Lincoln Index: Estimating Population Size | 林肯指数：种群数量估算考点突破

📚 Lincoln Index: Estimating Population Size | 林肯指数：种群数量估算考点突破

The Lincoln Index, also known as the Lincoln–Petersen method, provides a simple yet powerful way to estimate the size of a mobile animal population. Mastering its formula, assumptions, and common pitfalls is essential for scoring full marks in ecology-based exam questions. This guide breaks down everything you need to know for A-level Biology, whether you are tackling data analysis, evaluating experimental design, or explaining why the method might over- or underestimate population size.

林肯指数（也称林肯-彼得森法）为估算移动动物种群的大小提供了一种简单却有力的方法。掌握其公式、假设条件及常见误区，是生态类考题拿满分的关键。无论你是在做数据分析、评价实验设计，还是解释为什么该方法可能高估或低估种群数量，这篇指南都会帮你逐一突破。

1. Introduction to Population Estimation | 种群数量估算简介

Ecologists often need to know how many individuals of a species live in a particular habitat. Counting every organism directly is usually impossible, especially for animals that move, hide, or cover vast areas. Scientists therefore rely on sampling techniques and mathematical indices to produce reliable estimates without harming the ecosystem.

生态学家经常需要知道某一物种在特定栖息地中的个体数量。直接清点所有生物通常是不可能的，尤其是对于移动、隐蔽或分布范围广阔的动物。因此，科学家借助抽样技术和数学指标，在不对生态系统造成伤害的前提下得出可靠的估计值。

The Lincoln Index belongs to a family of mark-release-recapture (MRR) methods. It works by capturing a sample of animals, marking them in a harmless way, releasing them back into the population, and then taking a second sample later. The proportion of marked individuals in the second sample allows us to infer the total population size.

林肯指数属于标记-释放-再捕捉（MRR）方法的一种。它的原理是：先捕捉一组动物，用无害方式做标记，放回原种群；一段时间后再采集第二个样本。通过第二个样本中已标记个体的比例，便能推算出种群总数。

2. What is the Lincoln Index? | 什么是林肯指数？

The Lincoln Index is a single-mark, single-recapture method first used to estimate the number of ducks in a pond. It assumes a closed population where birth, death, immigration, and emigration do not occur during the study period. The model treats the population as a ‘bag’ of mixed marked and unmarked individuals, allowing a simple ratio calculation.

林肯指数是一种单次标记、单次重捕的方法，最初用于估算池塘中的鸭子数量。它假设种群是封闭的，研究期间不发生出生、死亡、迁入或迁出。该模型把种群看作一个装有标记和未标记个体的“袋子”，从而能够进行简单的比例计算。

Whenever you see a question asking about estimating the population of woodlice, beetles, snails, or small fish using a capture-recapture technique, you are almost certainly expected to apply or evaluate the Lincoln Index.

一旦考题要求用捕捉-重捕法估算潮虫、甲虫、蜗牛或小型鱼类的种群数量，你几乎可以肯定要用到或评价林肯指数。

3. The Lincoln Index Formula | 林肯指数公式

The most basic version of the Lincoln Index is:

最基本的林肯指数公式为：

N = (M × C) / R

Where N = estimated total population size, M = number of individuals caught in the first sample, marked, and released, C = total number of individuals caught in the second sample, and R = number of marked individuals recaptured in the second sample.

其中 N = 估算的总种群数量，M = 第一次捕获、标记并释放的个体数，C = 第二次捕获的个体总数，R = 第二次捕获中已标记的个体数（重捕标记数）。

Some exam boards use a slightly modified version to reduce bias when R is small: N = (M × C) / (R + 1). Always read the question carefully to see which formula is provided or expected.

某些考试局会使用略微修改的版本，以降低 R 较小时的偏差：N = (M × C) / (R + 1)。答题时务必仔细审题，看清题目提供或要求使用的是哪个公式。

The principle behind the formula is a simple proportion: M / N should be roughly equal to R / C. Rearranging gives N = (M × C) / R. This makes intuitive sense — if many marked animals are recaptured, the total population must be relatively small; if very few are recaptured, the population must be large.

