A unified framework for time‐to‐detection occupancy and abundance models

Time‐to‐detection (TTD) occupancy models are increasingly used to study site occupancy of organisms. Occupancy is a reduced representation of abundance (distinguishing only between 0 and >0 individuals), which is also often a quantity of interest. In this paper, we present a novel framework for TTD occupancy models that address limitations of existing approaches. Our approach incorporates factors that accommodate detection heterogeneity among sites/visits and inter‐visit dependency, allowing for the relaxation of some restrictive assumptions inherent in previous models. As a result, our framework offers a robust and versatile tool for analysing various ecological data sets. We employ a maximum likelihood approach to estimate model parameters and conduct inference for the proposed TTD occupancy models. A key feature of these models is the introduction of a community parameter. This parameter characterizes the similarity of detectabilities, ranging from complete independence to complete identity, across multiple site visits. This framework disentangles the detection rate, abundance and occupancy, similar to the popular N‐mixture model. For situations where abundance estimation is not the primary goal, a family of mixed gamma exponential TTD models is developed, which generally exhibit more stable numerical properties compared to N‐mixture type TTD models. The performance of the proposed models and some reduced models is evaluated through simulation studies. The results indicate that the N‐mixture TTD model tends to considerably overestimate the occupancy probability when the community parameter is less than one, a condition necessary to satisfy the strict closure population assumption. On the other hand, the standard exponential TTD occupancy model underestimates the occupancy probability in the presence of unobserved detection heterogeneity and inter‐visit dependency. An analysis of 63 bird species in the Karoo region of South Africa demonstrates the enhanced flexibility of the proposed TTD occupancy models for data fitting. This paper demonstrates the importance of employing more flexible and general models to accurately capture the complexities of ecological systems or survey data. We provide R‐code to fit all considered models to data. The proposed TTD model framework contributes to enhancing our understanding of species occupancy distributions.