Mark and recapture is a method commonly used in ecology to estimate an animal population's size where it is impractical to count every individual. A portion of the population is captured, marked, and released. Later, another portion will be captured and the number of marked individuals within the sample is counted. Since the number of marked individuals within the second sample should be proportional to the number of marked individuals in the whole population, an estimate of the total population size can be obtained by dividing the number of marked individuals by the proportion of marked individuals in the second sample. The method assumes, rightly or wrongly, that the probability of capture is the same for all individuals. and the Lincoln method.

Another major application for these methods is in epidemiology, where they are used to estimate the completeness of ascertainment of disease registers. Typical applications include estimating the number of people needing particular services (e.g. services for children with learning disabilities, services for medically frail elderly living in the community), or with particular conditions (e.g. illegal drug addicts, people infected with HIV, etc.).

thumb|Biologist is marking a [[Chittenango ovate amber snail to monitor the population.]]

Typically a researcher visits a study area and uses traps to capture a group of individuals alive. Each of these individuals is marked with a unique identifier (e.g., a numbered tag or band), and then is released unharmed back into the environment. A mark-recapture method was first used for ecological study in 1896 by C.G. Johannes Petersen to estimate plaice, Pleuronectes platessa, populations.

Sufficient time should be allowed to pass for the marked individuals to redistribute themselves among the unmarked population. Other organisms captured during the second visit, will not have been captured during the first visit to the study area. These unmarked animals are usually given a tag or band during the second visit and then are released. (also known as the Petersen–Lincoln index 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. An alternative less biased estimator of population size is given by the Chapman estimator:

Capture probability

thumb|Bank vole, [[Myodes glareolus, in a capture-release small mammal population study for London Wildlife Trust at Gunnersbury Triangle local nature reserve]]

The capture probability refers to the probability of a detecting an individual animal or person of interest, and has been used in both ecology and epidemiology for detecting animal or human diseases, respectively.

The capture probability is often defined as a two-variable model, in which f is defined as the fraction of a finite resource devoted to detecting the animal or person of interest from a high risk sector of an animal or human population, and q is the frequency of time that the problem (e.g., an animal disease) occurs in the high-risk versus the low-risk sector. For example, an application of the model in the 1920s was to detect typhoid carriers in London, who were either arriving from zones with high rates of tuberculosis (probability q that a passenger with the disease came from such an area, where q>0.5), or low rates (probability 1−q). It was posited that only 5 out of 100 of the travelers could be detected, and 10 out of 100 were from the high risk area. Then the capture probability P was defined as:

:<math>P = \frac{5}{10}fq+\frac{5}{90}(1-f)(1-q), </math>

where the first term refers to the probability of detection (capture probability) in a high risk zone, and the latter term refers to the probability of detection in a low risk zone. Importantly, the formula can be re-written as a linear equation in terms of f:

:<math>P = \left(\frac{5}{10}q-\frac{5}{90}(1-q)\right)f + \frac{5}{90}(1-q).</math>

Because this is a linear function, it follows that for certain versions of q for which the slope of this line (the first term multiplied by f) is positive, all of the detection resource should be devoted to the high-risk population (f should be set to 1 to maximize the capture probability), whereas for other value of q, for which the slope of the line is negative, all of the detection should be devoted to the low-risk population (f should be set to 0. We can solve the above equation for the values of q for which the slope will be positive to determine the values for which f should be set to 1 to maximize the capture probability:

:<math>\left( \frac{5}{10} q - \frac{5}{90}(1-q) \right) > 0, </math>

which simplifies to:

:<math>q > \frac{1}{10}. </math>

This is an example of linear optimization. A simple model which easily accommodates the three source, or the three visit study, is to fit a Poisson regression model. Sophisticated mark-recapture models can be fit with several packages for the Open Source R programming language. These include "Spatially Explicit Capture-Recapture (secr)", "Loglinear Models for Capture-Recapture Experiments (Rcapture)", and "Mark-Recapture Distance Sampling (mrds)". Such models can also be fit with specialized programs such as MARK or E-SURGE.

Other related methods which are often used include the Jolly–Seber model (used in open populations and for multiple census estimates) and Schnabel estimators (an expansion to the Lincoln–Petersen method for closed populations). These are described in detail by Sutherland.

Integrated approaches

Modelling mark-recapture data is trending towards a more integrative approach, which combines mark-recapture data with population dynamics models and other types of data. The integrated approach is more computationally demanding, but extracts more information from the data improving parameter and uncertainty estimates.

See also

  • German tank problem, for estimation of population size when the elements are numbered.
  • Tag and release
  • Abundance estimation
  • GPS wildlife tracking
  • Shadow Effect (Genetics)

References

Further reading

  • Petersen, C. G. J. (1896). "The Yearly Immigration of Young Plaice Into the Limfjord From the German Sea", Report of the Danish Biological Station (1895), 6, 5–84.
  • Schofield, J. R. (2007). "Beyond Defect Removal: Latent Defect Estimation With Capture-Recapture Method", Crosstalk, August 2007; 27–29.
  • A historical introduction to capture-recapture methods
  • Analysis of capture-recapture data