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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。

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，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。

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。

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>'''。即使到现在。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。

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。他们最初是由Richard Goodwin在1965或1967年应用过。

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。

 

 

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

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