📚 5 Principles of Population Ecology Applied to Human Populations | 应用于人类种群的人口生态学五原理
Population ecology studies how populations change over time. When applied to human populations, its principles become powerful mathematical models. In Edexcel A-Level Mathematics, tools such as exponential functions, differential equations and matrices allow us to analyse population growth, carrying capacity, density dependence, age structure and demographic transition. This article presents five core principles with worked examples and exam-style tips.
人口生态学研究种群如何随时间变化。将其原理应用于人类种群，就得到了强大的数学模型。在 Edexcel A-Level 数学中，指数函数、微分方程和矩阵等工具使我们能够分析人口增长、承载力、密度制约、年龄结构和人口转变。本文介绍五个核心原理，并配有例题和考试技巧。
1. Overview of Population Mathematics | 人口数学概述
In population ecology the size of a population, written N, changes through four processes: births (B), deaths (D), immigration (I) and emigration (E). If these are measured over a very short time interval, the instantaneous rate of change of the population is given by the balance equation below.
在人口生态学中，种群大小记作 N，通过四个过程发生变化：出生（B）、死亡（D）、迁入（I）和迁出（E）。若在极短的时间间隔内测量这些量，种群的瞬时变化率由下面的平衡方程给出。
dN/dt = B − D + I − E
− d)N. The difference b − d is called the intrinsic rate of increase and is denoted by r.
对于封闭种群，没有迁移，因此 I = E = 0。把人均出生率和人均死亡率分别记作 b 和 d，得到 dN/dt = (b − d)N。差值 b − d 称为内禀增长率，记作 r。
dN/dt = (b − d)N
r = b − d
Human populations are open systems, but for many modelling purposes we begin with the closed-population assumption. This allows us to use simple differential equations before adding migration terms. In A-Level questions, you are often asked to set up such equations from given birth and death rates, so check carefully whether the data are quoted per person or per 1000 people.
人类种群是开放系统，但在许多建模中我们先采用封闭种群假设。这样可以使用简单的微分方程，之后再加入迁移项。在 A-Level 题目中，经常要求根据给定的出生率和死亡率建立这类方程，因此要仔细判断数据是人均值还是每千人值。
| Symbol | Meaning | 符号 | 含义 |
| N, N₀ | Population size, initial size | N, N₀ | 种群大小、初始大小 |
| t | Time (usually in years) | t | 时间（通常以年为单位） |
| r | Intrinsic rate of increase | r | 内禀增长率 |
| K | Carrying capacity | K | 环境承载力 |
| λ | Discrete growth multiplier | λ | 离散增长乘数 |
2. Principle 1: Exponential Growth Model | 原理一：指数增长模型
is the initial population size at t = 0. The graph of N against t is an increasing exponential curve.
如果 r 恒定且为正，种群呈指数增长。微分方程 dN/dt = rN 的解为 N(t) = N₀ exp(rt)，其中 N₀ 是 t = 0 时的初始种群大小。N 对 t 的图像是一条递增的指数曲线。
dN/dt = rN ⟹ N(t) = N₀ eʳᵗ
d taking natural logarithms gives the doubling time formula, which is independent of the starting population. This is a very common one-mark question.
任何指数过程都有固定的倍增时间。令 N(t) = 2N₀ 并取自然对数，就得到倍增时间公式，它与初始人口无关。这是一道非常常见的单分题。
T = ln 2 / r ≈ 0.6931 / r
Worked example. .028 − 0.011 = 0.017 per year. After 20 years, N(20) = 5.0 × exp(0.017 × 20) = 5.0 × exp(0.34) ≈ 5.0 × 1.4049 ≈ 7.02 million. The doubling time is 0.6931 / 0.017 ≈ 40.8 years.
例题。某国人均出生率为每年 0.028，人均死亡率为每年 0.011，初始人口 500 万。则 r = 0.028 − 0.011 = 0.017（每年）。20 年后，N(20) = 5.0 × exp(0.017 × 20) = 5.0 × exp(0.34) ≈ 5.0 × 1.4049 ≈ 702 万。倍增时间为 0.6931 / 0.017 ≈ 40.8 年。
The weakness of this model is the hidden assumption of unlimited resources. Real human populations eventually meet limits such as arable land, fresh water, energy and housing, so pure exponential growth can only be a short-term approximation.
该模型的弱点在于隐含了资源无限的假设。现实中的人类种群最终会遇到耕地、淡水、能源和住房等限制，因此纯粹的指数增长只能是短期近似。
3. Principle 2: Logistic Growth and Carrying Capacity | 原理二：逻辑斯谛增长与承载力
The logistic model introduces the carrying capacity K, the maximum population that the environment can support sustainably. Growth is fast when the population is small and slows as N approaches K, because the factor (1 − N/K) tends to zero.
逻辑斯谛模型引入了环境承载力 K，即环境能够可持续支撑的最大种群数量。当种群较小时增长很快，当 N 趋近 K 时增长变慢，因为因子 (1 − N/K) 趋于零。
dN/dt = rN(1 − N/K)
Solving this separable differential equation by partial fractions gives the explicit solution below. The curve is sigmoid (S-shaped): it starts almost exponentially, then decelerates and approaches K asymptotically from below.
用部分分式求解这个可分离变量的微分方程，得到下面的显式解。曲线呈 S 形：起初近似指数增长，随后减速，并从下方渐近地趋近 K。
N(t) = K / (1 + ((K − N₀)/N₀) exp(−rt))
value back into the differential equation gives the maximum growth rate of the population, rK/4.
S 形曲线最陡的位置是拐点，恰好出现在 N = K/2 处。把这个值代回微分方程，就得到种群的最大增长率 rK/4。
N = K/2 ⟹ (dN/dt)ₘₐₓ = rK/4
Worked example. 897) ≈ 125. The maximum growth rate is 0.05 × 1000 / 4 = 12.5 individuals per unit time, reached when N = 500.
例题。若 dN/dt = 0.05N(1 − N/1000) 且 N₀ = 50，则 K = 1000，r = 0.05，因此 N(t) = 1000 / (1 + 19 exp(−0.05t))。当 t = 20 时，exp(−1) ≈ 0.3679，得 N ≈ 1000 / (1 + 6.9897) ≈ 125。最大增长率为 0.05 × 1000 / 4 = 12.5（每单位时间个体数），在 N = 500 时取得。
4. Principle 3: Density Dependence | 原理三：密度制约
Density dependence means that per capita birth and death rates are not constants but functions of population size. As a population becomes denser, competition for food, space and healthcare increases, so the birth rate b(N) falls and the death rate d(N) rises.
密度制约是指人均出生
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