集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织 - 用户贡献 [zh-cn]

感染率

2022-06-14T14:14:36Z

水手9303：

此词条暂由Potatozh初译，由水手9303校核整理，如发现问题，请见谅，并恳请告知。

An infection rate (or incident rate) is the probability or risk of an infection in a population. It is used to measure the frequency of occurrence of new instances of infection within a population during a specific time period.

感染率（或发病率）是一个人群的感染可能性或风险。它用于衡量特定时间段内某个人群中新感染病例的发生频率。

[[File:1.png|400px]]

The number of infections equals the cases identified in the study or observed. An example might by HIV infection during a specific time period in the defined population. The population at risk are the cases appearing in the population during the same time period. An example would be all the people in a city during a specific time period. The constant, or K is assigned a value of 100 to represent a percentage. An example would be to who are infected with HIV: 6,000 cases in March divided by the population of a city (one million) multiplied by the constant (K) would give an infection rate of 0.6%. Calculating the infection rate is used to analyze trends for the purpose of infection and disease control.<ref>{{cite web | title = Calculation of Infection Rates|url = http://health.utah.gov/epi/diseases/HAI/resources/Cal_Inf_Rates.pdf |publisher = Utah Department of Health|date= 2017| accessdate = 2017-01-09 }}</ref>

感染数量等于研究中确诊或观察到的病例。一个例子是，在特定时期内，在定义的人群中，可能的艾滋病感染者。高危人群，是同一时期内出现在某人群中的人数。例如，特定时间段内，某城市中的所有人。一个常数K，赋值为100，用来表示它是百分比。例如，一个城市中感染艾滋病毒的人口百分比：3月份，有6,000个例病例，除以该城市人口数目（一百万），再乘以常数（K，即100），得出感染率为0.6％。计算感染率，通常是用于分析趋势，以达到控制感染和疾病的目的<ref>{{cite web | title = Calculation of Infection Rates|url = http://health.utah.gov/epi/diseases/HAI/resources/Cal_Inf_Rates.pdf |publisher = Utah Department of Health|date= 2017| accessdate = 2017-01-09 }}</ref>。

An online infection rate calculator has been developed by the Centers for Disease Control and Prevention that allows the determination of the Streptococcal A infection rate in a population.<ref>{{cite web | title = Group A Streptococcus Calculator | publisher=Centers for Disease Control and Prevention |date= 4 July 2008 |url = https://www2a.cdc.gov/ncidod/dbmd/abcs/calc/calc_new/calculator.html | accessdate = 2017-01-09 }}</ref>

（美国）疾病预防控制中心开发了一个在线的感染率计算器，可以用来确定一个人群中的链球菌A的感染率<ref>{{cite web | title = Group A Streptococcus Calculator | publisher=Centers for Disease Control and Prevention |date= 4 July 2008 |url = https://www2a.cdc.gov/ncidod/dbmd/abcs/calc/calc_new/calculator.html | accessdate = 2017-01-09 }}</ref>。

== 临床应用 Clinical applications ==

Health care facilities routinely track their infection rates according to the guidelines issued by the Joint Commission.<ref>{{cite web | title= Basic Statistical Data Used in Acute Care Facilities | publisher = JB Learning | url = http://www.jblearning.com/samples/0763750344/45561_CH01.pdf | accessdate = 2017-01-09 }}</ref> The healthcare-associated infection (HAI) rates measure infection of patients in a particular hospital. This allows rates to compared with other hospitals. These infections can often be prevented when healthcare facilities follow guidelines for safe care. To get payment from Medicare, hospitals are required to report data about some infections to the Centers for Disease Control and Prevention's (CDC's) National Healthcare Safety Network (NHSN). Hospitals currently submit information on central line-associated bloodstream infections (CLABSIs), catheter-associated urinary tract infections (CAUTIs), surgical site infections (SSIs), MRSA Bacteremia, and C. difficile laboratory-identified events. The public reporting of these data is an effort by the Department of Health and Human Services.<ref>{{cite web | title = Healthcare-Associated Infections (HAI) data | publisher = Medicare.gov| url = https://www.medicare.gov/hospitalcompare/Data/Healthcare-Associated-Infections.html | accessdate = 2017-01-09 }}</ref>

