📚 Mathematical Models of Urbanisation Patterns since 1945 | 1945年以来城市化模式的数学模型
Since 1945, the proportion of the world’s population living in urban areas has risen from about 30% to more than 55%. For A-Level mathematicians, this global shift is not only a social process but also a rich data set that can be described, compared and predicted using functions, logarithms, differential equations and statistical measures.
自1945年以来，世界城市人口比例已从约30%升至55%以上。对A-Level数学而言，这一全球变化不仅是社会过程，也是丰富的数据集，可以用函数、对数、微分方程和统计量加以描述、比较和预测。
1. Urbanisation as a Mathematical Trend | 作为数学趋势的城市化
Urbanisation is usually measured by the percentage of a population living in urban areas, or by the absolute urban population. A time series of percentage urban can be treated as a function U(t), where t is years after 1945.
城市化通常用城市人口占总人口的百分比或城市绝对人口来衡量。城市人口百分比的时间序列可视为函数 U(t)，其中 t 为 1945 年后的年数。
The first task in modelling is to identify whether U(t) is increasing, decreasing, concave up or concave down. Since 1945, most countries show an increasing U(t), but the rate of increase is often fastest during industrialisation and then slows as saturation approaches.
建模的首要任务是判断 U(t) 是递增、递减、上凹还是下凹。1945 年以来，大多数国家的 U(t) 递增，但增长率通常在工业化期间最快，随后随着接近饱和而放缓。
Mathematically, this suggests an S-shaped or logistic pattern rather than a straight line.
从数学上看，这提示的是 S 形或逻辑斯蒂模式，而不是直线。
2. Variables and Data Types | 变量与数据类型
Key variables include t (years since 1945), U(t) (urbanisation rate as a percentage), P(t) (urban population), and possibly GDP per capita, birth rate or migration rate.
关键变量包括 t（自1945年起的年数）、U(t)（城市化率，百分比）、P(t)（城市人口），以及可能的人均 GDP、出生率或迁移率。
When given a table of values, always check whether the data are evenly spaced in time. If so, first differences and second differences can help decide whether a linear, quadratic or exponential model is appropriate.
给出数据表时，务必检查时间间隔是否均匀。若均匀，可用一阶差分和二阶差分帮助判断线性、二次或指数模型是否合适。
For percentage urban, values are bounded between 0 and 100, which is an important clue that unbounded exponential growth cannot continue indefinitely.
城市化率为百分比时，数值被限制在 0 到 100 之间，这是无界指数增长不能无限持续的重要线索。
3. Linear Growth Models | 线性增长模型
A linear model has the form U(t) = a + bt, where b is the constant absolute increase per year. For some countries over short periods, urbanisation appears roughly linear.
线性模型形式为 U(t) = a + bt，其中 b 为每年恒定的绝对增长量。对某些国家在较短时间内，城市化看起来近似线性。
U(t) = a + bt
Linear growth is suitable when the first differences, U(t+1) − U(t), are approximately constant. From two data points, the slope is calculated as b = (U₂ − U₁)/(t₂ − t₁).
当一阶差分 U(t+1) − U(t) 近似恒定时，线性增长是合适的。由两个数据点可计算斜率 b = (U₂ − U₁)/(t₂ − t₁)。
A linear model is easy to interpret, but it predicts unrealistic values if extrapolated far into the future: it would eventually exceed 100% or fall below 0%.
线性模型易于解释，但若向远期外推，会得出不现实的值：最终会超过100%或低于0%。
4. Exponential Growth and the Semi-log Plot | 指数增长与半对数图
If the urban population grows at a roughly constant percentage rate, an exponential model applies: P(t) = P₀e^(kt), where P₀ is the initial population and k is the continuous growth rate.
若城市人口以大致恒定的百分比增长，则适用指数模型：P(t) = P₀e^(kt)，其中 P₀ 为初始人口，k 为连续增长率。
P(t) = P₀e^(kt)
Taking natural logarithms gives ln P(t) = ln P₀ + kt, which is linear in t. Therefore, plotting ln P against t should produce a straight line with gradient k.
取自然对数得 ln P(t) = ln P₀ + kt，关于 t 是线性的。因此，绘制 ln P 对 t 的图应得到一条斜率为 k 的直线。
ln P(t) = ln P₀ + kt
In exams, a semi-log plot is often used to identify exponential growth. If the points lie close to a straight line, the exponential model is supported.
考试中常用半对数图识别指数增长。如果数据点近似在一条直线上，则支持指数模型。
However, for urbanisation rate U(t) rather than urban population, the bounded nature of percentages makes exponential growth only valid for an early phase.
然而，对城市化率 U(t) 而非城市人口，百分比的有限性使得指数增长只在早期阶段有效。
5. Logistic Growth and Carrying Capacity | 逻辑斯蒂增长与承载容量
A more realistic model for the urbanisation rate is the logistic model: dU/dt = rU(1 − U/K), where K is the carrying capacity, often set at 100% or an upper plateau.
对城市化率更现实的模型是逻辑斯蒂模型：dU/dt = rU(1 − U/K)，其中 K 为承载容量，通常设为100%或某一上限平台。
dU/dt = rU(1 − U/K)
The solution has an S-shaped curve. It starts with slow growth, accelerates, then slows as U approaches K.
其解呈 S 形曲线。开始增长缓慢，随后加速，当 U 接近 K 时又放缓。
A common form is U(t) = K / (1 + Ae^(−rt)), where A is a constant determined by the initial value.
常见形式为 U(t) = K / (1 + Ae^(−rt))，其中 A 是由初始值决定的常数。
U(t) = K / (1 + Ae^(−rt))
To find A, substitute U(0) = U₀: A = (K − U₀)/U₀.
求 A 时，代入 U(0) = U₀：A = (K − U₀)/U₀。
The logistic model is powerful because it captures the slowdown seen in mature urbanised countries such as the UK and Japan.
逻辑斯蒂模型很有效，因为它捕捉到了英国、日本等成熟城市化国家的放缓现象。
6. Rank-Size Rule and Zipf’s Law | 位序-规模法则与齐普夫定律
Urban systems can also be studied by ranking cities by population. Zipf’s law states that the population of the nth largest city is inversely proportional to its rank: Pₙ = P₁/n.
城市体系还可以通过按人口对城市排序来研究。齐普夫定律指出，第 n 大城市的人口与其位序成反比：Pₙ = P₁/n。
Pₙ = P₁/n
A more general form is Pₙ = P₁/n^q, where q is the Zipf exponent. If q = 1, the rank-size rule holds exactly.
更一般的形式为 Pₙ = P₁/n^q，其中 q 为齐普夫指数。若 q = 1，位序-规模法则严格成立。
Pₙ = P₁/n^q
Taking logarithms gives ln Pₙ = ln P₁ − q ln n. A log-log plot of population against rank should be linear with gradient −q.
取对数得 ln Pₙ = ln P₁ − q ln n。人口对位序的双对数图应呈线性，斜率为 −q。
ln Pₙ = ln P₁ − q ln n
Since 1945, many countries have seen q fall or rise depending on whether large cities grow faster or slower than smaller ones.
1945 年以来，许多国家的 q 值或降或升，取决于大城市比小城市增长更快还是更慢。
7. Urban Primacy Index | 城市首位度指数
Urban primacy measures the dominance of the largest city. A simple index is the ratio P₁/P₂, the population of the largest city divided by the second largest.
城市首位度衡量最大城市的支配地位。一个简单的指数是比值 P₁/P₂，即最大城市人口除以第二大城市人口。
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