1. Trends of animal populations are of great interest in ecology but cannot be directly observed owing to imperfect detection. Binomial mixture models use replicated counts to estimate abundance, corrected for detection, in demographically closed populations. Here, we extend these models to open populations and illustrate them using sand lizard Lacerta agilis counts from the national Dutch reptile monitoring scheme.

2. Our model requires replicated counts from multiple sites in each of several periods, within which population closure is assumed. Counts are described by a hierarchical generalized linear model, where the state model deals with spatio-temporal patterns in true abundance and the observation model with imperfect counts, given that true state. We used WinBUGS to fit the model to lizard counts from 208 transects with 1–10 (mean 3) replicate surveys during each spring 1994–2005.

3. Our state model for abundance contained two independent log-linear Poisson regressions on year for coastal and inland sites, and random site effects to account for unexplained heterogeneity. The observation model for detection of an individual lizard contained effects of region, survey date, temperature, observer experience and random survey effects.

4. Lizard populations increased in both regions but more steeply on the coast. Detectability increased over the first few years of the study, was greater on the coast and for the most experienced observers, and highest around 1 June. Interestingly, the population increase inland was not detectable when the observed counts were analysed without account of detectability. The proportional increase between 1994 and 2005 in total lizard abundance across all sites was estimated at 86% (95% CRI 35–151).

5. Synthesis and applications. Open-population binomial mixture models are attractive for studying true population dynamics while explicitly accounting for the observation process, i.e. imperfect detection. We emphasize the important conceptual benefit provided by temporal replicate observations in terms of the interpretability of animal counts.