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→Population equilibrium
===Population equilibrium===
=== Population equilibrium 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:
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:
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>x(\alpha - \beta y) = 0,</math>
:<math>-y(\gamma - \delta x) = 0.</math>
:<math>-y(\gamma - \delta x) = 0.</math>
The above system of equations yields two solutions:
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>
:<math>\{y = 0,\ \ x = 0\}</math>
and
and
及
及
:<math>\left\{y = \frac{\alpha}{\beta},\ \ x = \frac{\gamma}{\delta} \right\}.</math>
:<math>\left\{y = \frac{\alpha}{\beta},\ \ x = \frac{\gamma}{\delta} \right\}.</math>
Hence, there are two equilibria.
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.
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===