Population sizes of animals are often estimated by mark-recapture experiments.  There are many variations of these experiments, of various degrees of sophistication.  The simplest, which is also conceptually the most straight-forward, is described here.

     On day 1, a sample of individuals from the population is captured, marked in some way to indicated that it has been captured (for butterflies, a mark is often put on the underside of the wings with a sharpie marker), and released.  One to several days later, another sample of individuals is collected.  Each individual captured on this day is examined to determine whether they had been captured on day 1 ( whether it is marked).  From this procedure, three numbers are obtained:

         N_m = Number of individuals marked and released on day 1

         N_c  = Number of individuals captured on the second sampling day

         N_r  = Number of individuals captured on second day that had been marked on the previous day.

We wish to estimate N, the number of individuals in the population.

     Assuming that marking individuals does not affect their mortality rate or tendency to disperse, the proportion of marked individuals in the second-day sample should be a good estimate of the proportion of marked individuals in the entire population.  Another way of saying this is that we expect

          N_m /N = N_r /N_c  .
Rearranging this equation yields

          N = N_cN_m/N_r   ,

which provides an estimate of the population size from the three quantities obtained from the mark-recapture experiment.
 

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