卫生保健机构，根据联合委员会发布的准则，定期追踪他们负责跟踪的感染率<ref>{{cite web | title= Basic Statistical Data Used in Acute Care Facilities | publisher = JB Learning | url = http://www.jblearning.com/samples/0763750344/45561_CH01.pdf | accessdate = 2017-01-09 }}</ref>。医疗机构感染率（HAI），用于衡量特定医院中患者的感染情况。这样，可以将感染率与其他医院对比。当医疗机构遵循安全护理指南时，通常可以预防这些感染。想要从卫生管理部门获得拨款，医院必须向（美国）疾病预防控制中心（CDC）下属的国家医疗安全网络（NHSN）报告某些感染的数据。医院目前提交的信息包括：中央静脉导管相关性血流感染（CLABSI），导管相关性尿路感染（CAUTI），手术部位感染（SSI），耐甲氧西林金黄色葡萄球菌败血症，以及实验室难以鉴定的事件。这些数据的公开报告，是卫生与公众服务部努力实现的工作<ref>{{cite web | title = Healthcare-Associated Infections (HAI) data | publisher = Medicare.gov| url = https://www.medicare.gov/hospitalcompare/Data/Healthcare-Associated-Infections.html | accessdate = 2017-01-09 }}</ref>。

For meaningful comparisons of infection rates, populations must be very similar between the two or more assessments. However, a problem with mean rates is that they cannot reflect differences in risk between populations.<ref>{{cite web | title= NHSN Standardized Infection Ratio |url = https://www.cdc.gov/nhsn/pdfs/ps-analysis-resources/nhsn-sir-guide.pdf | accessdate = 2017-01-09 | date= December 2016 }}</ref>

为了有意义地比较感染率，两次或更多次评估之间的人群必须非常相似。然而，有一个问题是，平均感染率，无法反映不同人群之间的风险差异<ref>{{cite web | title= NHSN Standardized Infection Ratio |url = https://www.cdc.gov/nhsn/pdfs/ps-analysis-resources/nhsn-sir-guide.pdf | accessdate = 2017-01-09 | date= December 2016 }}</ref>。

== 参考文献 References ==

{{reflist}}

== 外部链接 External links ==

* [http://shea-online.org The Society for Healthcare Epidemiology of America] epidemiologists or physicians in infection control.
* [http://www.apic.org Association for Professionals in Infection Control and Epidemiology] infection prevention and control professionals.
* [http://www.cbic.org The Certification Board of Infection Control and Epidemiology, Inc.]

此页参考来源：维基百科：Infection rate

基本再生数

2022-06-14T12:35:57Z

水手9303：

此词条由Potatozh初译，由水手9303校核整理，若有错误，请见谅，并恳请告知。

[[基本再生数（basic reproduction number）]]是传染病研究中一个重要的参量，记为R0，下面给出它的定义。考虑疾病传播的最初阶段，此时只有少数的疾病案例，其他人都是易感个体，传染病学术语中，通常称为[[本地人口（native population）]]。假如，其中一个易感个体，在疾病爆发以后被感染，那么，基本再生数指的是，这个被传染的人，在他恢复之前（恢复健康，不再具有传染性），将疾病平均传染给了多少个人。例如，如果每个被感染者将疾病平均又传染给了其他两个人，则R0=2。如果初始感染者中，有一半的人，将疾病分别又传染给另一个人，而剩余的一半人没有传染给其他任何人，那么R0=1/2，依次类推。

如果R0=2，那么，每个得病的人，就会平均传染两个人，而这两个人中的每一个，又会再传染另外两个人。如此下去，每一轮，新的病例，都会加倍，从而出现指数增长规律。相反，如果R0=1/2，那么，疾病就会以指数形式衰减。R0=1，是增长和减少的分界点。因此，它也表示[[传播阀值（epidemic threshold）]]，偏离此值时，疾病要么增长，要么消亡。

以[[SIR模型]]为例【译注：将特定人群的人口结构，划分为了易感染者（S）、感染者（I）以及移出者（R），通过对三类人的动态变化的研究，来模拟流行病的发展规律】，其R0值，可以通过如下方式计算得到:<math> </math>

