Populations and Sustainability: Growth Models and Sustainable Yield | 人口与可持续性：增长模型与可持续产量

📚 Populations and Sustainability: Growth Models and Sustainable Yield | 人口与可持续性：增长模型与可持续产量

Population sustainability is a major application of differential equations in Edexcel Mathematics. A mathematical model describes how the size N(t) of a population changes over time, allowing us to predict future growth, identify stable equilibria, and determine how many individuals can be harvested without driving the population to extinction.

人口可持续性是 Edexcel 数学中微分方程的重要应用。数学模型描述种群规模 N(t) 随时间的变化，使我们能够预测未来增长、识别稳定平衡点，并确定在不导致种群灭绝的前提下可以收获多少个体。

1. Mathematical Models for Populations | 人口的数学模型

A population model represents the number of individuals N(t) as a function of time t. The central idea is to describe the rate of change dN/dt using biological assumptions such as birth rate, death rate, and resource limitation.

人口模型将个体数量 N(t) 表示为时间 t 的函数。核心思想是利用出生率、死亡率和资源限制等生物学假设来描述变化率 dN/dt。

In Edexcel questions, you are usually given a differential equation and asked to solve it, interpret its parameters, or find equilibrium population sizes. Population sustainability therefore combines calculus, algebra, and interpretation of real-world constraints.

在 Edexcel 考题中，通常会给出一个微分方程，要求求解、解释参数或找出平衡人口规模。因此，人口可持续性综合了微积分、代数以及对现实约束的解释。

2. The Exponential Growth Model | 指数增长模型

If resources are unlimited, the rate of population growth is proportional to the current population size. This gives the simplest model:

如果资源无限，人口增长率与当前人口规模成正比。这给出最简单的模型：

dN/dt = rN

Here r is the intrinsic growth rate, equal to birth rate minus death rate. The solution is:

这里 r 是内禀增长率，等于出生率减去死亡率。其解为：

N(t) = N₀ exp(rt)

This predicts unbounded exponential growth, so it can only be valid for a limited time before resources become limiting.

该模型预测无界的指数增长，因此只有在资源尚未成为限制因素之前的较短时间内才有效。

Assumes constant birth and death rates
Assumes unlimited resources
No migration or age structure

假设出生率和死亡率恒定
假设资源无限
不考虑迁移或年龄结构

3. Limitations of Exponential Growth | 指数增长的局限

In real populations, exponential growth cannot continue indefinitely. Food, space, and other resources become scarce, causing the growth rate to fall as the population increases.

在现实人口中，指数增长不可能无限持续。食物、空间和其他资源变得稀缺，导致增长率随着人口增加而下降。

This means a realistic model must include a feedback mechanism: as N gets large, dN/dt should approach zero or become negative. The carrying capacity K represents the maximum population size the environment can support.

这意味着现实模型必须包含反馈机制：当 N 增大时，dN/dt 应趋近于零或变为负值。环境承载力 K 表示环境能够支持的最大种群规模。

Ignoring this limitation can lead to overestimating sustainable harvest levels, which is dangerous for conservation policy.

忽视这一限制可能导致高估可持续收获水平，这对保护政策来说是危险的。

4. The Logistic Growth Model | 逻辑斯蒂增长模型

The logistic model introduces a carrying capacity K and reduces the growth rate by a factor 1 − N/K. The differential equation is:

逻辑斯蒂模型引入环境承载力 K，并通过因子 1 − N/K 降低增长率。其微分方程为：

dN/dt = rN(1 − N/K)

When N is much smaller than K, the factor is close to 1, so growth is nearly exponential. As N approaches K, the growth rate approaches zero.

当 N 远小于 K 时，该因子接近 1，因此增长近似指数增长。当 N 接近 K 时，增长率趋近于零。

The logistic model predicts an S-shaped curve. It is widely used in ecology and in sustainability calculations because it gives a stable carrying capacity instead of unbounded growth.

