The Lincoln-Peterson index is a method for deducing the population size of an elusive species. Many populations, such as those of many birds, insects, and ocean-going animals (from turtles to fish to whales), are very hard to count, due to their high mobility and wide dispersal. One common solution to this is to capture and tag a given number of individuals, re-release them into the wild, and then take random samples of the population at a later time. The simple way to think of this is that if you tag 100 animals, and later sample 100 animals, and you find only one tagged animal in your sample, you might guess that there are about 10,000 animals in the local population.
The Lincoln-Peterson index gives us a slightly more formal mathematical model for determining population size through our tagging and sampling process. It goes thusly:
M/N = m/n
- M = the number of individuals marked and released.
- N = the actual size of the study population.
- m = the number of marked individuals in the sample population.
- n = the total number of individuals in the sample.
We know M, m, and n, meaning that we can easily calculate N, the actual size of the study population. (To solve for N, we would write the equation as N=Mn/m).
As it happens, this index tends to overestimate the population size by a small amount. In order for this calculation to work accurately, no marked animals can be removed from the population (i.e., die) between the time they are tagged and the time that the sample is taken. As it happens, a population will constantly loose old (tagged) members, and gain new (untagged) members through the birth of new individuals. To account for this, a modified Lincoln-Peterson index was suggested:
N = M(n+1)/(m+1)
While the addition of the imaginary tagged animal to the sample size is a kludge that probably makes baby mathematician Jesus cry, it also results in more accurate estimations of population size, so it is frequently used. Another modification, known as the Schnabel index, is also often used: N=((M+1)x(n+1)/(m+1))-1.
For obvious reasons, the Lincoln-Peterson index is still the most popular when a quick and easy calculation is sufficient, but a modified form is more common in published works.