Lincoln Index (Mark-Recapture Method)

You catch a first group of animals, mark them inconspicuously and release them again. After a while, during which the marked animals mix evenly with the rest, you catch another group. In this second catch you count how many marked animals are present. The idea is a proportion: the fraction of marked animals in the second catch equals the fraction of marked animals in the whole population. From this follows the formula $N =\frac{ M\cdot{}C}{R}$ with the marked animals M, the entire second catch C and the marked ones recaptured in it R. The larger the fraction of marked animals in the second catch, the smaller the estimated population. This lets you estimate the size of populations you cannot count completely, such as fish in a pond.

You insert the three measured numbers into the formula $N =\frac{ M\cdot{}C}{R}$. M is the number of animals marked and released in the first catch, C the total number in the second catch and R the number of marked animals recaptured in the second catch. Example: you mark $M = 40$ fish. In the second catch there are $C = 50$ fish, of which $R = 8$ are marked. Then $N =\frac{ (40\cdot{}50)}{8} =\frac{ 2000}{8} = 250$ fish. It is important not to confuse C and R: C is the entire second catch, R only the marked part of it. For small R the estimate becomes inaccurate and tends to be too high; then you use the Chapman correction $N =\frac{ (M+1)(C+1)}{(R+1)} - 1$, which for the same example gives about 231.

The method assumes a closed population: between the two catches there must be no births, deaths, immigration or emigration, otherwise the proportion no longer holds. In addition, the marked animals must mix evenly with the unmarked ones so that the sample is representative; this needs enough time between the catches. The marking must affect neither survival nor catchability: marked animals must not be more conspicuous to predators, not shyer and not easier to recatch. Marks must also not be lost. Finally, every animal should have the same catch probability. If these conditions are violated, the estimate is biased. In practice one chooses markings and time intervals so that the assumptions are met as well as possible, and uses extended models if needed.

The recapture R is in the denominator of the formula $N =\frac{ M\cdot{}C}{R}$. If R is small, even small counting differences strongly affect the result: whether you recatch 2 or 3 marked animals changes the estimate considerably. In addition, the simple formula is systematically biased upward for small R, it overestimates the population on average. The reason lies in the statistics of drawing: with few recaptures the estimated fraction of marked animals is more random and biased in expectation. Therefore one uses the Chapman correction $N =\frac{ (M+1)(C+1)}{(R+1)} - 1$, which markedly reduces this bias and works even when $R = 0$. In principle you should choose M and C large enough to recatch enough marked animals, usually at least about ten, to obtain reliable estimates.

The method serves wherever populations cannot be counted completely because the animals are mobile, hidden or too numerous. In fisheries biology it estimates fish stocks in lakes and rivers to set catch quotas. In wildlife management it records mammal or bird populations for conservation, often with rings, ear tags or camera traps as marking. For insects and snails it determines local population sizes in ecological studies. Even in epidemiology the same principle is used to estimate the unknown number of disease cases by comparing two independent registers. The great advantage is that a complete count is unnecessary and even two samples yield a usable estimate. The precondition remains that the assumptions of the method are sufficiently met, otherwise the result is biased.