由于[[SIR模型]]中，任意时间间隔<math>\Delta t</math>内，个体恢复的概率为<math>\gamma\Delta t</math>，不能恢复的概率为<math>1-\gamma\Delta t</math>。因此，在总时间<math>t</math>后，个体仍然处于感染状态的概率为<math>
\begin{equation}
\lim_{\Delta t\to0}(1-\gamma\Delta t)^{t/\Delta t}=e^{-\gamma t},
\end{equation}</math>
同时，个体保持感染状态，然后在<math>t</math>到<math>t+dt</math>区间恢复的概率<math>p(t)dt</math>可以表示为上述概率与<math>\gamma dt</math>的乘积，即
<math>\begin{equation}
p(t)dt=\gamma e^{-\gamma t}dt,
\end{equation}</math>
通过上式我们得到，感染个体在恢复以前，处于感染态的时间长度t的分布。因为，[[基本再生数（basic reproduction number）]]R0指的就是，这个被传染的人，在他恢复之前，将疾病平均传染给了多少个人。所以，对于[[SIR模型]]的R0值，如果一个人在时间<math>t</math>内仍然具有传染性，则在这段时间内，与之接触的人的期望值为<math>\beta t</math>。R0的定义是针对[[本地人口]]给出的，而在[[本地人口]]中，他接触到的所有人都是易感者，他接触到的所有人，都有可能被传染。因此，<math>\beta t</math>也就是感染个体将传染的总人数。对<math>\beta t</math>中的t的分布取平均，就可以得到R0的平均值：<math>
\begin{equation}
R_0=\beta\gamma\int^\infty_0 t e^{-\gamma t}dt=\frac{\beta}{\gamma}.
\end{equation}</math>

此页参考来源：网络科学引论. M. E. J. Newman著郭世泽陈哲译电子工业出版社

基本再生数

2022-06-14T12:31:16Z

水手9303：

此词条由Potatozh初译，由水手9303校核整理，若有错误，请见谅，并恳请告知。

[[基本再生数（basic reproduction number）]]是传染病研究中一个重要的参量，记为R0，下面给出它的定义。考虑疾病传播的最初阶段，此时只有少数的疾病案例，其他人都是易感个体，传染病学术语中，通常称为[[本地人口（native population）]]。假如，其中一个易感个体，在疾病爆发以后被感染，那么，基本再生数指的是，这个被传染的人，在他恢复之前（恢复健康，不再具有传染性），将疾病平均传染给了多少个人。例如，如果每个被感染者将疾病平均又传染给了其他两个人，则R0=2。如果初始感染者中，有一半的人，将疾病分别又传染给另一个人，而剩余的一半人没有传染给其他任何人，那么R0=1/2，依次类推。

如果R0=2，那么，每个得病的人，就会平均传染两个人，而这两个人中的每一个，又会再传染另外两个人。如此下去，每一轮，新的病例，都会加倍，从而出现指数增长规律。相反，如果R0=1/2，那么，疾病就会以指数形式衰减。R0=1，是增长和减少的分界点。因此，它也表示[[传播阀值（epidemic threshold）]]，偏离此值时，疾病要么增长，要么消亡。

以[[SIR模型]]为例【译注：将特定人群的人口结构，划分为了易感染者（S）、感染者（I）以及移出者（R），通过对三类人的动态变化的研究，来模拟流行病的发展规律】，其R0值，可以通过如下方式计算得到:

由于[[SIR模型]]中，任意时间间隔<math>\Delta t</math>内，个体恢复的概率为<math>\gamma\Delta t</math>，不能恢复的概率为<math>1-\gamma\Delta t</math>。因此，在总时间t后，个体仍然处于感染状态的概率为<math>
\begin{equation}
\lim_{\Delta t\to0}(1-\gamma\Delta t)^{t/\Delta t}=e^{-\gamma t},
\end{equation}</math>
同时，个体保持感染状态，然后在t到<math>t+dt</math>区间恢复的概率<math>p(t)dt</math>可以表示为上述概率与<math>\gamma dt</math>的乘积，即
<math>\begin{equation}
p(t)dt=\gamma e^{-\gamma t}dt,
\end{equation}</math>
通过上式我们得到，感染个体在恢复以前，处于感染态的时间长度t的分布。因为，[[基本再生数（basic reproduction number）]]R0指的就是，这个被传染的人，在他恢复之前，将疾病平均传染给了多少个人。所以，对于[[SIR模型]]的R0值，如果一个人在时间t内仍然具有传染性，则在这段时间内，与之接触的人的期望值为<math>\beta t</math>。R0的定义是针对[[本地人口]]给出的，而在[[本地人口]]中，他接触到的所有人都是易感者，他接触到的所有人，都有可能被传染。因此，<math>\beta t</math>也就是感染个体将传染的总人数。对<math>\beta t</math>中的t的分布取平均，就可以得到R0的平均值：<math>
\begin{equation}
R_0=\beta\gamma\int^\infty_0 t e^{-\gamma t}dt=\frac{\beta}{\gamma}.
\end{equation}</math>

