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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:

 

:<math>

:<math>

\begin{align}

\begin{align}

  \frac{dx}{dt} &= \alpha x - \beta x y, \\

  \frac{dx}{dt} &= \alpha x - \beta x y, \\

  \frac{dy}{dt} &= \delta x y - \gamma y,

  \frac{dy}{dt} &= \delta x y - \gamma y,

\end{align}

\end{align}

</math>

</math>

:{{mvar|x}} is the number of prey (for example, [[rabbit]]s);

:{{mvar|x}} is the number of prey (for example, [[rabbit]]s);

:{{mvar|y}} is the number of some [[Predation|predator]] (for example, [[fox]]es);

:{{mvar|y}} is the number of some [[Predation|predator]] (for example, [[fox]]es);

:<math>\tfrac{dy}{dt}</math> and <math>\tfrac{dx}{dt}</math> represent the instantaneous growth rates of the two populations;

:<math>\tfrac{dy}{dt}</math> and <math>\tfrac{dx}{dt}</math> represent the instantaneous growth rates of the two populations;

:{{mvar|t}} represents time;

:{{mvar|t}} represents time;

:{{math|''α'', ''β'', ''γ'', ''δ''}} are positive real [[parameter]]s describing the interaction of the two [[species]].

:{{math|''α'', ''β'', ''γ'', ''δ''}} are positive real [[parameter]]s describing the interaction of the two [[species]].

are positive real parameters describing the interaction of the two species.

α, β, γ, δ are positive real parameters describing the interaction of the two species.

是描述两个物种相互作用的正实参数。

:{{math|''α'', ''β'', ''γ'', ''δ''}}是描述两个物种相互作用的正实参数。

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.

 

 

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>{{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 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.

洛特卡-沃尔泰拉捕食-被捕食模型最初是由阿尔弗雷德 · j · 洛特卡于1910年在自催化化学反应理论中提出的。这实际上就是逻辑方程式，最初由皮埃尔·弗朗索瓦·韦吕勒推导出来。1920年，Lotka 通过安德雷·柯尔莫哥洛夫，将这个模型扩展到“有机系统” ，以植物物种和草食动物物种为例，1925年，他在其关于生物数学的书中用这些方程分析了捕食者-猎物的相互作用。1926年，对数学生物学感兴趣的数学家和物理学家维托 · 沃尔泰拉发表了同样的方程组。沃尔泰拉的调查灵感来自于他与海洋生物学家翁贝托•安科纳(Umberto d’ ancona)的互动。安科纳当时正在追求自己的女儿，后来成了他的女婿。D’ ancona 研究了亚得里亚海的鱼类捕获量，并注意到在第一次世界大战期间(1914-1918年)捕获的捕食性鱼类的百分比有所增加。这使他感到困惑，因为在战争年代，捕鱼的努力大大减少了。沃尔泰拉独立于 Lotka 发展了他的模型，并用它来解释 d’ ancona 的观察。

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。这个模型最初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，洛特卡通过Andrey Kolmogorov将该模型扩展到了“有机系统”，以植物和草食性动物为例，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，后者当时正好向的女儿求婚，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。

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.

该模型后来被扩展到包括密度依赖的猎物生长和由 C.s. 霍林开发的功能性反应形式，这个模型后来被称为 Rosenzweig-MacArthur 模型。Lotka-Volterra 模型和 Rosenzweig-MacArthur 模型都被用来解释捕食者和猎物的自然种群动态，例如哈德逊湾公司的猞猁和雪鞋兔的数据，以及皇家岛国家公园的驼鹿和狼的种群数据。

该模型后来得到进一步扩展，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Rosenzweig–MacArthur模型。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。

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.

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.

洛特卡-沃尔泰拉方程在经济理论中的应用由来已久，它们最初的应用通常归功于理查德•古德温(Richard Goodwin)在1965年或1967年提出的理论。

Lotka-Volterra方程在经济理论中存在了很久。他们最初是由Richard Goodwin在1965或1967年应用过。

==Physical meaning of the equations==

== 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>{{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:

 

 

 

 

 

 

 

 

 

 

#Predators have limitless appetite.

* Predators have limitless appetite.

Predators have limitless appetite.

* Predators have limitless appetite.

食肉动物有无限的食欲。

* 食肉动物有无限的食欲。

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}}

===Prey===

=== Prey 猎物 ===

When multiplied out, the prey equation becomes

When multiplied out, the prey equation becomes

When multiplied out, the prey equation becomes

When multiplied out, the prey equation becomes

 

:<math>\frac{dx}{dt} = \alpha x - \beta x y.</math>

:<math>\frac{dx}{dt} = \alpha x - \beta x y.</math>

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.

被捕食的动物被假定有无限的食物供应并且以指数形式繁殖，除非被捕食; 这个指数增长在上面的方程中用术语 αx 来表示。捕食猎物的速率假定与捕食者和猎物相遇的速率成正比，这一点用 βxy 表示。如果其中一个或者等于零，那么就不存在捕食。

假定猎物具有无限的食物供应，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，则表示没有掠夺。

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.

===Predators===

=== Predators 捕食者===

The predator equation becomes

The predator equation becomes

The predator equation becomes

The predator equation becomes

 

:<math>\frac{dy}{dt} = \delta xy - \gamma y.</math>

:<math>\frac{dy}{dt} = \delta xy - \gamma y.</math>

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.

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==

== 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>{{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 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.

如果所有非负参数都不消失，三个参数可以被吸收到变量的归一化中，只留下一个参数: 由于第一个方程是齐次方程，第二个方程是齐次方程，参数 β/α 和 δ/γ 分别在正规化和归一化中被吸收，因此只保留任意性。这是影响解的性质的唯一参数。

如果该方程组中所有非负参数{{math|''α'', ''β'', ''γ'', ''δ''}}均存在，则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''}}中是齐次的，因此参数''β''/''α'' 和 ''δ''/''γ''分别在{{math|''y''}}和{{math|''x''}}中可以进行归一化处理，{{math|''γ''}}变成{{math|''t''}}的归一化，因此只有{{math|''α''/''γ''}}保持任意。并且它是影响解决方案性质的唯一参数。

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.

 

[[Image:Lotka Volterra dynamics.svg|center]]

[[Image:Lotka Volterra dynamics.svg|center]]