Estimation of abundance of fish populations by capture-recapture experiments

A natural fish population is assumed to be subjected to a number of fishing experiments, during which the captured fish are released after tagging all untagged fish. Assuming that, in thek ^th sampling survey, the tagged and untagged fish are subjected toP _tk andP _uk fishing rates, andS _tk andS _uk survival rates, respectively, the population size can be estimated from an estimate ofP _uk $$\begin{gathered} i.e., 1 - P_{uk} = \hfill \\ P_{tk \cdot } \frac{{S_t (k + 1)}}{{S_u (k + 1)}}\frac{\begin{gathered} Number of fish captured for the first time in \hfill \\ the (k + 1)^{th} survey plus fish belonging to this \hfill \\ group captured later \hfill \\ \end{gathered} }{\begin{gathered} Number of fish captured for the first time in \hfill \\ the k^{th} and captured again in the (k + 1)^{th} \hfill \\ \exp eriment plus fish belonging to this group \hfill \\ which are captured later \hfill \\ \end{gathered} } \hfill \\ \end{gathered}$$ All terms of the right hand side of the above equation are provided by capture-recapture experiments, except the survival rate of untagged fish which may be obtained by other information. This estimate of fishing rate of untagged fish is free from type (A) errors.