速率不一定等于其捕杀猎物的速率）。另外{{math|''γy''}}表示由于自然死亡或迁徙造成的捕食者数量减少率，它在没有猎物的情况下是呈指数型衰减的。

Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate.

Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate.

== Solutions to the equations 方程求解 ==

== Solutions to the equations 方程求解 ==

The equations have [[periodic function|periodic]] solutions and do not have a simple expression in terms of the usual [[trigonometric function]]s, although they are quite tractable.<ref>{{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>{{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>

The equations have [[periodic function|periodic]] solutions and do not have a simple expression in terms of the usual [[trigonometric function]]s, although they are quite tractable.<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>

The equations have periodic solutions and do not have a simple expression in terms of the usual trigonometric functions, although they are quite tractable.

The equations have periodic solutions and do not have a simple expression in terms of the usual trigonometric functions, although they are quite tractable.

If none of the non-negative parameters  vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in , and the second one in , the parameters β/α and δ/γ are absorbable in the normalizations of  and  respectively, and  into the normalization of , so that only  remains arbitrary. It is the only parameter affecting the nature of the solutions.

If none of the non-negative parameters  vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in , and the second one in , the parameters β/α and δ/γ are absorbable in the normalizations of  and  respectively, and  into the normalization of , so that only  remains arbitrary. It is the only parameter affecting the nature of the solutions.

如果该方程组中所有非负参数{{math|''α'', ''β'', ''γ'', ''δ''}}均存在，那么我们可以将其中三个变量进行归一化，仅留下一个参数：由于第一个方程在{{math|''x''}}中是齐次的，而第二个方程在{{math|''y''}}中也是齐次的，因此我们可以对分别在{{math|''y''}}和{{math|''x''}}中的参数''β''/''α'' 和 ''δ''/''γ''进行归一化处理，{{math|''γ''}}变成{{math|''t''}}的归一化，因此只有{{math|''α''/''γ''}}保持任意。它是影响解决方案性质的唯一参数。

如果该方程组中所有非负参数{{math|''α'', ''β'', ''γ'', ''δ''}}均存在，那么我们可以将其中三个变量进行归一化，仅留下一个参数：由于第一个方程对于{{math|''x''}}而言是齐次的，且第二个方程对于{{math|''y''}}而言也是齐次的，因此我们可以分别对{{math|''y''}}和{{math|''x''}}的参数归一化处理为''β''/''α'' 和 ''δ''/''γ''，再令{{math|''γ''}}对{{math|''t''}}作归一化，最后只有{{math|''α''/''γ''}}保持任意，它成为了影响解的性质的唯一参数。

A [[linearization]] of the equations yields a solution similar to [[simple harmonic motion]]<ref>{{cite book|last=Tong|first=H.|title=Threshold Models in Non-linear Time Series Analysis|location=|publisher=Springer–Verlag|year=1983}}</ref> with the population of predators trailing that of prey by 90° in the cycle.

A [[linearization]] of the equations yields a solution similar to [[simple harmonic motion]]<ref name=":22">{{cite book|last=Tong|first=H.|title=Threshold Models in Non-linear Time Series Analysis|location=|publisher=Springer–Verlag|year=1983}}</ref> with the population of predators trailing that of prey by 90° in the cycle.

A linearization of the equations yields a solution similar to simple harmonic motion with the population of predators trailing that of prey by 90° in the cycle.

A linearization of the equations yields a solution similar to simple harmonic motion with the population of predators trailing that of prey by 90° in the cycle.

Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 10 baboons and 10 cheetahs, one can plot the progression of the two species over time; given the parameters that the growth and death rates of baboon are 1.1 and 0.4 while that of cheetahs are 0.1 and 0.4 respectively. The choice of time interval is arbitrary.

Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 10 baboons and 10 cheetahs, one can plot the progression of the two species over time; given the parameters that the growth and death rates of baboon are 1.1 and 0.4 while that of cheetahs are 0.1 and 0.4 respectively. The choice of time interval is arbitrary.

假设有两种动物，即狒狒（猎物）和猎豹（捕食者）。如果初始条件是10只狒狒和10只猎豹，那么我们可以绘制出这两个物种随时间推移的数量。假设给定参数，狒狒的增长率和死亡率分别为1.1和0.4，而猎豹的增长率和死亡率分别为0.1和0.4，且间隔时间任意。

假设有两种动物，狒狒（猎物）和猎豹（捕食者）。如果初始条件是10只狒狒和10只猎豹，狒狒的增长率和死亡率分别为1.1和0.4，而猎豹的增长率和死亡率分别为0.1和0.4，且在时间轴上从初始点开始任选一段足够长的区间，那么我们可以绘制出这两个物种随时间推移的数量变化曲线。

One may also plot solutions parametrically as orbits in phase space, without representing time, but with one axis representing the number of prey and the other axis representing the number of predators for all times.

