The capture-recapture method lets you estimate the size of a wildlife population without counting every individual. You catch a sample, mark them, release them, then catch a second sample and count how many are marked. The proportion marked in the second sample estimates the proportion in the whole population.
What is the Lincoln-Petersen formula?
The Lincoln-Petersen formula puts that proportional reasoning into algebra. If you let:
- M = number caught and marked in the first sample
- C = total number caught in the second sample
- R = number in the second sample that were already marked (recaptured)
- N = estimated total population size
Then the key proportion is:
R / C = M / N
Rearranging for N:
N = (M × C) / R
The idea: the fraction of the second catch that is marked (R/C) should equal the fraction of the whole population that is marked (M/N).
How do you carry out the method step by step?
- First catch: catch a sample of the population, count them, and mark each one in a way that doesn't harm them (a tag, a dye, a small notch). Record M.
- Release: return all M marked individuals to the population and allow enough time for them to mix randomly with the rest.
- Second catch: catch a fresh sample. Count the total number caught (C) and the number that are already marked (R).
- Calculate: use N = (M × C) / R to estimate the population size.
Worked example: trout in a lake
A biologist is studying trout in a lake.
- She catches 40 trout, tags them, and releases them. So M = 40.
- Three days later she catches 50 trout. Of these, 8 have tags. So C = 50 and R = 8.
$$N = \frac{M \times C}{R} = \frac{40 \times 50}{8} = \frac{2000}{8} = 250$$
Estimated population: 250 trout.
What assumptions must hold for this estimate to be valid?
The Lincoln-Petersen formula relies on four key assumptions. An exam question often asks you to state and comment on them.
| Assumption | What it means | Why it matters |
|---|---|---|
| Random mixing | Marked individuals spread evenly through the population between catches | If they clump together, R over- or underestimates the true proportion |
| Population closed | No births, deaths, immigration, or emigration between the two catches | Any of these change N between the two samples |
| Equal catchability | Every individual is equally likely to be caught each time | If marks make recapture more (or less) likely, R is biased |
| Marks retained | No marks fall off or fade before the second catch | Lost marks reduce R artificially, inflating the estimate |
In your exam, if the question asks "give one reason why the estimate may not be reliable," pick one of these assumptions and explain how it might be violated in context.
How do you rearrange the formula if a different variable is unknown?
Occasionally a question gives you N and asks for M, C, or R. Treat the formula as a simple equation.
N = (M × C) / R
Multiply both sides by R: N × R = M × C
- To find M: M = (N × R) / C
- To find C: C = (N × R) / M
- To find R: R = (M × C) / N
Example: In a population estimated at 500 fish, a researcher tagged 60. In a second catch of 45 fish, how many would you expect to be tagged?
R = (M × C) / N = (60 × 45) / 500 = 2700 / 500 = 5.4
Since you can't have a fraction of a fish, round to the nearest whole number: expect 5 tagged fish.
How do you choose an appropriate sample size?
A larger second sample gives a more reliable estimate because R becomes more stable. In practice:
- If R = 0, the formula breaks down (division by zero) — the second sample was too small or the population too large.
- If R = C (all recaptured), either the first sample marked almost everyone or the second sample is too small.
- Aim for R to be at least 5 or 6 for a reasonable estimate.
This is a good discussion point for extended-answer questions: "Explain how a researcher could improve the reliability of the estimate."
What other contexts use capture-recapture?
Although wildlife ecology is the most common exam context, the same method applies whenever direct counting is impossible:
- Estimating the number of people in a city with a medical condition
- Estimating the number of defective items in a manufacturing batch
- Ecological surveys of insects, fish, birds
The maths is identical — only the biological details change.
Frequently asked questions
What does R stand for in the formula?
R stands for recaptured — the number of individuals in the second sample that were already marked in the first sample. It is not the total second sample size (that is C). Mixing up R and C is the most common error in capture-recapture questions.
Do I need to memorise the formula for the exam?
Yes. The formula N = (M × C) / R is not given on the GCSE formula sheet, so you need to learn it. However, if you forget the letters, you can rebuild it from the proportion R/C = M/N and rearrange.
What does it mean if my estimate of N is very large?
A very large N usually means R was very small — only a few marked individuals turned up in the second catch. A small R can arise because the population is genuinely large, or because the marked animals moved away, or because the second catch happened too soon after release. Comment on these possibilities if asked to evaluate the reliability.
Is capture-recapture on the Foundation tier?
Capture-recapture can appear on both Foundation and Higher, but it is more common at Higher. Foundation questions tend to be straightforward substitutions into the formula; Higher questions additionally ask you to evaluate assumptions and comment on reliability.
Work through capture-recapture problems interactively with Professor Pi — add the AI Tutors connector at aitutors.me.