Multiseason occupancy models for correlated replicate surveys

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1 Methods in Ecology and Evolution 2014, 5, doi: / X Multiseason occupancy models for correlated replicate surveys James E. Hines 1 *, James D. Nichols 1 and Jaime A. Collazo 2 1 U. S. Geological Survey, Patuxent Wildlife Research Center, Laurel, MD 20708, USA; and 2 U. S. Geological Survey, North Carolina Cooperative Fish and Wildlife Research Unit and Department of Applied Ecology, North Carolina State University, Raleigh, NC 27695, USA Summary 1. Occupancy surveys collecting data from adjacent (sometimes correlated) spatial replicates have become relatively popular for logistical reasons. Hines et al. (2010) presented one approach to modelling such data for single-season occupancy surveys. Here, we present a multiseason analogue of this model (with corresponding software) for inferences about occupancy dynamics. We include a new parameter to deal with the uncertainty associated with the first spatial replicate for both single-season and multiseason models. We use a case study, based on the brown-headed nuthatch, to assess the need for these models when analysing data from the North American Breeding Bird Survey (BBS), and we test various hypotheses about occupancy dynamics for this species in the south-eastern United States. 2. The new model permits inference about local probabilities of extinction, colonization and occupancy for sampling conducted over multiple seasons. The model performs adequately, based on a small simulation study and on results of the case study analysis. 3. The new model incorporating correlated replicates was strongly favoured by model selection for the BBS data for brown-headed nuthatch (Sitta pusilla). Latitude was found to be an important source of variation in local colonization and occupancy probabilities for brown-headed nuthatch, with both probabilities being higher near the centre of the species range, as opposed to more northern and southern areas. 4. We recommend this new occupancy model for detection nondetection studies that use potentially correlated replicates. Key-words: Breeding Bird Surveys, brown-headed nuthatch, correlated replicate surveys, imperfect detection, multiseason models, occupancy Introduction Occupancy modelling has become a widely used approach for drawing inferences about a number of interesting ecological topics including species ranges, range dynamics, specieshabitat associations and multi-species associations. (MacKenzie et al. 2006; Bailey, MacKenzie & Nichols 2014). One of the reasons for the widespread use of these models is their flexibility with respect to a number of aspects of sampling design. Designs can be readily adapted to address a variety of ecological questions and to incorporate a number of logistical constraints. One aspect of logistical flexibility is the nature of the replication required to adequately deal with detection probability (MacKenzie et al. 2006: ). One approach to replication is to select a number of smaller plots within the sample unit and to conduct surveys at these plots. This approach is typically most appropriate when such plots are selected randomly and with replacement (MacKenzie et al. 2006; Kendall & White 2009). However, study logistics may preclude use of this recommended approach to replicate *Correspondence author. jhines@usgs.gov plot selection. In some such cases minimal, or even no, bias is induced (Guillera-Arroita 2011), whereas in other cases, bias can be substantial (e.g. Hines et al. 2010). The approach of searching along trails for sign (footprints, scat) of carnivores and their prey has become common (e.g. Karanth & Nichols 2002), and this approach has been adapted to occupancy modelling by using successive trail segments as spatial replicates (e.g. Karanth et al. 2011). However, this logistical approach yields replicates with spatial correlation, such that local occupancy of adjacent trail segments is likely to be positively correlated. Such correlation can induce a large negative bias in occupancy estimates obtained using standard models (Hines et al. 2010), leading to the need to either abandon such survey designs or else develop models to accommodate the correlation. Models developed to incorporate such correlation include the spatially discrete Markov model of Hines et al. (2010) and the continuous point process model of Guillera-Arroita et al. (2011). The models of Hines et al. (2010) and Guillera-Arroita et al. (2011) were developed for single-season occupancy modelling, the goal of which is to estimate occupancy for one season or point in time. Our primary objective in this Published This article is a U.S. Government work and is in the public domain in the USA.

