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此词条Jie翻译。

{{About|the predator-prey equations|the competition equations|Competitive Lotka–Volterra equations}}

{{About|the predator-prey equations|the competition equations|Competitive Lotka–Volterra equations}}

The '''Lotka–Volterra equations''', also known as the '''predator–prey equations''', are a pair of first-order [[nonlinear]] [[differential equation]]s, frequently used to describe the [[dynamical system|dynamics]] of [[Systems biology|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 equation]]s, frequently used to describe the [[dynamical system|dynamics]] of [[Systems biology|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'''捕食者-猎物方程'''，以下简称为'''捕猎方程'''）是一对一阶非线性微分方程组，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：

'''<font color="#ff8000"> Lotka–Volterra方程式</font>'''（洛特卡-沃尔泰拉方程式，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：

:{{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}}是捕食者（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>代表两个种群数量的瞬时增长率；

:<math>\tfrac{dy}{dt}</math> 和 <math>\tfrac{dx}{dt}</math>代表这两个种群的瞬时增长率；

:{{mvar|t}}代表时间；

:{{mvar|t}}代表时间；

:{{math|''α'', ''β'', ''γ'', ''δ''}}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 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 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]].

Lotka–Volterra方程组是Kolmogorov模型的一个示例，而Kolmogorov模型<ref name=":0" /><ref name=":1" /><ref name="scholarpedia" /> 具有更一般的模型框架，可以用来刻画捕食者与猎物之间因猎食，竞争，疾病和共生等关系而形成的生态动力系统。

Lotka–Volterra方程组是Kolmogorov模型（柯尔莫戈洛夫模型）的一个示例，而Kolmogorov模型<ref name=":0" /><ref name=":1" /><ref name="scholarpedia" /> 具有更一般的模型框架，可以用来刻画捕食者与猎物之间因猎食，竞争，疾病和共生等关系而形成的生态动力系统。

== 历史 ==

== 历史 ==