公式背后的原理是简单的比例关系：M / N 应约等于 R / C。移项后便得到 N = (M × C) / R。这很符合直觉——如果重捕到的标记动物很多，说明总种群相对较小；如果重捕到的极少，说明种群数量庞大。

4. Assumptions of the Lincoln Index | 林肯指数的假设条件

The reliability of the Lincoln Index depends on several crucial assumptions. Breaking any one of them can lead to inaccurate estimates. Make sure you can list and explain each assumption, because evaluation questions are very common.

林肯指数的可靠性取决于几个关键假设。任何一项假设被打破都会导致估算失准。务必能够逐一列出并解释，因为评价类考题非常普遍。

The marking does not affect survival or behaviour. Marks must be non-toxic, harmless, and not make the animal more visible to predators or less likely to be recaptured.
标记不影响生存或行为。标记必须无毒、无害，且不能使动物更容易被捕食者发现或降低其被重捕的概率。
The marking remains visible for the entire study period. If marks wash off, fade, or are shed during moulting, R will be underestimated, leading to an overestimate of N.
标记在整个研究期间保持可见。如果标记被洗掉、褪色或在蜕皮时脱落，R 会被低估，从而导致 N 被高估。
The population is closed. No births, deaths, immigration, or emigration occur between the two sampling events. If recruitment occurs, the population estimate is skewed.
种群是封闭的。两次取样之间没有出生、死亡、迁入或迁出。如果有个体补充，种群估计值会出现偏差。
Marked and unmarked individuals mix randomly. After release, marked animals must disperse evenly throughout the population so the second sample is representative.
标记个体与未标记个体随机混合。释放后，标记动物必须在整个种群中均匀分布，以确保第二次样本具有代表性。
The probability of capture is equal for all individuals. There should be no ‘trap-happy’ or ‘trap-shy’ effects.
所有个体的被捕捉概率相等。不应出现“喜爱入陷阱”或“害怕入陷阱”的倾向性差异。

5. Step-by-step Procedure | 逐步操作步骤

A typical field investigation using the Lincoln Index follows these steps:

使用林肯指数进行野外考察一般遵循以下步骤：

1. Capture a sample of the target species using a suitable method (pitfall traps, sweep nets, live traps). Count and record the number as M.

1. 使用适当的方法（如陷阱、扫网、活捕笼）捕捉目标物种的一个样本。清点数量并记录为 M。

2. Mark each individual in a way that is temporary but lasting enough for the study. Small dots of non-toxic paint, nail varnish, or numbered tags are common. Ensure marks do not hinder movement or attract predators.

2. 以暂时但足够持久的方式标记每个个体。常用无毒颜料、指甲油小点或编号标签。确保标记不妨碍运动，也不吸引捕食者。

3. Release the marked animals at the exact point of capture and allow sufficient time for them to mix back into the population. This period may range from a few hours to a day, depending on the species.

3. 将标记动物在捕捉点就地释放，并给予足够时间使其重新融入种群。这一等待期从几小时到一天不等，取决于物种特性。

4. Conduct a second capture using the same method and effort. Count the total number caught as C. Inspect each individual for marks and record the number of marked individuals as R.

4. 使用相同的方法和努力程度进行第二次捕捉。记录捕捉总数 C，并逐一检查有无标记，记录标记个体数 R。

5. Plug the values into the Lincoln Index formula to estimate N. Calculate any measures of uncertainty if required.

5. 将数值代入林肯指数公式估算 N。如有需要，计算不确定度。

6. Worked Example: Estimating Woodlouse Population | 例题：估算潮虫种群

A student sets pitfall traps in a garden to estimate the woodlouse population. On day 1, she collects 58 woodlice, marks them with a tiny dot of acrylic paint, and releases them. Two days later, she traps again and catches 71 woodlice, of which 12 bear the paint mark.