2 584 J. E. Hines et al. paper is to extend this modelling to include multiple seasons for the purpose of drawing inferences about occupancy dynamics, including estimation of time-specific occupancy and the vital rates that drive changes in this state variable, local probabilities of extinction and colonization. A secondaryobjectiveistoaskwhethermodelswithspatialcorrelation are needed for the modelling of data from the North American Breeding Bird Survey (BBS). Hines et al. (2010) speculated that their model might be useful for BBS data, which are collected from 50 discrete points that occur along roads used as travel routes, and we wanted to assess the degree to which this was true, at least for one species. Another secondary objective is to use this BBS analysis to test hypotheses about occupancy dynamics for a focal species, brown-headed nuthatch. A final third objective is to introduce a new parameterization to deal with uncertainty that may accompany the treatment of the initial replicate plot in trail- or road-based surveys. Materials and methods MODEL DEVELOPMENT Single-season model The basic sampling situation entails sampling s sample units. At each sample unit, spatial replicates (e.g. trail segments, sample points along a road) are surveyed sequentially, beginning at one segment or survey point on the trail or road, conducting surveys at each subsequent segment or point and ending at a final segment or survey point (denoted as replicate K), after which no more surveys are conducted. The singleseason model of Hines et al. (2010) was developed to deal with the potential for positive correlation between local occupancy of successive segments or points (spatial replicates). This model first divided the detection process into two components: (1) animal presence on a given replicate (referred to as local presence) and (2) animal detection, given local presence. The first component was then modelled as a first-order Markov process with parameters: h ij = Pr(species present on replicate j sample unit i occupied and species not present on previous replicate j 1); h 0 ij = Pr(species present on replicate j sample unit i occupied and species present on previous replicate j 1). For the situations that motivated model development, h < h 0. The second component of the detection process is denoted as: p ij = Pr(detection at replicate j sampleuniti occupied and species locally present on replicate j). This detection parameter should not be confused with the detection probability of standard occupancy modelling which represents the product of the probabilities of local presence and detection given local presence. Inference will typically not be possible under the most general parameterization of this detection process model (h ij, h 0 ij, p ij ). Instead, replicate-specific parameters may be modelled as functions of replicatespecific covariates or else treated as constants across all replicates (Hines et al. 2010). Single-season detection histories can be modelled using the above parameters. Hines et al. (2010) noted that the modelling of the initial spatial replicate was problematic, in the sense that there was no information from a prior replicate (j 1). If the first replicate in the sequence represented a natural starting point (e.g. a trailhead), then they recommended modelling local presence for this replicate using h ij. If the first replicate in the sequence was not a natural starting point, they proposed the ad hoc approach of modelling local presence for the initial replicate as a finite mixture, with mixture probability defined as the equilibrium of the Markov spatial process defined by the h ij and h 0 ij. Subsequently, we realized that we did not need to specify the mixture parameter, but could incorporate it into the model and estimate it directly. We have thus modified the initial model of Hines et al. (2010) by defining the following parameter: p i = Pr(species local presence in unsurveyed replicate 0 sample unit i occupied). Unsurveyed replicate 0 simply refers to the spatial replicate that would have preceded replicate 1, had the surveys had one additional replicate at the beginning of the sequence. In order to illustrate this single-season model, consider the modelling of the following detection history for sample unit i, consisting of four replicates or stops, 0101: Prðh i ¼ 0101Þ¼w½ð½ph 0 þð1 pþhšð1 pþh 0 þ½pð1 h 0 Þ þð1 pþð1 hþšhþp½ð1 h 0 Þhþh 0 ð1 pþh 0 ŠpŠ eqn1 We dropped the subscripts specifying sample unit and replicate for ease of presentation. The appearance of a 1 in the above detection history indicates occupancy of the sample unit and thus the need for the initial occupancy parameter. Additive terms in the above expression represent uncertainty about local presence that is modelled using mixtures that account for both possibilities (locally present or not). Detection histories of all 0 s indicate uncertainty about sample unit occupancy and require mixture modelling of sample unit occupancy as well. For example, the probability associated with detection history h i = 00 can be written as: Prðh i ¼ 00Þ¼ð1 wþþwfð½ph 0 þð1 pþhšð1 pþh 0 þ½pð1 h 0 Þ þð1 pþð1 hþšhþð1 pþþ½ph 0 þð1 pþhšð1 pþð1 h 0 Þ þ½pð1 h 0 Þþð1 pþð1 hþšð1 hþg eqn2 The first term in this expression represents the probability that the sample unit is unoccupied. The remainder of line 1 includes the probability that the unit is occupied, the uncertainty about local presence at replicate 1 and the probability that replicate 2 is occupied. The second line includes uncertainty about local presence at replicate 1, as well as the probability that replicate 2 is not occupied. Basedontheabove,wecanwritethe likelihood for this new singleseason model as the product of probabilities associated with the detection histories of all sample units: Lðfw i g; fh ij g; fh 0 ij g; fp ig; fp ij gjh 1 ; h 2 ; ; h s Þ¼ Ys i¼1 Prðh i Þ eqn 3 The above likelihood is too general for inference, but a variety of model constraints, including covariate modelling and constancy across spatial replicates, yields model structures that are useful for inference. With the exception of the new p parameterization, the basic model above had been tested via simulation and performed acceptably (Hines et al. 2010). Parameter estimates can be obtained either via maximum likelihood (as implemented in PRESENCE, Hines 2006) or via MCMC (Hines et al. 2010). A small simulation study was conducted to test the efficacy of the new mixture model parameterization for the initial spatial replicate (Appendix S1).