One may also plot solutions parametrically as orbits in phase space, without representing time, but with one axis representing the number of prey and the other axis representing the number of predators for all times.

This corresponds to eliminating time from the two differential equations above to produce a single differential equation

This corresponds to eliminating time from the two differential equations above to produce a single differential equation

relating the variables x and y. The solutions of this equation are closed curves. It is amenable to separation of variables: integrating

relating the variables x and y. The solutions of this equation are closed curves. It is amenable to separation of variables: integrating

仅包含关联变量''x'' 和 ''y''。该方程的解是个闭合曲线，可以分离变量：对以下式子进行积分

该方程仅包含变量''x'' 和 ''y''。方程的解是一条闭合曲线，可以通过分离变量法求解：对以下式子进行积分

where V is a constant quantity depending on the initial conditions and conserved on each curve.

where V is a constant quantity depending on the initial conditions and conserved on each curve.

其中''V''是取决于初始条件的定量，并且在每条曲线上均守恒。

其中''V''是取决于初始条件的常量，并且在每条曲线上均守恒。

An aside: These graphs illustrate a serious potential problem with this ''as a biological model'': For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs as well. This modelling problem has been called the "atto-fox problem", an [[atto-]]<nowiki>fox</nowiki> being a notional 10⁻¹⁸ of a fox.<ref name="LobrySari2015">{{cite journal |last1=Lobry |first1=Claude |last2=Sari |first2=Tewfik |title=Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem |journal=Arima |date=2015 |volume=20 |pages=95–125 |url=http://arima.inria.fr/020/pdf/vol.20.pp.95-125.pdf}}</ref><ref>{{cite journal |last=Mollison |first=D. |url=http://www.ma.hw.ac.uk/~denis/epi/velocities.pdf |title=Dependence of epidemic and population velocities on basic parameters |journal=Math. Biosci. |volume=107 |issue=2 |pages=255–287 |year=1991 |doi=10.1016/0025-5564(91)90009-8|pmid=1806118 }}</ref>

An aside: These graphs illustrate a serious potential problem with this ''as a biological model'': For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs as well. This modelling problem has been called the "atto-fox problem", an [[atto-]]<nowiki>fox</nowiki> being a notional 10⁻¹⁸ of a fox.<ref name="LobrySari2015">{{cite journal |last1=Lobry |first1=Claude |last2=Sari |first2=Tewfik |title=Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem |journal=Arima |date=2015 |volume=20 |pages=95–125 |url=http://arima.inria.fr/020/pdf/vol.20.pp.95-125.pdf}}</ref><ref name=":23">{{cite journal |last=Mollison |first=D. |url=http://www.ma.hw.ac.uk/~denis/epi/velocities.pdf |title=Dependence of epidemic and population velocities on basic parameters |journal=Math. Biosci. |volume=107 |issue=2 |pages=255–287 |year=1991 |doi=10.1016/0025-5564(91)90009-8|pmid=1806118 }}</ref>

An aside: These graphs illustrate a serious potential problem with this as a biological model: For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs as well. This modelling problem has been called the "atto-fox problem", an atto-<nowiki>fox</nowiki> being a notional 10⁻¹⁸ of a fox.

An aside: These graphs illustrate a serious potential problem with this as a biological model: For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs as well. This modelling problem has been called the "atto-fox problem", an atto-<nowiki>fox</nowiki> being a notional 10⁻¹⁸ of a fox.

另外值得注意的是，这些图说明了该方程式作为生物学模型的严重潜在问题：因为这种特定的参数选择，在每个周期中，狒狒的数量都被减少到极低的数量，但却有能力恢复（事实上，在极低的狒狒密度下，猎豹的数量仍然很大）。这在现实中显然是不太可能的，离散个体的偶然性波动，以及狒狒的家庭结构和生命周期都有可能导致狒狒种族灭绝，结果也就造成了猎豹的灭绝。按照此类方法建模出现的问题被称为“atto-fox问题”，“atto“这里指的是fox的十之负十八次方。

