Mark-Recapture Population Calculator

How mark-recapture estimates population size

Mark-recapture estimates how many animals live in an area without counting every one. Catch a first sample of M animals, mark each one and release it. Later, catch a second sample of C animals and count the R of them that carry a mark. If the marked animals have mixed back in at random, the marked share of the second sample estimates the marked share of the whole population, R ÷ C = M ÷ N, so the population size is N = M × C ÷ R.

That is the Lincoln-Petersen estimate, also called the Lincoln index. It is named after the Danish fisheries biologist C. G. J. Petersen, who tagged plaice in the 1890s, and the American ornithologist Frederick Lincoln, who estimated waterfowl numbers from bird-banding returns in 1930. British specifications call the fieldwork mark-release-recapture, and some books write n₁, n₂ and m₂ for M, C and R. This calculator gives the Lincoln-Petersen figure beside Chapman’s less biased version, the standard error and an approximate 95 percent interval, and draws the second sample so that the ratio behind the answer can be seen.

Using the calculator

Enter the three counts. M is every animal marked and released on the first visit. C is everything caught on the second visit, marked or not, and R is how many of those carry a mark. The slider for R runs only up to the smaller of M and C, because the recaptures belong to both groups.

The diagram draws the second sample, a filled dot for each marked animal and a hollow ring for each unmarked one, with the marked dots spread evenly through the grid, as the method assumes marked animals are spread through the population. Above 400 animals each dot stands for 1 percent of the sample. The readouts give both estimates, the standard error and interval of Chapman’s estimate, the marked share R ÷ C, and M + C − R, the number of different animals caught, which is the least the population can be. A note appears under the readouts when the counts are too few to trust, and the working sets out every step with your numbers.

For the rest of an ecology write-up, the Simpson’s diversity index calculator turns quadrat and sweep-net counts into a diversity index, and for plant physiology the water potential calculator covers osmosis in plant cells.

Worked example: 30 marked, 40 caught, 12 recaptured

These are the values loaded above. A student marks 30 woodlice with a dot of non-toxic paint, releases them where they were found and comes back the next day to catch 40, of which 12 are marked.

• Marked share of the second sample: 12 ÷ 40 = 0.3.
• Lincoln-Petersen: N = 30 × 40 ÷ 12 = 100, the same as 30 ÷ 0.3.
• Chapman: N = 31 × 41 ÷ 13 − 1 = 96.77.
• Variance: 31 × 41 × 18 × 28 ÷ (13² × 14) = 640,584 ÷ 2,366 = 270.75.
• Standard error: √270.75 = 16.454.
• 95 percent interval: 96.77 ± 1.96 × 16.454 = 96.77 ± 32.25, which is 64.5 to 129.0.
• Different animals caught: 30 + 40 − 12 = 58, so there are at least 58 woodlice.

So the best single figure is about 97 woodlice, and the counts are consistent with anything from about 65 to 129. Chapman’s figure is 3.2 percent below the Lincoln-Petersen 100, and its standard error is 17 percent of the estimate.

Chapman’s estimator and the 95 percent interval

The Lincoln-Petersen formula overestimates on average, and with small samples it can fail outright: if no marked animal turns up in the second sample, R = 0 and it divides by zero. Chapman (1951) adjusted it to N = (M + 1)(C + 1) ÷ (R + 1) − 1. This is exactly unbiased whenever M + C is at least the true population size, stays finite at R = 0, and comes very close to the Lincoln-Petersen figure once R is large, so it is the better single number to report.

The variance usually quoted with it is Seber’s (1970): Var = (M + 1)(C + 1)(M − R)(C − R) ÷ [(R + 1)²(R + 2)]. Its square root is the standard error, and adding and subtracting 1.96 standard errors gives an approximate 95 percent interval. This is a normal approximation. When R is small the real spread of possible estimates is lopsided, with a long tail towards large populations, so the symmetric interval is rough, and its lower end can even fall below M + C − R, the number of animals you know exist. The calculator says when that happens. For what a standard error and a confidence interval mean in general, the standard deviation calculator works them out for a set of repeated measurements.

How many animals to mark

Precision depends mostly on the number of recaptures. From the variance above, the standard error as a share of the estimate is about √[(1 − R ÷ M)(1 − R ÷ C) ÷ R], which is close to 1 ÷ √R when the recaptures are a small part of both samples. So the error shrinks only with the square root of R: halving it takes about four times as many recaptures, and since the number expected is M × C ÷ N, that means doubling both samples.

In a population of about 1,000, marking 100 and catching 100 should find about 10 marked animals, and the standard error of Chapman’s estimate is then 238.6. Marking and catching 200 each should find about 40, and the standard error falls to 121.0, close to half. A widely used rule of thumb is to plan for at least 7 recaptures; with fewer, the estimate is too uncertain to be much use.

What this model leaves out

• Births, deaths and movement. The method assumes a closed population, with nothing joining or leaving between the samples. Births and immigrants add unmarked animals and push the estimate up. Leave long enough for marked animals to mix back in, but no longer. Over seasons and years numbers rise and fall, as the predator-prey simulator shows for a pair of species.
• Lost or harmful marks. A mark that wears off, or one that makes an animal easier for predators to spot, leaves too few marked animals in the second sample, so the estimate comes out too high.
• Trap-happy and trap-shy animals. If being caught once makes an animal more likely to be caught again, perhaps because the trap was baited, R is too high and N too low. If it makes the animal warier, the error goes the other way.
• Unequal catchability. Some individuals are simply easier to catch, because of their size, sex or behaviour. They turn up in both samples more than their numbers warrant, which raises R and biases the estimate low.
• More than two samples. Repeated sampling, as in the Schnabel method, and models for open populations such as Jolly-Seber use more information than two counts hold.
• Exact small-sample intervals. The normal approximation is the simple textbook interval. Methods built on the binomial or Poisson distribution of R give lopsided intervals that suit a small R better.

Common mistakes

• Swapping C and R. 30 × 12 ÷ 40 = 9, fewer than the 30 animals already marked, which cannot be right. The estimate can never be smaller than M or C.
• Leaving the marked animals out of C. C is the whole second catch. Using only the 28 unmarked animals gives 30 × 28 ÷ 12 = 70 instead of 100.
• Sampling too soon, or in the same spot. Marked animals released at one place stay near it for a while, so a second sample taken there finds too many of them and the estimate comes out low.
• Reporting too many figures. With 12 recaptures the 95 percent interval runs about 33 percent either side of the estimate, so 96.77 is better reported as about 97, between 65 and 129.
• Reading R = 0 as an infinite population. It means the samples were too small to find a marked animal. Mark and catch more, rather than quoting the formula’s division by zero.