4 Population Change | 人口变化

📚 4 Population Change | 人口变化

Population change is a core application of differential equations in A-Level Mathematics. It shows how the size of a population grows or declines over time and how we can model that behaviour using rates of change.

人口变化是 A-Level 数学中微分方程的核心应用之一。它展示了人口规模如何随时间增长或减少，以及我们如何用变化率来描述这种变化。

1. What is a population model? | 什么是人口模型？

A population model describes how the number of individuals P in a population changes over time t. In A-Level Mathematics, we usually model population change with differential equations because the rate of change dP/dt often depends on the current population size.

人口模型描述一个种群中个体数量 P 如何随时间 t 变化。在 A-Level 数学中，我们通常用微分方程来建立人口变化模型，因为变化率 dP/dt 往往取决于当前的人口规模。

Continuous population models assume that P can be treated as a differentiable function of t, even though real populations are discrete. This approximation works well when the population is large.

连续人口模型假设 P 可以看作 t 的可微函数，尽管真实种群是离散的。当种群规模很大时，这种近似效果很好。

2. The exponential growth equation | 指数增长方程

If each individual contributes to growth at a constant rate, the population satisfies the differential equation dP/dt = kP, where k is the net growth rate per unit time.

如果每个个体以恒定速率贡献增长，人口满足微分方程 dP/dt = kP，其中 k 是单位时间净增长率。

This is a first-order separable differential equation. The term kP means the growth rate is proportional to the population size, so larger populations grow faster in absolute terms.

这是一个一阶可分离变量微分方程。kP 这一项表示增长速率与人口规模成正比，因此人口越多，绝对增长越快。

For example, if k = 0.02 per year, a population of 10,000 grows at an instantaneous rate of 200 individuals per year.

例如，如果 k = 0.02 / 年，则 10,000 人口的瞬时增长速率为每年 200 人。

3. Solving dP/dt = kP | 求解 dP/dt = kP

Separating variables gives (1/P) dP = k dt. Integrating both sides yields ln|P| = kt + C. Writing P(0) = P₀ gives the solution P(t) = P₀eᵏᵗ.

分离变量得到 (1/P) dP = k dt。两边积分得到 ln|P| = kt + C。代入初始条件 P(0) = P₀，得到解 P(t) = P₀eᵏᵗ。

If k > 0, the population grows without bound. If k < 0, the population decays towards zero. The sign of k determines whether the model represents growth or decline.

如果 k > 0，人口无限增长；如果 k

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