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→Stability of the fixed points
第一种解明显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绝种状态。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。达到此平衡的总体水平取决于参数''α'', ''β'', ''γ'', 和 ''δ''的选定值。
第一种解明显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绝种状态。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。达到此平衡的总体水平取决于参数''α'', ''β'', ''γ'', 和 ''δ''的选定值。
===Stability of the fixed points===
=== Stability of the fixed points 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.
The stability of the fixed point at the origin can be determined by performing a [[linearization]] using [[partial derivative]]s.
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.
不动点在原点处的稳定性可以通过使用偏导数将其线性化来确定。
The Jacobian matrix of the predator–prey model is
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}
:<math>J(x, y) = \begin{bmatrix}
\alpha - \beta y & -\beta x \\
\alpha - \beta y & -\beta x \\
\delta y & \delta x - \gamma
\delta y & \delta x - \gamma
\end{bmatrix}.</math>
\end{bmatrix}.</math>
and is known as the [[community matrix]].
and is known as the [[community matrix]].
and is known as the community matrix.
and is known as the community matrix.
该矩阵也可以称为群落矩阵。
====First fixed point (extinction) 第一不动点（灭绝）====
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
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
当在(0, 0)的稳态下求值时，雅可比矩阵''J''变为
:<math>J(0, 0) = \begin{bmatrix}
:<math>J(0, 0) = \begin{bmatrix}
\alpha & 0 \\
\alpha & 0 \\
0 & -\gamma
0 & -\gamma
\end{bmatrix}.</math>
\end{bmatrix}.</math>
The eigenvalues of this matrix are
The eigenvalues of this matrix are
该矩阵的特征值是:
:<math>\lambda_1 = \alpha, \quad \lambda_2 = -\gamma.</math>
:<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.
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。继而得到原点的固定点是鞍点。
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.
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（非灭绝）物种群体才有可能趋向于消失。因此，即使种群处于初期状态下，其动力系统仍然可能导致两种物种都灭绝。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种物种都灭绝并不容易。（实际上，只有在人为地彻底消灭猎物，导致捕食者饿死之后，才会发生这种情况。如果相反消灭了捕食者，那么在这个简单的模型中，猎物将不受限制地增长。）捕猎模型（捕食者-猎物）中的种群可以无限地接近零，且仍然可以恢复。
====Second fixed point (oscillations) 第二不动点（震荡）====
Evaluating ''J'' at the second fixed point leads to
Evaluating ''J'' at the second fixed point leads to
Evaluating J at the second fixed point leads to
Evaluating J at the second fixed point leads to
在第二个不动点求''J''可得：
:<math>J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix}
:<math>J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix}
0 & -\frac{\beta \gamma}{\delta} \\
0 & -\frac{\beta \gamma}{\delta} \\
\frac{\alpha \delta}{\beta} & 0
\frac{\alpha \delta}{\beta} & 0
\end{bmatrix}.</math>
\end{bmatrix}.</math>
The eigenvalues of this matrix are
The eigenvalues of this matrix are
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>\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 <math>\omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma}</math> and period <math>T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2})</math>.
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 <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 orbits 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>.
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 <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>循环并振荡且无阻尼。
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.
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
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固定点。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>
:<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
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
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
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>
:<math>K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma,</math>
:<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]].
where ''e'' is [[e (mathematical constant)|Euler's number]].
where e is Euler's number.
where e is Euler's number.
其中 e 是欧拉数。
其中''e''是欧拉数
==See also==
==See also==