# Lotka–Volterra方程式 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:

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：

\displaystyle{ \begin{align} \frac{dx}{dt} &= \alpha x - \beta x y, \\ \frac{dy}{dt} &= \delta x y - \gamma y, \end{align} }

Where，

x is the number of prey (for example, rabbits);
y is the number of some predator (for example, foxes);
$\displaystyle{ \tfrac{dy}{dt} }$ and $\displaystyle{ \tfrac{dx}{dt} }$ represent the instantaneous growth rates of the two populations;
t represents time;
α, β, γ, δ are positive real parameters describing the interaction of the two species.

x是猎物（例如兔子）的数量；
y是捕食者（例如狐狸）的数量；
$\displaystyle{ \tfrac{dy}{dt} }$$\displaystyle{ \tfrac{dx}{dt} }$代表个体的瞬时增长率；
t代表时间；
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.

Lotka–Volterra方程组是Kolmogorov模型的一个示例，而Kolmogorov模型 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 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 。这个模型实际上是一类逻辑方程 ，源自于皮埃尔·弗朗索瓦·韦吕勒（Pierre François Verhulst） 。1920年，洛特卡以植食和草食性动物为例，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 。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 ，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

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.

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.

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 与1967年

## 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:

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

• 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 and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.

### 猎物

When multiplied out, the prey equation becomes

$\displaystyle{ \frac{dx}{dt} = \alpha x - \beta x y. }$

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 x or y is zero, then there can be no predation.

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.

### 捕食者

The predator equation becomes

$\displaystyle{ \frac{dy}{dt} = \delta xy - \gamma y. }$

In this equation, 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 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.

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.

## 方程求解

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 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 x, and the second one in y, the parameters β/α and δ/γ are absorbable in the normalizations of y and x respectively, and γ into the normalization of t, so that only α/γ remains arbitrary. It is the only parameter affecting the nature of the solutions.

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.

### 简单示例

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.

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

$\displaystyle{ \frac{dy}{dx} = - \frac{y}{x} \frac{\delta x - \gamma}{\beta y -\alpha} }$

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

$\displaystyle{ \frac{\beta y - \alpha}{y} \,dy + \frac{\delta x - \gamma}{x} \,dx = 0 }$

yields the implicit relationship

$\displaystyle{ V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y), }$

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

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-fox being a notional 10−18 of a fox.

### 相空间图的进一步示例

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

### 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:

$\displaystyle{ x(\alpha - \beta y) = 0, }$
$\displaystyle{ -y(\gamma - \delta x) = 0. }$

The above system of equations yields two solutions:

$\displaystyle{ \{y = 0,\ \ x = 0\} }$

and

$\displaystyle{ \left\{y = \frac{\alpha}{\beta},\ \ x = \frac{\gamma}{\delta} \right\}. }$

Hence, there are two equilibria.

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.

### 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 derivatives.

The Jacobian matrix of the predator–prey model is

$\displaystyle{ J(x, y) = \begin{bmatrix} \alpha - \beta y & -\beta x \\ \delta y & \delta x - \gamma \end{bmatrix}. }$

and is known as the community matrix.

#### 第一不动点（灭绝）

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

$\displaystyle{ J(0, 0) = \begin{bmatrix} \alpha & 0 \\ 0 & -\gamma \end{bmatrix}. }$

The eigenvalues of this matrix are

$\displaystyle{ \lambda_1 = \alpha, \quad \lambda_2 = -\gamma. }$

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.

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.

#### 第二不动点（震荡）

Evaluating J at the second fixed point leads to

$\displaystyle{ J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix} 0 & -\frac{\beta \gamma}{\delta} \\ \frac{\alpha \delta}{\beta} & 0 \end{bmatrix}. }$

The eigenvalues of this matrix are

$\displaystyle{ \lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}. }$

As the eigenvalues are both purely imaginary and conjugate to each others, this fixed point is elliptic, so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency $\displaystyle{ \omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma} }$ and period $\displaystyle{ T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2}) }$.

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 $\displaystyle{ \omega = \sqrt{\alpha \gamma} }$.

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

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。

$\displaystyle{ y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max\limits_{x,y\gt 0}. }$

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

where e is Euler's number.

## 另请参见

• 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
• 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

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## 参考文献

• Leigh, E. R. (1968). "The ecological role of Volterra's equations". Some Mathematical Problems in Biology.  – a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903.
• Murray, J. D. (2003). Mathematical Biology I: An Introduction. New York: Springer. ISBN 978-0-387-95223-9.

## 相关链接

• 来自Wolfram 演示项目— 需要CDF 播放器 (免费):
• 猎捕方程
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• 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）

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