此页参考来源：网络科学引论. M. E. J. Newman著郭世泽陈哲译电子工业出版社

感染率

2022-06-14T12:22:08Z

水手9303：/* 临床应用 Clinical applications */

此词条暂由Potatozh初译，正在由水手9303校核整理中，带来阅读不便，请见谅。

An infection rate (or incident rate) is the probability or risk of an infection in a population. It is used to measure the frequency of occurrence of new instances of infection within a population during a specific time period.

感染率（或发病率）是一个人群的感染可能性或风险。它用于衡量特定时间段内某个人群中新感染病例的发生频率。

[[File:1.png|400px]]

The number of infections equals the cases identified in the study or observed. An example might by HIV infection during a specific time period in the defined population. The population at risk are the cases appearing in the population during the same time period. An example would be all the people in a city during a specific time period. The constant, or K is assigned a value of 100 to represent a percentage. An example would be to who are infected with HIV: 6,000 cases in March divided by the population of a city (one million) multiplied by the constant (K) would give an infection rate of 0.6%. Calculating the infection rate is used to analyze trends for the purpose of infection and disease control.<ref>{{cite web | title = Calculation of Infection Rates|url = http://health.utah.gov/epi/diseases/HAI/resources/Cal_Inf_Rates.pdf |publisher = Utah Department of Health|date= 2017| accessdate = 2017-01-09 }}</ref>

感染数量等于研究中确诊或观察到的病例。一个例子是，在特定时期内，在定义的人群中，可能的艾滋病感染者。高危人群，是同一时期内出现在某人群中的人数。例如，特定时间段内，某城市中的所有人。一个常数K，赋值为100，用来表示它是百分比。例如，一个城市中感染艾滋病毒的人口百分比：3月份，有6,000个例病例，除以该城市人口数目（一百万），再乘以常数（K，即100），得出感染率为0.6％。计算感染率，通常是用于分析趋势，以达到控制感染和疾病的目的<ref>{{cite web | title = Calculation of Infection Rates|url = http://health.utah.gov/epi/diseases/HAI/resources/Cal_Inf_Rates.pdf |publisher = Utah Department of Health|date= 2017| accessdate = 2017-01-09 }}</ref>。

An online infection rate calculator has been developed by the Centers for Disease Control and Prevention that allows the determination of the Streptococcal A infection rate in a population.<ref>{{cite web | title = Group A Streptococcus Calculator | publisher=Centers for Disease Control and Prevention |date= 4 July 2008 |url = https://www2a.cdc.gov/ncidod/dbmd/abcs/calc/calc_new/calculator.html | accessdate = 2017-01-09 }}</ref>

（美国）疾病预防控制中心开发了一个在线的感染率计算器，可以用来确定一个人群中的链球菌A的感染率<ref>{{cite web | title = Group A Streptococcus Calculator | publisher=Centers for Disease Control and Prevention |date= 4 July 2008 |url = https://www2a.cdc.gov/ncidod/dbmd/abcs/calc/calc_new/calculator.html | accessdate = 2017-01-09 }}</ref>。

== 临床应用 Clinical applications ==

Health care facilities routinely track their infection rates according to the guidelines issued by the Joint Commission.<ref>{{cite web | title= Basic Statistical Data Used in Acute Care Facilities | publisher = JB Learning | url = http://www.jblearning.com/samples/0763750344/45561_CH01.pdf | accessdate = 2017-01-09 }}</ref> The healthcare-associated infection (HAI) rates measure infection of patients in a particular hospital. This allows rates to compared with other hospitals. These infections can often be prevented when healthcare facilities follow guidelines for safe care. To get payment from Medicare, hospitals are required to report data about some infections to the Centers for Disease Control and Prevention's (CDC's) National Healthcare Safety Network (NHSN). Hospitals currently submit information on central line-associated bloodstream infections (CLABSIs), catheter-associated urinary tract infections (CAUTIs), surgical site infections (SSIs), MRSA Bacteremia, and C. difficile laboratory-identified events. The public reporting of these data is an effort by the Department of Health and Human Services.<ref>{{cite web | title = Healthcare-Associated Infections (HAI) data | publisher = Medicare.gov| url = https://www.medicare.gov/hospitalcompare/Data/Healthcare-Associated-Infections.html | accessdate = 2017-01-09 }}</ref>

