How Do You Use the Lincoln Index (Capture-Mark-Release-Recapture) to Estimate Population Size?

For SL and HL, Subtopic 2.1, the Lincoln index estimates the population size of a mobile species by comparing the number initially marked with the proportion of marked individuals found in a second sample.

The method is called capture-mark-release-recapture:

1. Capture a first sample using an appropriate sampling method.
2. Count and mark the organisms harmlessly, then release them.
3. Allow enough time for them to mix randomly throughout the population.
4. Capture a second sample and count how many individuals carry marks.
5. Apply the Lincoln index:

$$N = \frac{n_1 \times n_2}{n_3}$$

Here, $N$ is the estimated population size, $n_1$ is the number marked and released initially, $n_2$ is the total number in the second capture, and $n_3$ is the number of marked individuals recaptured.

For example, suppose $40$ grasshoppers are marked and released. A second sample contains $50$ grasshoppers, including $10$ marked individuals:

$$N = \frac{40 \times 50}{10} = 200$$

The estimated population is therefore $200$ grasshoppers. This works because the proportion marked in the second sample should approximate the proportion marked in the whole population: $\frac{n_3}{n_2} \approx \frac{n_1}{N}$. The approach developed from animal-marking work such as Frederick C. Lincoln's studies of North American waterfowl.

A common misconception is that more marked recaptures mean a larger population. In fact, if $n_1$ and $n_2$ stay constant, a larger $n_3$ produces a smaller population estimate.

Exam technique: Define every variable, show substitution and working, and state that the result is an estimate. For an evaluation question, explain how violating specific assumptions affects reliability rather than merely listing them.