## Introduction

Much ecological research seeks to understand drivers of species distributions across space and time. Examples include studies of metapopulation ecology (Hanski 1994), population viability (Beissinger & Westphal 1998), community composition and dynamics (Mordecai, Cooper & Justicia 2009; Zipkin, Dewan & Royle 2009), resource selection (MacKenzie 2006) and disease spread (Thompson 2007). Modelling distribution of species based on presence–absence data using occupancy models offers flexibility in addressing such diverse questions with relatively simple sampling designs that account for detectability (Mackenzie & Royle 2005). Understanding the distribution and ecology of episodic or mobile species, however, requires us to address multiple challenges related to sampling biases (Pollock *et al.* 2004; Kéry & Schmidt 2008; Kéry *et al.* 2009). In particular when studying vocal species, challenges include (i) individuals may be more detectable in acoustically favourable environments (Pacifici *et al.* 2008; Mattsson & Marshall 2009), (ii) individuals periodically become unavailable for detection within a sample unit (Farnsworth *et al.* 2002; Diefenbach *et al.* 2007; Rota *et al.* 2009) and (iii) spatial clustering of survey locations may induce spatial dependence among nearby points (for review see Campomizzi *et al.* 2008).

The first challenge (imperfect detection) can be addressed by simultaneously estimating occupancy and detection probabilities based on repeated detection/non-detection data (MacKenzie *et al.* 2002; Mattsson & Marshall 2009). If unaddressed, variation in detectability can produce misleading inferences regarding species distribution (Williams, Nichols & Conroy 2002; Gu & Swihart 2004). Detection bias has been recognized and addressed in several applications that investigate distributions of species (Wintle *et al.* 2005; O’Connell *et al.* 2006; Bailey *et al.* 2007; Kéry & Schmidt 2008).

A second challenge is that periodic unavailability for detection due to species movement or phenology may violate the closure assumption of occupancy models and may generate biased estimates of patch occupancy (Pollock *et al.* 2004; Kéry & Schmidt 2008). The robust design is a sampling design comprised of nested primary and secondary surveys and allows application of models that account for (i) variation in detectability during each secondary survey and (ii) violation of the closure assumption among primary surveys (Pollock 1982). In addition to providing a means to account for potential biases, the robust design offers an opportunity to distinguish occupancy at two nested scales (Fig. 2). At a coarser scale, we can estimate the probability that a site is usable (i.e. that a species may use the site), which we define here as occupancy (ψ). Given species occupancy at the coarser scale, we can then estimate the probability that a species uses a site during each primary survey, which we define here as use (θ). Taken together, this multi-scale modelling approach allows examination of species distribution at two scales simultaneously.

Multi-scale occupancy models may be fit to detection data collected using the robust design, and they simultaneously provide estimates of occupancy, use and detection (Mordecai 2007; Nichols *et al.* 2008). As such, these models may be particularly useful for investigators that are interested in examining patch occupancy across multiple primary surveys and during each of >2 primary surveys (e.g. days or weeks). Use (i.e. availability for detection) and detection are often separated when estimating local species abundance (for review see Johnson 2008), and such estimates can be provided by generalized Horvitz–Thompson estimators (Pollock *et al.* 2004; Diefenbach *et al.* 2007). In contrast, distinguishing patterns in species distribution (i.e. occupancy) from use and detection has received little attention (Mordecai 2007; Nichols *et al.* 2008).

The third, and perhaps the least addressed, challenge is that survey locations are often clustered and conspecifics may aggregate or occupy areas covering multiple sampling locations, which induces spatial dependence and therefore underestimation of variation among nearby sample units (Sauer, Link & Royle 2005). A solution to this dependence is to apply a random effect that references a coarser, aggregate sampling unit when predicting distribution at finer spatial scales (Royle & Dorazio 2006; Royle *et al.* 2007). Although this may be accomplished through maximum-likelihood estimation and linear mixed modelling, hierarchical Bayes models offer flexible and robust approaches to modelling distributions of species based on sparse detections while accounting for spatiotemporal dependencies and detectability (Royle & Dorazio 2008, pp. 106–124). Applying a hierarchical Bayes approach to multi-scale occupancy models offers a robust and extensible solution for dealing with multiple challenges of studying nested patterns of distribution or resource use by mobile or episodic species.

Here, we describe a multi-scale site occupancy model that integrates existing occupancy modelling approaches by simultaneously estimating site occupancy (ψ) and use (θ) while accounting for detectability (*P*) and spatial dependence through the use of random effects. In particular, this model addresses challenges to studying episodic or mobile species by employing a Bayesian state-space modelling approach and is an extension of existing multi-scale occupancy models that assume no spatial dependence among sample units (Mordecai 2007; Nichols *et al.* 2008). We first demonstrate that multi-scale occupancy models are a generalization of single-scale occupancy models, and then we describe sampling designs necessary to simultaneously estimate occupancy and temporal or spatial patterns of use while accounting for detectability. We then present an analysis based on bird data collected in southern West Virginia as part of a long-term monitoring programme administered by the National Park Service (NPS). In particular, we examine occupancy and temporal patterns of use by the Louisiana waterthrush *Seiurus motacilla* Vieillot, a riparian obligate passerine, based on catchment area and a measure of benthic macroinvertebrate community composition. Finally, we discuss the importance and potential extensions of hierarchical Bayes multi-scale occupancy modelling for addressing many questions in ecology, management and conservation biology.