Mark and recapture: estimating a population with Lincoln-Petersen and Chapman, and why a few recaptures are not enough

How the Lincoln-Petersen estimate works

Mark a known number of animals, let them mix back in, catch a second sample, and assume the share of marked animals in that sample equals their share in the whole population. If 200 marked beetles are mixed into a population and 12 of the next 150 caught are marked, then 200 is to N as 12 is to 150, so N = 200 × 150 ÷ 12 = 2,500.

The idea is older than ecology. W. E. Ricker's handbook for Canada's Fisheries Research Board, Computation and Interpretation of Biological Statistics of Fish Populations (Bulletin 191, 1975), traces it to John Graunt's Observations on the London Bills of Mortality of 1662, in which children born in a year were the "marked" individuals. C. G. J. Petersen applied it to marked fish from 1896; Knut Dahl used it for trout in Norwegian tarns about ten years later; and Lincoln brought it to birds in 1930, estimating duck numbers from band returns. Hence the two names on one formula.

The Chapman estimator, and why it is the one to report

Chapman's estimator, N = (M + 1)(C + 1) ÷ (R + 1) − 1, corrects the plain formula's tendency to overestimate, and is the version to use unless recaptures are in the hundreds. Ricker reports that Bailey (1951) and Chapman (1951) showed the simple estimate "tends to overestimate the true population", and that Chapman's version gives "an unbiased estimate in most situations". In Charles Krebs's Ecological Methodology, chapter 2 (University of British Columbia, draft third edition), the same formula is the one Seber (1982) recommends: unbiased if M + C is greater than N, and nearly unbiased with at least seven recaptures.

Three details trip people up:

• The −1. Krebs keeps it; Ricker drops it as "of no practical significance". Either way the answer rounds to the same estimate, but say which you used when you compare with someone else.
• No recaptures. With R = 0, Lincoln-Petersen divides by zero. Chapman still returns a number, but it only says the samples were too small.
• Sightings instead of captures. If the second sample is a count of sightings, so one animal can be counted twice, Krebs gives Bailey's version, N = M(C + 1) ÷ (R + 1), which assumes marked and unmarked animals are equally likely to be seen.

The gap between the two shrinks as R grows: 16% at four recaptures, under 1% at 167. Ricker reports the trout as 337 from 110 × 178 ÷ 58 = 337.6, without the −1; with it, 336.6 rounds to 337.

Worked example: ground beetles in a field margin

A survey marks ground beetles caught in pitfall traps along a field margin with a dot of paint and releases them. A week later the traps are run again with the same effort. The numbers are made up for the example; the method is Krebs's and Ricker's.

1. First sample. 200 beetles marked and released: M = 200.
2. Second sample. 150 caught: C = 150. Of these, 12 carry paint: R = 12.
3. Lincoln-Petersen. 200 × 150 ÷ 12 = 2,500.
4. Chapman. 201 × 151 ÷ 13 − 1 = 30,351 ÷ 13 − 1 = 2,334.7 − 1 = 2,334.
5. Choose the interval. R ÷ C = 12 ÷ 150 = 0.08, below 0.10, and R is under 50, so Seber's guide says Poisson limits (see the next section).
6. Limits for R. Krebs's Table 2.1 gives 95% Poisson limits of 6.686 and 20.335 for 12.
7. Put each limit in place of R. 30,351 ÷ 21.335 − 1 = 1,422 and 30,351 ÷ 7.686 − 1 = 3,948.
8. Report it. About 2,300 beetles, 95% confidence interval 1,400 to 3,900.

Two checks follow. The bias condition in Ricker, M × C greater than four times N, holds: 200 × 150 = 30,000, more than four times 2,334. And the interval is wide: the true number could be about 60% of the estimate or two thirds above it, because 12 recaptures is a small count.

Which confidence interval to use

Put confidence limits on the number of recaptures, then convert each limit into a population size; which limits to use depends on how large R is and how large a share of the second sample it makes. Krebs sets out Seber's decision guide:

For the binomial case, the limits apply to the marked share. Krebs's bird example converts them as N = M ÷ (R ÷ C): with the Wilson limits, 50 ÷ 0.803 = 62 and 50 ÷ 0.430 = 116. The snowshoe hare data in his Box 2.1, from Green and Evans's live-trapping in Minnesota in the winter of 1932-33, have R ÷ C = 0.397, so they too take binomial limits, by the Wilson method: 2,383 hares, 95% interval 2,135 to 2,700. Ricker's Example 3.1 does the same for Dahl's trout, 337 with limits of 272 and 417.

Note the shape of every interval: it reaches further above the estimate than below. Because N depends on 1 ÷ R, a recapture count that is low by chance raises the estimate more than a count high by the same amount lowers it. An estimate written as "2,334 ± 1,000" misstates that.