卫生保健机构，根据联合委员会发布的准则，定期追踪他们负责跟踪的感染率<ref>{{cite web | title= Basic Statistical Data Used in Acute Care Facilities | publisher = JB Learning | url = http://www.jblearning.com/samples/0763750344/45561_CH01.pdf | accessdate = 2017-01-09 }}</ref>。医疗机构感染率（HAI），用于衡量特定医院中患者的感染情况。这样，可以将感染率与其他医院对比。当医疗机构遵循安全护理指南时，通常可以预防这些感染。想要从卫生管理部门获得拨款，医院必须向（美国）疾病预防控制中心（CDC）下属的国家医疗安全网络（NHSN）报告某些感染的数据。医院目前提交的信息包括：中央静脉导管相关性血流感染（CLABSI），导管相关性尿路感染（CAUTI），手术部位感染（SSI），耐甲氧西林金黄色葡萄球菌败血症，以及实验室难以鉴定的事件。这些数据的公开报告，是卫生与公众服务部努力实现的工作<ref>{{cite web | title = Healthcare-Associated Infections (HAI) data | publisher = Medicare.gov| url = https://www.medicare.gov/hospitalcompare/Data/Healthcare-Associated-Infections.html | accessdate = 2017-01-09 }}</ref>。

For meaningful comparisons of infection rates, populations must be very similar between the two or more assessments. However, a problem with mean rates is that they cannot reflect differences in risk between populations.<ref>{{cite web | title= NHSN Standardized Infection Ratio |url = https://www.cdc.gov/nhsn/pdfs/ps-analysis-resources/nhsn-sir-guide.pdf | accessdate = 2017-01-09 | date= December 2016 }}</ref>

为了有意义地比较感染率，两次或更多次评估之间的人群必须非常相似。然而，有一个问题是，平均感染率，无法反映不同人群之间的风险差异<ref>{{cite web | title= NHSN Standardized Infection Ratio |url = https://www.cdc.gov/nhsn/pdfs/ps-analysis-resources/nhsn-sir-guide.pdf | accessdate = 2017-01-09 | date= December 2016 }}</ref>。

== 参考文献 References ==

{{reflist}}

== 外部链接 External links ==

* [http://shea-online.org The Society for Healthcare Epidemiology of America] epidemiologists or physicians in infection control.
* [http://www.apic.org Association for Professionals in Infection Control and Epidemiology] infection prevention and control professionals.
* [http://www.cbic.org The Certification Board of Infection Control and Epidemiology, Inc.]

此页参考来源：维基百科：Infection rate

此词条Jie翻译。
由CecileLi初步审校，由潮升阶精校

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

{{short description|Pair of equations modelling predator-prey cycles in biology}}

[[文件:Milliers fourrures vendues en environ 90 ans odum 1953 en.jpg|缩略图|右|卖给哈德逊湾公司的雪鞋野兔毛皮（后方黄色区域）和加拿大山猫毛皮（前方黑色线条）。注：加拿大山猫会以雪兔为食。]]

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:

''' Lotka–Volterra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：

:<math>
\begin{align}
\frac{dx}{dt} &= \alpha x - \beta x y, \\
\frac{dy}{dt} &= \delta x y - \gamma y,
\end{align}
</math>

Where，
:{{mvar|x}} is the number of prey (for example, [[rabbit]]s);
:{{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;
:{{mvar|t}} represents time;
:{{math|''α'', ''β'', ''γ'', ''δ''}} are positive real [[parameter]]s describing the interaction of the two [[species]].

其中：
:{{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；
:<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；
:{{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。

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。

== 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 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>

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 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>

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''' Rosenzweig–MacArthur模型<ref name=":10" />''' 。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 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>

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''' Arditi–Ginzburg模型<ref name=":13" />'''。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 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>

Lotka-Volterra方程在理论经济学中有很长的应用历史，最早由理查德·古德温（Richard Goodwin）应用于1965<ref name=":15" /> 与1967年<ref name=":16" /><ref name=":17" />。

== 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 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>

Lotka–Volterra模型对捕猎双方的环境和种群数量演化做出了许多假设，这些假设显得过于理想化，在自然界中显得并不实际<ref name=":18" />：

* The prey population finds ample food at all times.
* The food supply of the predator population depends entirely on the size of the prey population.
* The rate of change of population is proportional to its size.
* During the process, the environment does not change in favour of one species, and genetic adaptation is inconsequential.
* Predators have limitless appetite.

