Editing Mark and recapture (section)

== Lincoln–Petersen estimator ==
{{Main article|Lincoln index}}
The '''Lincoln–Petersen method'''<ref name="seber">{{cite book |last=Seber |first=G. A. F. |title=The Estimation of Animal Abundance and Related Parameters |location=Caldwel, New Jersey |year=1982 |publisher=Blackburn Press |isbn=1-930665-55-5 }}</ref> (also known as the Petersen–Lincoln index<ref name="Southwood" /> or [[Lincoln index]]) can be used to estimate population size if only two visits are made to the study area.  This method assumes that the study population is "closed". In other words, the two visits to the study area are close enough in time so that no individuals die, are born, or move into or out of the study area between visits.  The model also assumes that no marks fall off animals between visits to the field site by the researcher, and that the researcher correctly records all marks.

Given those conditions, estimated population size is:

:<math>\hat{N} = \frac{nK}{k},</math>

=== Derivation ===

It is assumed<ref name=Krebs1999>{{cite book |title=Ecological Methodology |author=Charles J. Krebs |edition=2nd |date=1999 |publisher=Benjamin/Cummings |url=https://books.google.com/books?id=1GwVAQAAIAAJ |isbn=9780321021731 }}</ref> that all individuals have the same probability of being captured in the second sample, regardless of whether they were previously captured in the first sample (with only two samples, this assumption cannot be tested directly).

This implies that, in the second sample, the proportion of marked individuals that are caught (<math>k/K</math>) should equal the proportion of the total population that is marked (<math>n/N</math>). For example, if half of the marked individuals were recaptured, it would be assumed that half of the total population was included in the second sample.

In symbols,
:<math>\frac{k}{K} = \frac{n}{N}.</math>
A rearrangement of this gives
:<math>\hat{N}=\frac{nK}{k}, </math>
the formula used for the Lincoln–Petersen method.<ref name=Krebs1999 />

=== Sample calculation ===
In the example (n, K, k) = (10, 15, 5) the Lincoln–Petersen method estimates that there are 30 turtles in the lake.
: <math>\hat{N} = \frac{nK}{k} = \frac{10\times 15}{5} = 30</math>