逻辑斯蒂模型预测 S 形曲线。由于它给出稳定的环境承载力而不是无界增长，因此被广泛用于生态学和可持续性计算。

5. Equilibria and Stability | 平衡点与稳定性

An equilibrium occurs when the population size stops changing, so dN/dt = 0. For the logistic model:

当种群规模停止变化时，即 dN/dt = 0，就出现平衡。对于逻辑斯蒂模型：

dN/dt = 0 ⇒ N* = 0 or N* = K

N = 0 is an unstable equilibrium because a small positive population will grow away from it. N = K is stable because small disturbances return the population to K.

N = 0 是不稳定平衡点，因为一个小的正种群会从零增长离开。N = K 是稳定平衡点，因为小扰动会使种群回到 K。

Stability is important for sustainability: if harvesting pushes the population below a critical level, recovery may not occur. Therefore, managers aim to keep N close to or above certain thresholds.

稳定性对可持续性很重要：如果收获将种群推到临界水平以下，种群可能无法恢复。因此，管理者力求将 N 保持在某个阈值以上或附近。

6. Sustainable Harvesting Concepts | 可持续收获概念

Sustainability means removing individuals from a population without causing long-term decline. In a harvested logistic model, a constant harvest rate h is subtracted:

可持续性意味着在不导致长期下降的情况下从种群中移除个体。在带收获的逻辑斯蒂模型中，减去恒定的收获率 h：

dN/dt = rN(1 − N/K) − h

If h is small, the population settles at a new equilibrium below K. If h is too large, dN/dt is always negative and the population collapses to extinction.

如果 h 很小，种群会在低于 K 的某个新平衡点稳定下来。如果 h 太大，dN/dt 始终为负，种群将崩溃至灭绝。

This model is the foundation of fisheries and wildlife management. It shows that sustainability is not simply about taking a fixed number, but about taking no more than the population can replace.

该模型是渔业和野生动物管理的基础。它表明可持续性并不是简单地取一个固定数量，而是收获量不能超过种群能够自我补充的数量。

7. Maximum Sustainable Yield | 最大可持续产量

At equilibrium, harvest equals population growth, so:

在平衡状态下，收获量等于种群增长量，因此：

h = rN(1 − N/K)

To maximise h, we differentiate with respect to N:

为了最大化 h，我们对 N 求导：

dh/dN = r(1 − 2N/K)

Setting dh/dN = 0 gives N = K/2. Substituting this back gives the maximum sustainable yield:

令 dh/dN = 0，得到 N = K/2。代回得到最大可持续产量：

h_max = rK/4

Therefore, MSY occurs when the population is held at half the carrying capacity. This is a classic result in population mathematics and appears frequently in exam questions.

因此，当种群保持在环境承载力的一半时，可获得最大可持续产量。这是人口数学中的经典结论，经常出现在考题中。

8. Yield–Effort Models | 产量–努力量模型

Instead of a fixed harvest rate, many real-world harvests depend on fishing effort or hunting effort E. A common model assumes:

与固定收获率不同，许多现实中的收获依赖于捕捞努力量或狩猎努力量 E。常用模型假设：

h = qEN

Here q is the catchability coefficient. At equilibrium with the logistic growth model:

这里 q 是可捕系数。在与逻辑斯蒂增长模型达到平衡时：

N* = K(1 − qE/r)

The sustainable yield is then:

那么可持续产量为：

Y = qE N* = qEK(1 − qE/r)

Maximising Y with respect to E gives the optimal effort E* = r/(2q) and the same maximum yield Y* = rK/4.

对 E 最大化 Y 可得到最优努力量 E* = r/(2q)，以及相同的最大产量 Y* = rK/4。

This yield–effort curve is often used to explain overfishing: beyond E*, greater effort produces lower yields and risks collapse.