* 猎物随时都有充足的食物。
* 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。
* 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。
* 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 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>

在这种情况下，微分方程的解是确定并连续的。反过来，这也意味着捕食者和猎物的世代是持续重叠的<ref name=":19" />。

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

=== 猎物 ===

When multiplied out, the prey equation becomes

当猎物数量成倍增趋势时，猎物方程可表示为:

:<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 {{mvar|x}} or {{mvar|y}} 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.

基于这两个假设，上述方程式可以理解为：猎物种群的变化率等于其自身数量的增长率减去被捕食数量的增长率。

=== 捕食者===

The predator equation becomes

捕食者方程可表示为:

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

In this equation, {{math|''δxy''}} 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 {{math|''γy''}} 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.

因此，上述方程式可以理解为，捕食者种群的变化率取决于其捕杀猎物的速率减去其内在死亡（包括迁徙）率。

== 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 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" /><ref name=":21" />。

If none of the non-negative parameters {{math|''α'', ''β'', ''γ'', ''δ''}} vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in {{math|''x''}}, and the second one in {{math|''y''}}, the parameters ''β''/''α'' and ''δ''/''γ'' are absorbable in the normalizations of {{math|''y''}} and {{math|''x''}} respectively, and {{math|''γ''}} into the normalization of {{math|''t''}}, so that only {{math|''α''/''γ''}} 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 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.

方程的线性化后求得的解类似于简谐运动曲线<ref name=":22" /> ，捕食者的数量比猎物的数量落后大概四分之一个周期。

[[文件:Lotka Volterra dynamics.svg|缩略图|居中|Linearized version]]

{{further|Limit cycle}}

{{anchor|Atto-fox}}

=== 简单示例 ===

[[文件:WeChat Screenshot 20210111180355.png|缩略图|右|前文提到关于狒狒和猎豹问题的种群动态。]]

[[文件:WeChat Screenshot 20210111180609.png|缩略图|右|当捕食者种群在不同初始条件下，其捕猎问题的相空间图。]]

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，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 [[orbit (dynamics)|orbit]]s 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.

或者也可以在相空间轨道中将其解进行参数化处理，此时就可以略去时间轴。仅用其中一个轴代表全时间段猎物的数量，而另一轴代表全时间段捕食者的数量。

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

对应于上面的两个微分方程，此方法可以得出约掉时间参数的一个全新的微分方程

:<math>\frac{dy}{dx} = - \frac{y}{x} \frac{\delta x - \gamma}{\beta y -\alpha}</math>

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

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

:<math>\frac{\beta y - \alpha}{y} \,dy + \frac{\delta x - \gamma}{x} \,dx = 0</math>

yields the implicit relationship

得到隐性关系

: <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y),</math>

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 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>

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" />。

=== 相空间图的进一步示例 ===

[[文件:WeChat Screenshot 20210112125549.png|缩略图|左|捕猎相空间图]]

A less extreme example covers:
{{mvar|α}} = 2/3, {{mvar|β}} = 4/3, {{mvar|γ}} = 1 = {{mvar|δ}}. Assume {{math|''x'', ''y''}} quantify thousands each. Circles represent prey and predator initial conditions from {{mvar|x}} = {{mvar|y}} = 0.9 to 1.8, in steps of 0.1. The fixed point is at (1, 1/2).

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，不动点为（1，1/2）。

== 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.

在该模型系统中，捕食者在有大量猎物的情况下肆意增长，但最终它们会因为食物供应不足而下降。随即捕食者数量变低，猎物数量将再次增加。这样形成的动力学系统以增长和下降为周期持续下去。

=== 种群平衡===

Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0:

当捕猎双方种群数量都没有变化时，即当式中两个导数都等于0时，模型就会出现种群平衡：

:<math>x(\alpha - \beta y) = 0,</math>

:<math>-y(\gamma - \delta x) = 0.</math>

The above system of equations yields two solutions:

对上面的方程组进行求解，得到：

:<math>\{y = 0,\ \ x = 0\}</math>

and

及

:<math>\left\{y = \frac{\alpha}{\beta},\ \ x = \frac{\gamma}{\delta} \right\}.</math>

Hence, there are two equilibria.