3 Multiseason models for correlated replicates 585 Multiseason model MacKenzie et al. (2003) described a basic Markov model for occupancy dynamics focusing on changes in occupancy status of a site over seasons or years. This model requires two kinds of parameters in addition to those needed for their single-season model (MacKenzie et al. 2002), probabilities of local extinction and local colonization. Define these parameters as follows: e i,t = Pr (sample unit i not occupied in season t+1 occupied in t), c i,t =Pr(sampleuniti occupied in season t+1 not occupied in t), where seasons are separated by sufficient time that changes in occupancy status are possible (often sampling seasons will occur each year). The underlying process model for community dynamics can be written as follows (e.g. MacKenzie et al. 2003, 2006): w i;tþ1 ¼ w i;t ð1 e i;t Þþð1 w i;t Þc i;t eqn 4 Under this model, the probability that a site i is occupied at time t + 1 is written as the sum of two components: the product of probability of persistence (complement of extinction) and probability of occupancy at time t, and the product of probability of colonization and the probability that the site was not occupied at time t. The data underlying a multiseason model are simply the detection histories for multiple seasons. For example, h i = uses the two single-season detection histories of eqns 1 and 2 above. The initial 0101 indicates that four segments (spatial replicates) were visited for site i in the first year, 1, with detections occurring on the second and fourth segments. The subsequent 00 indicates that only two segments for site i werevisitedinyear2,withnodetectionsoneither visit. The modelling of this detection history in a 2-season model based on process model (eqn 4) is given below, including only subscripts for year: Prðh i ¼ Þ ¼w 1 ½½p 1 h 0 1 þð1 p 1Þh 1 Šð1 p 1 Þh 0 1 þ½p 1ð1 h 0 1 Þ þð1 p 1 Þð1 h 1 ÞŠh 1 Þp 1 ½ð1 h 0 1 Þh 1 þ h 0 1 ð1 p 1Þh 0 1 Šp 1Š ½e 1 þð1 e 1 Þfð½p 2 h 0 2 þð1 p 2Þh 2 Šð1 p 2 Þh 0 2 þ½p 2ð1 h 0 2 Þ þð1 p 2 Þð1 h 2 ÞŠh 2 Þð1 p 2 Þþ½p 2 h 0 2 þð1 p 2Þh 2 Šð1 p 2 Þ ð1 h 0 2 Þþ½p 2ð1 h 0 2 Þþð1 p 2Þð1 h 2 ÞŠð1 h 2 ÞgŠ eqn 5 The first component (lines 1 and 2) of eqn 5 simply repeats the model of the detection process for the first year (eqn 1). This history includes two detections, so we know the site was occupied in year 1. There are two possible true states in year 2, unoccupied or occupied but not detected. The e 1 (in line 3 above) is the probability that the site is unoccupied in year 2, given that it was occupied in year 1. The (1 e 1 )specifies the alternative probability that the site was indeed occupied in year 2. The terms following (1 e 1 ) repeat much of eqn 2 and specify the probability of no detections on the two segments in year 2, given that thesitewasoccupied. Inference is thus based on the set of site-specific detection histories for all sampled sites (the data), and the probabilities associated with each of these histories (e.g. eqn 5). The detection histories differ from those of the single-season modelling in having multiple years of detection data, and the modelling requires the two additional parameters reflecting local extinction, e t, and colonization, c t. The likelihood is obtained in the same manner as for the single-season model, by multiplying all of the probabilities for the observed detection histories: Lðfw it g; fe it g; fc it g; fh ijt g; fh 0 ijt g; fp itg; fp ijt gjh 1 ; h 2 ;;h s Þ¼ Ys i¼1 Prðh i Þ eqn 6 As is the case for most of the existing occupancy models, the models based on (eqn 6) can deal with missing observations and, as in the example, different numbers of replicate segments in different sites and years. Parameters associated with both the ecological and detection processes can also be modelled as functions of site- and timespecific covariates (as in MacKenzie et al. 2006). A small simulation study confirmed that the above model performs adequately, in the sense of producing approximately unbiased estimates (Appendix S1, Table S1). EXAMPLE OF APPLICATION: BROWN-HEADED NUTHATCH IN THE SOUTH ATLANTIC COASTAL PLAIN Study area The study area spanned across the South Atlantic Coastal Plain, specifically the South Atlantic Migratory Bird Initiative region, henceforth SAMBI (latitudes North; Fig. 1). The northern portion of this region is dominated by longleaf pine (Pinus palustris) and loblolly pine (Pinus taeda). The southern SAMBI is in a transitional zone of northeastern Florida where tropical mangroves meet coastal plain plant communities; this region is dominated by pond pine (Pinus erotina) and Atlantic white