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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，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着时间的推移而发展的图表; 假定参数是狒狒的生长率和死亡率分别为1.1和0.4，而猎豹的生长率和死亡率分别为0.1和0.4。时间间隔的选择是任意的。

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。

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

 

:<math>\frac{dy}{dx} = - \frac{y}{x} \frac{\delta x - \gamma}{\beta y -\alpha}</math>

:<math>\frac{dy}{dx} = - \frac{y}{x} \frac{\delta x - \gamma}{\beta y -\alpha}</math>

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

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

关联变量 x 和 y。这个方程的解是封闭曲线。它可以通过分离变量法: 集成

仅包含关联变量''x'' 和 ''y''。该方程的解是个闭合曲线，可以分离变量：对以下式子进行积分

 

:<math>\frac{\beta y - \alpha}{y} \,dy + \frac{\delta x - \gamma}{x} \,dx = 0</math>  

:<math>\frac{\beta y - \alpha}{y} \,dy + \frac{\delta x - \gamma}{x} \,dx = 0</math>  

yields the implicit relationship  

yields the implicit relationship  

: <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y),</math>

: <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y),</math>

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.

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.

其中 v 是一个常量，取决于初始条件，在每条曲线上都是守恒的。

其中''V''是取决于初始条件的定量，并且在每条曲线上均守恒。

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.

旁白: 这些图表说明了作为生物模型的一个严重的潜在问题: 对于这个特定的参数选择，在每个周期中，狒狒的数量减少到极低的数量，但是恢复了(而猎豹的数量在狒狒密度最低时仍然相当可观)。然而，在现实生活中，个体数量的随机波动，以及狒狒的家庭结构和生命周期，可能会导致狒狒实际上灭绝，结果，猎豹也会灭绝。这个建模问题被称为“ atto-fox 问题” ，也就是狐狸的一个概念。

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。

===Phase-space plot of a further example===

===Phase-space plot of a further example===