更改

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  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.

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.

如果该方程组中所有非负参数{{math|''α'', ''β'', ''γ'', ''δ''}}均存在，则可以将其中三个变量进行归一化，仅留下一个参数：由于第一个方程在{{math|''x''}}中是齐次的，而第二个方程在{{math|''y''}}中是齐次的，因此参数''β''/''α'' 和 ''δ''/''γ''分别在{{math|''y''}}和{{math|''x''}}中可以进行归一化处理，{{math|''γ''}}变成{{math|''t''}}的归一化，因此只有{{math|''α''/''γ''}}保持任意。并且它是影响解决方案性质的唯一参数。

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：由于第一个方程在{{math|''x''}}中是齐次的，而第二个方程在{{math|''y''}}中也是齐次的，因此我们可以对分别在{{math|''y''}}和{{math|''x''}}中的参数''β''/''α'' 和 ''δ''/''γ''进行归一化处理，{{math|''γ''}}变成{{math|''t''}}的归一化，因此只有{{math|''α''/''γ''}}保持任意。它是影响解决方案性质的唯一参数。

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.

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.

假设有两种动物，即狒狒（猎物）和猎豹（捕食者）。如果初始条件是10只狒狒和10只猎豹，则可以绘制出这两个物种随时间推移的数量。假设给定参数，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，且时间间隔任意。

假设有两种动物，即狒狒（猎物）和猎豹（捕食者）。如果初始条件是10只狒狒和10只猎豹，那么我们可以绘制出这两个物种随时间推移的数量。假设给定参数，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，且间隔时间任意。

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.

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.

另外值得注意的是，这些图说明了作为生物学模型的严重潜在问题：因为这种特定的参数选择，在每个周期中，狒狒的数量都被减少到极低的数量，但又有能力恢复（事实上，在极低的狒狒密度下，猎豹的数量仍然很大）。这显然在现实中是不太可能的，离散个体的偶然性波动，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fox的十之负十八次方。

另外值得注意的是，这些图说明了该方程式作为生物学模型的严重潜在问题：因为这种特定的参数选择，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（事实上，在极低的狒狒密度下，猎豹的数量仍然很大）。这在现实中显然是不太可能的，离散个体的偶然性波动，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fox的十之负十八次方。

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).

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).

{{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，且固定点在（1，1/2）。

{{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，且固定点在（1，1/2）。