cedar (Chamaecyparis thyoides; Watson & McWilliams 2005). We adopted the spatial extent defined by Iglecia, Collazo & McKerrow (2012), which extended the SAMBI boundary by 30 km to avoid edge effects and to encompass the habitats surrounding BBS routes that straddle the limits of the region. The SAMBI is bounded on its western edge at the fall line where alluvial and marine sediments meet the exposed continental bedrock of the piedmont region (Hupp 2000), and on the eastern edge by the Atlantic Ocean. The total area of the buffered SAMBI is ha with ha of water and elevation ranges from 0 m to 219 m (Iglecia, Collazo & McKerrow 2012). Avian survey data and focal species We used North American Breeding Bird Survey (BBS) data from 1997 to 2007 to illustrate application of the analytical framework described in this paper. The BBS is a national, standardized, volunteer-based survey, with over 4000 routes sampling breeding birds across North America each year (Flather & Sauer 1996). Each survey is a ~394 km long route composed of 50 stops spaced ~08 km apart. Observers conduct three-minute point counts at each stop, recording all birds seen and heard within a 400 m radius. By treating the stops as spatial replicates, the area sampled is the sum of the 50 circles centred at the stop points and of 400 m radius. We included 120 of 130 possible routes in the analysis. We included routes in the analyses if (1) route lengths were within 10% of the standard 394 km length when created (i.e. largely unchanged) and (2) routes did not overlap other active routes (Iglecia, Collazo & McKerrow 2012). We used data from the brown-headed nuthatch (hereafter nuthatch or BHNU) to assess the utility of the correlated spatial replicates model for BBS data and to test predictions about rate parameters of the species along the latitudinal gradient in SAMBI. The nuthatch is an endemic, resident species in the south-eastern United States (Slater et al. 2013). It ranges from eastern Texas to Florida and northward to the southern tip of Delaware. Its range in

4 586 J. E. Hines et al. Fig. 1. Map of the South Atlantic Bird Initiative region (solid black line) in the South Atlantic Coastal Plain, south-eastern United States. The region was divided into three geographic bands: south (Latitude ), central (Latitude ), and north (Latitude ) to test hypotheses about occupancy dynamics of the brown-headed nuthatch in the region. We used multiseason occupancy models and 11 years of the North American Breeding Bird Survey data. SAMBI hits its northern boundary in southern Virginia. Nuthatches are most abundant in pine forests, especially open longleaf pine savannas in the coastal plain, predominant in the northern portion of SAMBI (Hamel 1992). This passerine excavates nests in the cavities of snags, dead pine trees and some deciduous trees. Increased fire suppression leads to understorey growth, making habitat unsuitable forthespecies(slateret al. 2013). Hypotheses and predictions Predictions were based on general principles of population dynamics and expectations about rate parameters given the location of the species within this region (Brown 1984; Lawton 1993). The relatively narrow, strip-like shape of the SAMBI provided a unique opportunity to formulate such predictions because the study area spanned over eight degrees of latitude, including the northern (latitude 37 N) boundary and close to the southern boundary (latitude 29 N) of the BHNU, within the same physiographic region (South Atlantic Coastal Plain; Fig. 1). This geographic setting allowed us to ask questions about the species rate parameters in the northern (35 37 N), southern (29 31 N) and central portions (32 34 N) of the SAMBI. The number of routes within each of these latitudinal regions of SAMBI was roughly the same (Iglecia, Collazo & McKerrow 2012). Our general prediction (e.g. Brown 1984; Lawton 1993; Brown, Stevens & Kaufman 1996; Gaston 2000) for mean route occupancy ( w ) was that it would be higher in the central portion of the species range (C) relative to the northern (N) and southern (S) portions of its distribution in SAMBI ( w C [ w N ; w S ). We also expected that route extinction probabilities would be higher at the southern and northern portions of the species distribution than in the core centre (e N ;:e S [ e C ) because of both the likelihood of suboptimal environmental conditions, and lower occupancy and resulting reduced sources of potential immigrants for a rescue effect (Brown & Kodric-Brown 1977). Lastly, we predicted that route colonization rates would be highest at the central portion of the species