A continuous-time formulation for spatial capture-recapture models

Abstract

Spatial capture-recapture (SCR) models are relatively new but have become the standard approach used to estimate animal density from capture-recapture data. It has in the past been impractical to obtain sufficient data for analysis on species that are very difficult to capture such as elusive carnivores that occur at low density and range very widely. Advances in technology have led to alternative ways to virtually “capture" individuals without having to physically hold them. Some examples of these new non-invasive sampling methods include scat or hair collection for genetic analysis, acoustic detection and camera trapping. In traditional capture-recapture (CR) and SCR studies populations are sampled at discrete points in time leading to clear and well defined occasions whereas the new detector types mentioned above sample populations continuously in time. Researchers with data collected continuously currently need to define an appropriate occasion and aggregate their data accordingly thereby imposing an artificial construct on their data for analytical convenience. This research develops a continuous-time (CT) framework for SCR models by treating detections as a temporal non homogeneous Poisson process (NHPP) and replacing the usual SCR detection function with a continuous detection hazard function. The general CT likelihood is first developed for data from passive (also called “proximity") detectors like camera traps that do not physically hold individuals. The likelihood is then modified to produce a likelihood for single-catch traps (traps that are taken out of action by capturing an animal) that has proven difficult to develop with a discrete-occasion approach. The lack of a suitable single-catch trap likelihood has led to researchers using a discrete-time (DT) multi-catch trap estimator to analyse single-catch trap data. Previous work has found the DT multi-catch estimator to be robust despite the fact that it is known to be based on the wrong model for single-catch traps (it assumes that the traps continue operating after catching an individual). Simulation studies in this work confirm that the multi-catch estimator is robust for estimating density when density is constant or does not vary much in space. However, there are scenarios with non-constant density surfaces when the multi-catch estimator is not able to correctly identify regions of high density. Furthermore, the multi-catch estimator is known to be negatively biased for the intercept parameter of SCR detection functions and there may be interest in the detection function in its own right. On the other hand the CT single-catch estimator is unbiased or nearly so for all parameters of interest including those in the detection function and those in the model for density. When one assumes that the detection hazard is constant through time there is no impact of ignoring capture times and using only the detection frequencies. This is of course a special case and in reality detection hazards will tend to vary in time. However when one assumes that the effects of time and distance in the time-varying hazard are independent, then similarly there is no information in the capture times about density and detection function parameters. The work here uses a detection hazard that assumes independence between time and distance. Different forms for the detection hazard are explored with the most flexible choice being that of a cyclic regression spline. Extensive simulation studies suggest as expected that a DT proximity estimator is unbiased for the estimation of density even when the detection hazard varies though time. However there are indirect benefits of incorporating capture times because doing so will lead to a better fitting detection component of the model, and this can prevent unexplained variation being erroneously attributed to the wrong covariate. The analysis of two real datasets supports this assertion because the models with the best fitting detection hazard have different effects to the other models. In addition, modelling the detection process in continuous-time leads to a more parsimonious approach compared to using DT models when the detection hazard varies in time. The underlying process is occurring in continuous-time and so using CT models allows inferences to be drawn about the underlying process, for example the timevarying detection hazard can be viewed as a proxy for animal activity. The CT formulation is able to model the underlying detection hazard accurately and provides a formal modelling framework to explore different hypotheses about activity patterns. There is scope to integrate the CT models developed here with models for space usage and landscape connectivity to explore these processes on a finer temporal scale. SCR models are experiencing a rapid growth in both application and method development. The data generating process occurs in CT and hence a CT modelling approach is a natural fit and opens up several opportunities that are not possible with a DT formulation. The work here makes a contribution by developing and exploring the utility of such a CT SCR formulation.