一名学生在花园中设置陷阱来估算潮虫种群。第一天，她收集了58只潮虫，用丙烯颜料点上小点做标记后释放。两天后再次用陷阱捕捉，得到71只潮虫，其中有12只带有标记。

Using N = (M × C) / R: M = 58, C = 71, R = 12.

使用公式 N = (M × C) / R：M = 58, C = 71, R = 12。

N = (58 × 71) / 12 = 4118 / 12 ≈ 343

The estimated population size is approximately 343 woodlice. If the student had used a corrected formula, it might be N = (58 × 71) / (12 + 1) = 4118 / 13 ≈ 317.

估算的种群大小约为343只潮虫。如果学生使用了修正公式，结果可能为 N = (58 × 71) / (12 + 1) = 4118 / 13 ≈ 317。

When presenting your answer, always round to a sensible number — in this case, 343 or ‘about 340’. Never claim a spurious degree of accuracy by giving decimal places.

作答时，应合理取整——例如 343 或“约340”。切勿提供虚假精度的小数位。

7. Sources of Error and Bias | 误差与偏差来源

Examiners love to ask why an estimate might be higher or lower than the true population. Systematic errors often stem from violated assumptions.

考官喜欢问为什么估算值可能高于或低于真实种群。系统误差往往源于假设被违反。

If marks are lost or become unrecognisable (e.g., due to rain or moulting), R is too low, making N too high. If marking increases predation risk, marked individuals are removed from the population before the second capture, again lowering R and inflating N.

如果标记丢失或变得无法辨认（例如因雨水或蜕皮），R 会偏低，导致 N 偏高。如果标记增加了被捕食风险，标记个体在第二次捕捉前就被移除，同样会降低 R 并夸大 N。

If the population is not closed and immigration occurs, the proportion of marked individuals in the second sample drops, leading to an overestimate. Conversely, if marking makes animals more likely to be recaptured (trap-happy), R rises and N is underestimated.

如果种群不封闭且有迁入，第二次样本中标记个体的比例下降，造成高估。反之，若标记使动物更容易被重捕（喜陷阱），R 增加，N 被低估。

Timing matters too: a short mixing interval may mean the second sample is taken before marked individuals have dispersed evenly, resulting in unrepresentative R values.

时间安排也很重要：混合间隔过短会导致第二次取样时标记个体尚未均匀散布，产生不具代表性的 R 值。

8. Modifications for Improved Accuracy | 改进准确性的修正

To improve reliability, ecologists often use a longer sampling period, multiple capture-recapture cycles, or the Schnabel method (multiple marks over successive days). For the A-level, you mainly need to recognise that the Lincoln Index is simplest but may require a correction factor like the +1 adjustment for small R.

为提高可靠性，生态学家常采用更长的采样期、多次捕捉-重捕循环或施纳贝尔法（连续数日多次标记）。就 A-level 而言，你主要需要认识到林肯指数最为简单，但在 R 较小时可能需要 +1 等校正因子。

Always recommend a pilot study to check whether marks persist and whether behaviour is altered. Use harmless, quick-drying materials and consider the species’ biology when deciding on marking location.

始终建议先做预试验，检查标记是否持久、行为是否改变。使用无害、速干的材料，并根据物种生物学特性决定标记位置。

Where possible, combine Lincoln Index data with other ecological measures such as mean density from quadrats to cross-validate conclusions.

在可能的情况下，将林肯指数数据与其他生态测量（如样方平均密度）相结合，交叉验证结论。

9. Ethical Considerations | 伦理考量

All field studies involving live animals must minimise stress and harm. The mark-release-recapture procedure must be justified, and the marking technique must be approved by an ethical committee if required. In exam answers, always mention that handling time should be kept brief, animals should be released promptly, and the number of captured individuals should not deplete the population.