distribution (c C [ c N ; c S ). The primary basis for this prediction is the existence of more potential colonists in regions with higher occupancy (Hanski 1999; Bled, Royle & Cam 2011; Yackulic et al. 2012). These predictions about vital rates are also based on general ideas about greater variation and dynamics near edges of a species range than near its range centre (e.g. Gaston 2000; Doherty, Boulinier & Nichols 2003; Karanth et al. 2006). Although our use of three discrete latitude classes was best suited for addressing our hypotheses, other modelling approaches are possible. For example, we could have included route latitude as a continuous covariate and fit quadratic models to the parameters for which we had hypotheses that contrasted range centre vs. periphery. Modelling The BBS data for each route were modelled by treating each of the 50 stops as a geographic replicate at which BHNU was either detected (detection history entry 1) or not (detection history entry 0).

5 Multiseason models for correlated replicates 587 Our initial question was whether these BBS data would require the model with correlated replicates, or instead whether the standard models assuming independent replicates would be adequate. Note that spatial independence of replicates (standard models) can be viewed as a constraint on our Markov model, such that h ijt = h 0 ijt. Thus, our initial model set included standard multiseason models (MacKenzie et al. 2003, 2006) as well as models using the correlated replicates structure presented here. The new models contain a number of model parameters, and each has the potential to be modelled in multiple ways with respect to covariates and potential sources of variation. Our pragmatic approach to dealing with a very large number of potential models was to first specify likely structures for both sampling and ecological processes. We then used model selection (Burnham & Anderson 2002) to resolve our uncertainty in this step and to address the fundamental question about need (or not) for the correlated replicate models for this data set. Following this initial model selection, we resolved to take the best model(s) from this effort and to then explore alternative sources of variation in parameters governing the detection processes. If this step did not lead us to modify the best model(s) identified in the initial model set, then we would return to this model(s) and explore alternative sources of variation in parameters governing the ecological dynamic processes. We make no claim that this approach is optimal or even good, only that it seems reasonable and permits the revisiting of tentative decisions made in the initial step of the process. As explained above, our hypotheses about ecological processes focused primarily on latitudinal variation among the three geographic bands. Multiseason occupancy models include the occupancy for the initial year of study as a parameter, and we expected this parameter to vary as a function of latitude (model notation, w 97 (l)). We expected probabilities of colonization and/or extinction to vary by latitude as well, but this was a key uncertainty. Thus, we included in the initial model set four possible ways to model extinction and colonization. Both vital rates could vary with latitude (e (l), c (l)), only one might vary ((e (l), c (.)) or (e (.), c (l))), or neither (e (.), c (.)). We expected the probabilities of local presence on the spatial replicates to be higher in the centre of the BHNU range and smaller to the north and south, so we modelled these parameters as a function of latitude as well (model notation, h(l), h 0 (l)). We modelled conditional (on local presence) detection probability as a function of both year and latitude. We expected the range centre to contain higher BHNU abundances (e.g. Sagarin & Gaines 2002) and thus detection probabilities (Royle & Nichols 2003). Evidence of temporal variation in detection probabilities of individual birds based on past BBS studies (e.g. Sauer, Peterjohn & Link 1994; Link & Sauer 1998) led us to believe that this source of variation would be relevant for us as well. However, we were uncertain about the joint operation of both latitude and annual effects so explored both additive (p (l+t)) and interactive (p (l 9 t)) models. We knew from use of this data set to develop the new p parameterization for single-season modelling that this parameter was estimated to be near 0 for these data, so we included neither time nor latitude as possible sources of variation. However, in order to deal with possible variation in local occupancy of initial stops on the various