Editing Mark and recapture (section)

== Capture probability ==
[[File:Bank Vole GT Jo Hodges.jpg|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,<ref>{{Cite journal|last=Drenner|first=Ray|date=1978|title=Capture probability: the role of zooplankter escape in the selective feeding of planktivorous fish|journal=Journal of the Fisheries Board of Canada|volume=35|issue=10|pages=1370–1373|doi=10.1139/f78-215}}</ref> and has been used in both ecology and [[epidemiology]] for detecting animal or human diseases,<ref>{{Cite journal|last=MacKenzie|first=Darryl|date=2002|title=How should detection probability be incorporated into estimates of relative abundance?|journal=Ecology|volume=83|issue=9|pages=2387–2393|doi=10.1890/0012-9658(2002)083[2387:hsdpbi]2.0.co;2}}</ref> 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.<ref name=":0">{{Cite book|title=Ecological Models and Data in R|last=Bolker|first=Benjamin|publisher=Princeton University Press|year=2008|isbn=9781400840908}}</ref> 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'').<ref>{{Cite journal|last=Unknown|date=1921|title=The Health of London|journal=Hosp Health Rev|volume=1|issue=3|pages=71–2|pmid=29418259|pmc=5518027}}</ref> 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]].<ref name=":0" /> In more complex cases, where more than one resource ''f'' is devoted to more than two areas, multivariate [[Mathematical optimization|optimization]] is often used, through the [[simplex algorithm]] or its derivatives.