因此我们得到了两个平衡点（对应于上式的两个解）。

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。

=== 不动点的稳定性 ===

The stability of the fixed point at the origin can be determined by performing a [[linearization]] using [[partial derivative]]s.

不动点在原点处的稳定性可以通过求偏导数将其线性化来确定。

The [[Jacobian matrix]] of the predator–prey model is

可以得出捕猎模型的雅可比矩阵为：

:<math>J(x, y) = \begin{bmatrix}
\alpha - \beta y & -\beta x \\
\delta y & \delta x - \gamma
\end{bmatrix}.</math>

and is known as the [[community matrix]].

该矩阵也可以称为群落矩阵。

====第一不动点（灭绝）====

When evaluated at the steady state of (0, 0), the Jacobian matrix ''J'' becomes

当在(0, 0)的稳态下求值时，雅可比矩阵''J''为

:<math>J(0, 0) = \begin{bmatrix}
\alpha & 0 \\
0 & -\gamma
\end{bmatrix}.</math>

The [[eigenvalue]]s of this matrix are

该矩阵的特征值是:

:<math>\lambda_1 = \alpha, \quad \lambda_2 = -\gamma.</math>

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.

第一不动点的稳定性非常重要。只有当它是稳定的，非零物种群体才有可能趋向于灭绝。并且在此条件下，即使种群处于初期状态，根据该动力学系统仍然可能得出两种物种都灭绝的结果。但是，由于第一不动点位于原点处是一个鞍点，因此很不稳定，进而说明模型中的两个物种都灭绝并没有那么容易。（实际上，只有在人为地彻底消灭猎物，进而导致捕食者饿死之后，才会发生这种情况。相反地，如果消灭了捕食者，那么在这个简单的模型中，猎物将不受限制地增长。）捕食者和猎物的种群数量可以无限地接近零，且仍然可以恢复。

====第二不动点（震荡）====

Evaluating ''J'' at the second fixed point leads to

在第二个不动点求雅可比矩阵''J''可得：

:<math>J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix}
0 & -\frac{\beta \gamma}{\delta} \\
\frac{\alpha \delta}{\beta} & 0
\end{bmatrix}.</math>

The eigenvalues of this matrix are

该矩阵的特征值是

:<math>\lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}.
</math>

As the eigenvalues are both purely imaginary and conjugate to each others, this fixed point is [[Dynamical system#Conjugation results|elliptic]], so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency <math>\omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma}</math> and period <math>T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2})</math>.

由于特征值既是纯虚数又是共轭的，因此该不动点是椭圆的并且其解具有周期性，即在不动点周围的椭圆环上以一定频率振荡：<math>\omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma}</math> ，其周期为：<math>T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2})</math>。

As illustrated in the circulating oscillations in the figure above, the level curves are closed [[orbit (dynamics)|orbit]]s 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>循环并且无阻尼震荡。

The value of the [[constant of motion]] ''V'', or, equivalently, ''K'' = exp(''V''), <math>K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x}</math>, can be found for the closed orbits near the fixed point.

对于不动点附近的闭合轨道，不难找到其运动常数''V''的值，同样包括，''K'' = exp(''V''), <math>K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x}</math>。

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。

:<math>y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max\limits_{x,y>0}.</math>

The maximal value of ''K'' is thus attained at the stationary (fixed) point <math>\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)</math> and amounts to

因此，在固定点（即不动点）<math>\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)</math>处可以获得''K''的最大值，该值等于<math>K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma,</math>

where ''e'' is [[e (mathematical constant)|Euler's number]].