产量–努力量曲线常被用来解释过度捕捞：超过 E* 后，更大的努力量反而导致更低产量，并可能造成种群崩溃。

9. Predator–Prey Oscillations | 捕食者–猎物振荡

When two species interact, population sustainability becomes a coupled system. A simple Lotka–Volterra predator–prey model is:

当两个物种相互作用时，种群可持续性变为一个耦合系统。简单的 Lotka–Volterra 捕食者–猎物模型为：

dx/dt = ax − bxy

dy/dt = cxy − dy

Here x is the prey population, y is the predator population, and a, b, c, d are positive constants. Solutions typically show periodic oscillations rather than a single stable equilibrium.

这里 x 是猎物数量，y 是捕食者数量，a、b、c、d 是正常数。解通常表现为周期性振荡，而不是单一稳定平衡。

Although this model is simplified, it highlights that sustainability in an ecosystem can depend on interactions between species, not just on one population in isolation.

尽管该模型已被简化，但它强调生态系统中的可持续性可能取决于物种之间的相互作用，而不仅仅是孤立的一个种群。

10. Human Populations and Sustainability Indicators | 人类人口与可持续性指标

Human population data can be modelled using the same ideas. The average growth rate between two census years can be estimated by:

人类人口数据可以用相同的思想建模。两个普查年份之间的平均增长率可以通过以下公式估算：

r ≈ (ln N₂ − ln N₁)/(t₂ − t₁)

If the growth rate declines as population rises, the logistic model may be more appropriate than the exponential model. The estimated carrying capacity K then becomes a sustainability indicator.

如果增长率随人口上升而下降，逻辑斯蒂模型可能比指数模型更合适。估算出的环境承载力 K 随后成为可持续性指标。

Governments and organisations use such models to plan food supply, housing, energy, and environmental protection. However, human behaviour and technology make K less fixed than in animal populations.

政府和组织使用此类模型来规划食品供应、住房、能源和环境保护。然而，人类行为和技术使 K 不像动物种群那样固定。

11. Limitations and Model Refinements | 模型局限与改进

Simple logistic and harvest models ignore age structure, migration, seasonality, and random environmental events. They also assume r and K are constant, which is rarely true over long periods.

简单的逻辑斯蒂和收获模型忽略了年龄结构、迁移、季节性以及随机环境事件。它们还假设 r 和 K 恒定，这在长期内很少成立。

More advanced models introduce time delays, spatial spread, stochastic terms, and multiple species interactions. Such refinements make the mathematics more complex but produce more realistic sustainability predictions.

更高级的模型引入时间延迟、空间扩散、随机项和多物种相互作用。这些改进使数学更复杂，但能产生更现实的可持续性预测。

In exam contexts, you should be able to state the assumptions behind a model and explain why a prediction may differ from real data.

在考试情境中，你应该能够陈述模型背后的假设，并解释为什么预测可能与实际数据存在差异。

12. Exam-Style Summary | 考试型总结

Key formulae to remember for Edexcel population sustainability questions include:

Edexcel 人口可持续性题目需要记住的关键公式包括：

Exponential growth: dN/dt = rN, N(t) = N₀ exp(rt)
Logistic growth: dN/dt = rN(1 − N/K)
Constant harvesting: dN/dt = rN(1 − N/K) − h
Maximum sustainable yield: N = K/2, h_max = rK/4
Yield–effort model: Y = qEK(1 − qE/r), E* = r/(2q)

指数增长：dN/dt = rN，N(t) = N₀ exp(rt)
逻辑斯蒂增长：dN/dt = rN(1 − N/K)
恒定收获：dN/dt = rN(1 − N/K) − h
最大可持续产量：N = K/2，h_max = rK/4
产量–努力量模型：Y = qEK(1 − qE/r)，E* = r/(2q)

Always interpret your answer in the context of sustainability, and check whether the harvest level is safe or likely to cause extinction.

始终在可持续性背景下解释你的答案，并检查收获水平是安全的还是可能导致灭绝。

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Populations and Sustainability: Growth Models and Sustainable Yield | 人口与可持续性：增长模型与可持续产量