所有涉及活体动物的野外研究都必须将压力和伤害降至最低。标记-释放-重捕程序须有正当理由，标记技术必要时须经伦理委员会批准。作答时一定要提到：操作时间要短，动物应尽快释放，捕获数量不应造成种群耗竭。

Many species are protected by law; check local regulations and consider using invertebrates to reduce ethical concerns. A-level practicals nearly always use woodlice, snails, or brine shrimp for this reason.

许多物种受法律保护；应查阅当地法规，并考虑使用无脊椎动物以减少伦理顾虑。正因如此，A-level 实验几乎都用潮虫、蜗牛或卤虫。

10. Alternative Methods | 替代方法

The Lincoln Index is not the only way to estimate population size. For sessile or slow-moving organisms, quadrat sampling and transects are more appropriate. For large mammals, camera traps and DNA fingerprinting from hair or faeces are increasingly common.

林肯指数并非估算种群大小的唯一方法。对于固着或缓慢移动的生物，样方和样带更为合适。对于大型哺乳动物，红外相机陷阱及通过毛发或粪便进行DNA指纹识别的方法日渐普及。

Relate your knowledge of the Lincoln Index to other sampling strategies: it is particularly useful for mobile, cryptic animals in a defined area, but less suitable for highly migratory species or those with extremely low recapture rates.

将你对林肯指数的了解与其他抽样策略相联系：它特别适用于某确定区域内移动而隐秘的动物，但对于高度迁徙的物种或重捕率极低的物种则不太适用。

In exams, you might be asked to choose the most appropriate method for a given scenario and justify your choice. Use the Lincoln Index when a mark-release-recapture design is feasible and the population is reasonably closed.

考试中可能要求你为给定情景选择最合适的方法并说明理由。只要标记-释放-重捕方案可行且种群基本封闭，就选择林肯指数。

11. Exam Tips and Common Pitfalls | 考试技巧与常见陷阱

Always use the exact formula provided in the question. If a formula is given, do not substitute your own version. Show your working step by step.
始终使用题目给出的准确公式。若题目提供了公式，切勿替换成自己的版本。计算过程要逐步展示。
Be specific when evaluating reliability. Instead of saying ‘the result might be wrong’, explain that ‘if marks fade, R is reduced, causing an overestimate of N’. Show a chain of logic.
评价可靠性时要具体。别说“结果可能不对”，而要解释“如果标记褪色，R 减少，导致 N 被高估”。展示因果逻辑链。
Link assumptions to biological reality. Mention that snails may retract making marks invisible, or that small mammals may groom off tags.
将假设与生物学实际相联系。例如蜗牛可能缩回壳内使标记不可见，小型哺乳动物可能梳理毛发弄掉标签。
Units are not needed for N as it is a count of individuals, but you can write ‘individuals’ to clarify.
N 不需要单位，因为它是个体数，但可以写上“只”以示清晰。
Comment on sample size. A very small R (e.g., 1 or 2) makes the estimate unreliable due to chance effects. Increasing trapping effort improves accuracy.
评论样本量。非常小的 R（例如 1 或 2）会因为偶然效应使得估算不可靠。增加捕捉努力可提高准确性。

12. Conclusion: Mastering the Lincoln Index | 结语：掌握林肯指数

The Lincoln Index is a beautiful intersection of mathematics, ecology, and experimental design. Once you can confidently explain the formula, list and link the assumptions to potential sources of error, and evaluate the reliability of an estimate, you are fully equipped for top marks. Practice applying the formula to fresh data sets, and always think critically about why a given estimate might differ from the true value.

林肯指数是数学、生态学与实验设计的精彩交汇。一旦你能自信地解释公式、列出假设并将其与潜在误差来源相联系，并能评价估算的可靠性，你就完全有能力拿到高分。多在新鲜数据上练习应用公式，并始终批判性地思考为什么某个估计值可能与真实值不同。

Use this guide alongside past exam questions, and remember — understanding the ‘why’ behind each step is far more valuable than rote memorisation.

将本指南与历年真题结合使用，并记住——理解每一步背后的“为什么”，远比死记硬背更有价值。

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Lincoln Index: Estimating Population Size | 林肯指数：种群数量估算考点突破