Martingales in mark-recapture experiments with constant recruitment and survival

Abstract

The method known as mark-recapture has been used for almost one hundred years in assessing animal populations. For many years, these models were restricted to closed populations; no changes to the population were assumed to occur through either migration or births and deaths. Numerous estimators for the closed population have been proposed through the years, some of the most recent by Paul Yip which make use of martingales to derive the necessary estimates. The independently derived Jolly-Seber model (1965) was the first to address the open population situation. That method as originally proposed is cumbersome mathematically due to the large number of parameters to be estimated as well as the inability to obtain estimates until the end of a series of capture events since some of the "observed" variables necessary are prospective. It also is cumbersome for the biologist in the field as individual marks and capture histories are required for each animal. Variations have been proposed through the years which hold survival and/or capture probabilities constant across capture occasions. Models based on log-linear estimators have also been proposed (Cormack 1989). This paper builds on the closed population work of Yip in using martingale-based conditional least squares to estimate population parameters for an open population where it is assumed recruitment of new individuals into the population is constant from one capture occasion to the next, and capture and survival probabilities are constant across capture occasions. It is an improvement over most other methods in that no detailed capture histories are needed; animals are simply noted as marked or unmarked. Performance of the estimator proposed is studied through computer simulation and comparison with classical estimators on actual data sets.