Why a few recaptures are not enough

Below about ten recaptures the estimate is too imprecise to be much use, even when it is not biased: with four recaptures the 95% range runs from about half to about double. Two separate problems meet here, and the sources give a threshold for each:

• Bias. Ricker says the probability of the zero-recapture kind of bias "can be ignored if recaptures number 3-4 or more". Seber, as Krebs reports him, calls Chapman's estimate nearly unbiased from seven recaptures.
• Precision. The interval depends on how many recaptures you have, and it narrows slowly, as the table shows.

The limits are Krebs's Table 2.1, from Crow and Gardner (1959); each multiple is (R + 1) divided by (limit + 1). This is why Robson and Regier's charts, which Ricker reproduces, call for roughly 25 to 75 recaptures to land within ±25% of the true population 19 times in 20, depending on how large the population is.

Planning: how many to mark and how many to catch

Decide the accuracy you need, guess the population on the high side, and choose M and C so the expected number of recaptures, M × C ÷ N, is large enough. Krebs passes on Robson and Regier's three standard levels, each 95% of the time: ±50% for a preliminary survey, ±25% for management and ±10% for research.

A worked plan: you expect about 2,000 animals and want management accuracy, so aim for at least 25 recaptures. M × C ÷ 2,000 = 25 means M × C = 50,000, for example 225 marked and 225 caught. Krebs notes that keeping M and C about equal handles the fewest animals for a given precision, and that "it is often necessary to mark 50% or more of the individuals in a population" for even moderate accuracy. If 450 handlings is beyond the project, the honest choice is a preliminary-survey accuracy, or a different method, not a confident number from ten recaptures.

Ricker's simpler check for bias, M × C greater than four times N, is a floor, not a target: for 2,000 animals it needs only 8,000, such as 90 × 90, which would give about four recaptures.

The five assumptions, and what breaking each does

The estimate is only as good as five assumptions, which Krebs lists for the Petersen method; most failures push the estimate up. The direction follows from N = M × C ÷ R: anything that lowers the marked share of the second sample inflates N, and anything that raises it deflates N.

Krebs adds that deaths during marking leave the estimate valid for the population after release, that N always means the catchable population (nets miss small fish), and that the best defence against recruitment is to take both samples over a short period. Random sampling is the other half: he advises dividing the area into equal subareas and sampling randomly chosen ones with the same effort, rather than relying on marked animals mixing evenly.

Lost marks can be measured. Krebs's example marks 500 beetles with two kinds of paint and finds, in the second sample, 23 with only the first mark, 6 with only the second and 127 with both. His formula estimates 157 recaptures against the 156 seen: one beetle lost both marks.

Counting marked and unmarked in the field

Record M exactly when you release, and in the second sample count marked and unmarked animals as two separate tallies; C is their sum. Counting only the marked ones and estimating the total afterwards is how C goes wrong.

Simple Online Counter is a free, browser-based set of counters that keeps its counts in your browser's own storage. In the multi counter, name two counters "marked" and "unmarked" and tap one per animal as you check it; add a third for animals that died in the trap if you need to subtract them. The tally counter does the first sample with one tap per animal marked. The yes/no counter shows the marked share of the second sample as you go, with a normal-approximation range once you pass 30 animals; for small numbers of recaptures use the Poisson or binomial limits above instead. None of these marks or identifies animals: the counts are yours, and the formulas turn them into an estimate. Round the result to the precision the interval supports, as the flock counting guide argues for any estimate.

Beyond two samples: Schnabel and open populations

With more than two capture sessions, the Schnabel method pools them: N = Σ(Cₜ × Mₜ) ÷ ΣRₜ, where each session t catches Cₜ, finds Rₜ already marked, and Mₜ animals carried marks beforehand. Krebs describes it as a weighted average of Petersen estimates, from Schnabel (1938), with every unmarked animal marked and released at each session. When fewer than a tenth of the population is caught or marked per session, he gives a better form with ΣRₜ + 1 below the line.

If the study runs long enough for births, deaths and movement, the population is open and the Petersen family no longer applies. Krebs covers the Cormack-Jolly-Seber method for that case, and programs such as MARK and CAPTURE for designs that test whether catchability varies.

Mistakes the worked examples repeat

What this guide does not cover

This guide covers the two-sample Petersen family of estimates for closed populations: Lincoln-Petersen, Chapman and Bailey, their confidence intervals, sample sizes and assumptions. It mentions, but does not teach, the Schnabel method, open-population models such as Cormack-Jolly-Seber, and likelihood methods in programs such as MARK, which a study with several sessions or uncertain catchability should use. It does not cover how to capture, mark or handle animals: that depends on the species, and in many countries on permits and animal welfare rules, which your institution or wildlife authority sets.

On the sources: Krebs's chapter is the author's own draft of the third edition, posted by the University of British Columbia; Ricker's Bulletin 191 is the 1975 edition held by Fisheries and Oceans Canada. The figures quoted from Seber, Chapman, Robson and Regier and Crow and Gardner are as those two sources report them.