另外值得注意的是，这些图说明了该方程组作为生物学模型的严重潜在问题：因为某种特定的参数选择，在每个周期中，狒狒的数量都被减少到极低的数量，但却有能力恢复（并且在极低的狒狒种群密度之下，猎豹的数量仍然很大）。这在现实中显然是不可能的，离群索居的狒狒的数量的波动，以及狒狒的家庭结构和生命周期都有可能导致狒狒种族灭绝，结果也就造成了猎豹的灭绝。按照此类方法建模出现的问题被称为"atto-fox问题"，"atto"这里指的是十的负十八次方<ref name="LobrySari2015" /><ref name=":23" />。

一个较为合理的例子是：

一个较为合理的例子是：

{{mvar|α}} = 2/3, {{mvar|β}} = 4/3, {{mvar|γ}} = 1 = {{mvar|δ}}. 假设{{math|''x'', ''y''}}处于“千”级别还不到“万“。圆圈代表从{{mvar|x}} = {{mvar|y}} = 0.9 到 1.8时猎物和捕食者的初始条件，步长为0.1，且固定点在（1，1/2）。

{{mvar|α}} = 2/3, {{mvar|β}} = 4/3, {{mvar|γ}} = 1 = {{mvar|δ}}. 假设{{math|''x'', ''y''}}处于“千”级别还不到“万“。圆圈代表从{{mvar|x}} = {{mvar|y}} = 0.9 到 1.8时猎物和捕食者的初始条件，步长为0.1，不动点为（1，1/2）。

== Dynamics of the system 系统动力学 ==

== Dynamics of the system 系统动力学 ==

In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a cycle of growth and decline.

In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a cycle of growth and decline.

The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters α, β, γ, and δ.

The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters α, β, γ, and δ.

第一种解明显代表了两种物种的灭绝。当它们的解都为0时，它们将继续无限期地保持绝种状态。第二种解则表示两个种群都维持其当前固定数量（非零点），在简化模型中，它们无限期地保持状态不变。达到此平衡的总体水平取决于参数''α'', ''β'', ''γ'', 和 ''δ''的选定值。

第一种解实际上代表了两种物种的灭绝。当它们的解都为0时，它们将继续无限期地保持灭绝状态。第二种解则表示两个种群都维持其当前固定数量（非零点），在简化模型中，它们无限期地保持状态不变。达到平衡时的种群数量水平取决于参数''α'', ''β'', ''γ'', 和 ''δ''的给定值。

The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives.

The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives.

When evaluated at the steady state of (0,&nbsp;0), the Jacobian matrix J becomes

When evaluated at the steady state of (0,&nbsp;0), the Jacobian matrix J becomes

当在(0,&nbsp;0)的稳态下求值时，雅可比矩阵''J''变为

当在(0,&nbsp;0)的稳态下求值时，雅可比矩阵''J''为

In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point.

In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point.

在模型中，由于''α'' 和 ''γ''始终大于零，因此上述特征值的符号将始终不同。继而得到原点的固定点是鞍点。

在模型中，由于''α'' 和 ''γ''始终大于零，因此上述特征值的符号将始终不同。继而得到原点的不动点是鞍点。

The stability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.

The stability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.

Evaluating J at the second fixed point leads to

Evaluating J at the second fixed point leads to

在第二个不动点求''J''可得：

在第二个不动点求雅可比矩阵''J''可得：

As illustrated in the circulating oscillations in the figure above, the level curves are closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with frequency <math>\omega = \sqrt{\alpha \gamma}</math>.

As illustrated in the circulating oscillations in the figure above, the level curves are closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with frequency <math>\omega = \sqrt{\alpha \gamma}</math>.

如上图的循环振荡所示，其等高线围绕不动点形成闭合轨道：因此捕食者和猎物的种群数量在不动点处以频率<math>\omega = \sqrt{\alpha \gamma}</math>循环并振荡且无阻尼。

如上图的循环振荡所示，其等高线围绕不动点形成闭合轨道：因此捕食者和猎物的种群数量在不动点处以频率<math>\omega = \sqrt{\alpha \gamma}</math>循环并且无阻尼震荡。

Increasing K moves a closed orbit  closer to the fixed point. The largest value of the constant K is obtained by solving the optimization problem

Increasing K moves a closed orbit  closer to the fixed point. The largest value of the constant K is obtained by solving the optimization problem

''K''的增加会将闭合轨道移近固定点。另外通过解决优化问题还可以获得常数''K''的最大值。

''K''的增加会将闭合轨道移近不动点。另外通过解决优化问题还可以获得常数''K''的最大值。