# 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:

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（又称为猎食方程）是一对一阶非线性微分方程，它经常被用来描述两个物种间因相互作用，而产生的生物系统动力学反应。其中一个作为捕食者，而另一个作为猎物。其人口数量会随时间变化遵循如下一对方程组：

\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.

where x is the number of prey (for example, rabbits); y is the number of some predator (for example, foxes); {\displaystyle {\tfrac {dy}{dt}}}{\tfrac {dy}{dt}} and {\displaystyle {\tfrac {dx}{dt}}}{\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是某些捕食者（例如狐狸）的数量；
$\displaystyle{ \tfrac{dy}{dt} }$$\displaystyle{ \tfrac{dx}{dt} }$代表两个人口的瞬时增长率；
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.

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方程式系统是Kolmogorov模型的一个示例，该模型是一个更通用的框架，可以利用捕食者与猎物之间的猎食，竞争，疾病和共生关系来模拟生态系统的动力学。

## History 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.

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.

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.

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 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:

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进化做出了许多假设，这些假设在自然界过于理想而不一定能实现：

• 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.
• 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，并且无关于遗传适应性。
• 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.

### Prey 猎物

When multiplied out, the prey equation becomes

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.

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.

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 捕食者

The predator equation becomes

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.

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 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.

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.

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.

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.

### A simple example 简单举例

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.

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.

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

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

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

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

[数学，数学]

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

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

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

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

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

yields the implicit relationship

yields the implicit relationship

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


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.

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.

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.

### Phase-space plot of a further example

A less extreme example covers:

300x300pxA less extreme example covers:

300x300pxA 不那么极端的例子包括:

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).

= 2/3,  = 4/3,  = 1 = . Assume  quantify thousands each. Circles represent prey and predator initial conditions from  =  = 0.9 to 1.8, in steps of 0.1. The fixed point is at (1, 1/2).

= 2/3,  = 4/3,  = 1 = .假设每个数量都是几千。圆圈表示猎物和捕食者的初始条件，从 = = 0.9到1.8，步长为0.1。定点位于(1,1/2)。


## Dynamics of the system

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.

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.

### Population equilibrium

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:

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:

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

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

X (alpha-beta y) = 0

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

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

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

The above system of equations yields two solutions:

The above system of equations yields two solutions:

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

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

{ y = 0，x = 0}

and

and

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

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

Hence, there are two equilibria.

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.

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.

### Stability of the fixed points

The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives.

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

The Jacobian matrix of the predator–prey model is

$\displaystyle{ J(x, y) = \begin{bmatrix} \lt math\gt J(x, y) = \begin{bmatrix} \lt math \gt j (x，y) = begin { bmatrix } \alpha - \beta y & -\beta x \\ \alpha - \beta y & -\beta x \\ Alpha-beta y &-beta x \delta y & \delta x - \gamma \delta y & \delta x - \gamma Delta y & delta x-gamma \end{bmatrix}. }$

\end{bmatrix}.[/itex]

and is known as the community matrix.

and is known as the community matrix.

#### First fixed point (extinction)

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

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

$\displaystyle{ J(0, 0) = \begin{bmatrix} \lt math\gt J(0, 0) = \begin{bmatrix} (0,0) = begin { bmatrix } \alpha & 0 \\ \alpha & 0 \\ 阿尔法 & 0 0 & -\gamma 0 & -\gamma 0 &-gamma \end{bmatrix}. }$

\end{bmatrix}.[/itex]

The eigenvalues of this matrix are

The eigenvalues of this matrix are

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

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

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.

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.

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.

#### Second fixed point (oscillations)

Evaluating J at the second fixed point leads to

Evaluating J at the second fixed point leads to

$\displaystyle{ J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix} \lt math\gt J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix} J left (frac { gamma }{ delta } ，frac { alpha }{ beta }右) = begin { bmatrix } 0 & -\frac{\beta \gamma}{\delta} \\ 0 & -\frac{\beta \gamma}{\delta} \\ 0 &-frac { beta gamma }{ delta } \frac{\alpha \delta}{\beta} & 0 \frac{\alpha \delta}{\beta} & 0 0.0.0 \end{bmatrix}. }$

\end{bmatrix}.[/itex]

The eigenvalues of this matrix are

The eigenvalues of this matrix are

$\displaystyle{ \lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}. \lt math\gt \lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}. 1 = i sqrt { alpha gamma } ，quad lambda 2 =-i sqrt { alpha gamma }. }$

[/itex]

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

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.

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

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

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

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

< math > y ^ alpha ^ e ^ {-beta y } x ^ gamma ^ {-delta x } = frac { y ^ alpha ^ gamma }{ e ^ { delta x + beta y }}{ longright tarrow max limits { x，y > 0}。 </math >

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

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

$\displaystyle{ K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma, }$

$\displaystyle{ K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma, }$

< math > k ^ * = left (frac { alpha }{ beta e } right) ^ alpha left (frac { gamma }{ delta e } right) ^ gamma，</math >

where e is Euler's number.

where e is Euler's number.