其中''e''是欧拉数

== 另请参见 ==

*[[Competitive Lotka–Volterra equations]]

*[[Generalized Lotka–Volterra equation]]

*[[Mutualism and the Lotka–Volterra equation]]

*[[Community matrix]]

*[[Population dynamics]]

*[[Population dynamics of fisheries]]

*[[Nicholson–Bailey model]]

*[[Reaction–diffusion system]]

*[[Paradox of enrichment]]

* [[Lanchester's laws]], a similar system of differential equations for military forces

* ''' 竞争性Lotka–Volterra方程Competitive Lotka–Volterra equations'''
* ''' 广义Lotka–Volterra方程Generalized Lotka–Volterra equation'''
* ''' 互惠主义与Lotka–Volterra方程Mutualism and the Lotka–Volterra equation'''
* ''' 群落矩阵Community matrix'''
* ''' 种群动态Population dynamics'''
* ''' 渔业种群动态Population dynamics of fisheries'''
* ''' Nicholson–Bailey模型Nicholson–Bailey model'''
* ''' 反应扩散系统Reaction–diffusion system'''
* ''' 富集悖论Paradox of enrichment'''
* ''' 兰切斯特定律，有关于军事战术的微分方程组Lanchester's laws, a similar system of differential equations for military forces'''

== 备注==

{{Reflist|30em}}

== 参考文献==

*{{cite book|first=E. R.|last=Leigh|year=1968|chapter=The ecological role of Volterra's equations|title=Some Mathematical Problems in Biology}} – a modern discussion using [[Hudson's Bay Company]] data on [[lynx]] and [[hare]]s in [[Canada]] from 1847 to 1903.

*{{cite book|title=Understanding Nonlinear Dynamics|url=https://archive.org/details/understandingnon0000kapl|url-access=registration|first1=Daniel|last1=Kaplan|first2=Leon|last2=Glass|authorlink2=Leon Glass|location=New York|publisher=Springer|year=1995|isbn=978-0-387-94440-1}}

*{{cite book|first=J. D.|last=Murray|title=Mathematical Biology I: An Introduction|location=New York|publisher=Springer|year=2003|isbn=978-0-387-95223-9}}

*{{cite journal|first1=James A.|last1=Yorke|first2=William N. Jr.|last2=Anderson|title=Predator-Prey Patterns (Volterra-Lotka equations)|journal=[[PNAS]]|volume=70|issue=7|pages=2069–2071|year=1973|jstor=62597|doi=10.1073/pnas.70.7.2069|pmid=16592100|pmc=433667|url=http://www.pnas.org/content/70/7/2069.full.pdf}}

*{{cite journal|first1=J.|last1=Llibre|first2=C.|last2=Valls|author2-link= Clàudia Valls |title=Global analytic first integrals for the real planar Lotka-Volterra system|journal=J. Math. Phys.|volume=48|issue=3|pages=033507|year=2007|doi=10.1063/1.2713076|bibcode=2007JMP....48c3507L}}

== 相关链接 ==

{{Commons category}}

*From the ''[[Wolfram Demonstrations Project]]'' — requires [http://demonstrations.wolfram.com/download-cdf-player.html CDF player (free)]:

**[http://demonstrations.wolfram.com/PredatorPreyEquations Predator–Prey Equations]

**[http://demonstrations.wolfram.com/PredatorPreyModel Predator–Prey Model]

**[http://demonstrations.wolfram.com/PredatorPreyDynamicsWithTypeTwoFunctionalResponse Predator–Prey Dynamics with Type-Two Functional Response]

**[http://demonstrations.wolfram.com/PredatorPreyEcosystemARealTimeAgentBasedSimulation Predator–Prey Ecosystem: A Real-Time Agent-Based Simulation]

* [https://espadrine.github.io/lotka-volterra/ Lotka-Volterra Algorithmic Simulation] (Web simulation).

* 来自Wolfram 演示项目— 需要CDF 播放器 (免费):
** 猎捕方程
** 猎捕模型
** 具有第二类功能响应的捕猎动力系统
** 捕猎生态系统：基于实时主体的仿真
* Lotka-Volterra算法仿真（网络仿真）

{{modelling ecosystems|expanded=other}}

{{DEFAULTSORT:Lotka-Volterra Equation}}

[[Category:Predation]]

Category:Predation

类别: 掠夺

[[Category:Ordinary differential equations]]

Category:Ordinary differential equations

类别: 常微分方程

[[Category:Fixed points (mathematics)]]

Category:Fixed points (mathematics)

类别: 定点(数学)

[[Category:Population models]]

Category:Population models

类别: 人口模型

[[Category:Mathematical modeling]]

Category:Mathematical modeling

类别: 数学建模

[[Category:Community ecology]]

Category:Community ecology

类别: 群落交错区

<noinclude>

This page was moved from [[wikipedia:en:Lotka–Volterra equations]]. Its edit history can be viewed at [[Lotka–Volterra方程式/edithistory]]</noinclude>

[[Category:待整理页面]]