routes, we did fit a model with a separate p parameter for each route (model notation p (i)). Our initial model set thus included models with these sources of variation, as well as corresponding analogues of standard multiseason models with no parameters to deal with correlation among replicates. As noted above, subsequent model sets explored different sources of variation for model parameters dealing with sampling and process variation, respectively. All models were fit using maximum likelihood implemented in programme PRESENCE (Hines 2006). Results Based on the initial model set (Table 1), the low-aic model was model (w 97 (l), h(l), h 0 (l), c(l), ɛ(.), ). Model selection results strongly favoured the models with correlated replicates (summed model weights 1) over the standard occupancy models. The nature of the spatial correlation was as predicted, with ^h 0 [ ^h (Table 2). Local (replicate level) occupancy probabilities tended to be higher in the central latitude band as predicted. However, conditional detection probabilities were not always higher for routes in the central latitude band. There was substantial support for models in which colonization was a function of latitude (summed weights = 079). As predicted, colonization probabilities were higher in the central latitude band (range centre, Table 2). There was virtually no support for models in which extinction was also a function of latitude (summed weights < 001). In addition, the point estimates of local extinction under models that did include latitude did not follow predictions, as probabilities of extinction for the three latitudinal bands were not only similar, but were not consistently lower in the central area. Estimates of initial and subsequent occupancy were larger in the central area as predicted (Table 2, Fig. 2). The second model set (Table 3) used the top model from the initial model set (Table 1) as a basis for exploring various more Table 1. Model selection results for our apriorimodel set for brownheaded nuthatch in the SAMBI region of the South Atlantic Coastal Plain. This initial model set explores latitudinal variation in the vital rates, latitudinal and temporal variation in detection probabilities, and the need for models with correlated replicates. AIC is Akaike s information criterion, DAIC is the difference between the AIC value of a focal model and the low-aic model in the set, K is the number of model parameters and 2Loglik is the negative of twice the logarithm of the likelihood function evaluated at the maximum likelihood estimates Model AIC DAIC K 2Loglik w 97 (l), h(l), h 0 (l), c(.), e(.), w 97 (l), h(l), h 0 (l), c(.), e(.), w 97 (l), h(l), h 0 (l), c(l), e(l), w 97 (l), h(l), h 0 (l), c(.), e(l), w 97 (l), h(l), h 0 (l), c(.), e(l), w 97 (l), h(l), h 0 (l), c(l), e(l), w 97 (l), c(l), e(.), p(l + t) w 97 (l), c(.), e(.), p(l + t) w 97 (l), c(.), e(l), p(l + t) w 97 (l), c(l), e(l), p(l + t) w 97 (l), c(.), e(.), p(l 9 t) w 97 (l), c(.), e(l), p(l 9 t) w 97 (l), c(l), e(l), p(l 9 t) w 97 (l), c(l), e(.), p(l 9 t)

6 588 J. E. Hines et al. and less general models of parameters associated with the detection process. None of these models appeared to provide any improvement over the better models of Table 1. A third Table 2. Parameter estimates (SE) for brown-headed nuthatch in the SAMBI region of the South Atlantic Coastal Plain under model (w 97 ðlþ; hðlþ; h 0 ðlþ; cðlþ; eð:þ; pðl tþ; pð:þþ:w l 97 is the occupancy in the initial study year, 1997, for sample units in latitude band l, h l is the probability of local occupancy for a replicate in latitude band l not preceded by a locally occupied replicate, h 0l is the probability of local occupancy in latitude band l for a replicate preceded by a locally occupied replicate, p is the probability that the initial replicate is preceded by an occupied local area, c l is the colonization probability for unoccupied sample units in latitude band l, e is the local extinction probability for occupied sample units (similar for all latitude bands) and p l t is the conditional (on local occupancy) detection probability for a replicate in a sampleunitoflatitudebandl and year t Parameter Latitude band (l) South Central North Estimate (^SE ) Estimate (^SE ) Estimate (^SE) w l (0104) 066 (0085) 046 (0145) h l 005 (0007) 007 (0007) 006 (0009) h 0l 092 (0014) 094 (0011) 084 (0039) p <001 ( ) <001 ( ) <001 ( ) c l 027 (0059) 032 (0056) 014 (0047) e 009 (0017) 009 (0017) 009 (0017) p l (0046) 017 (0024) 010 (0049) p l (0026) 013 (0017) 017 (0053) p l (0027) 011 (0016) 020 (0054) p l (0029) 017 (0019) 024 (0065) p l (0018) 019 (0023) 018 (0046) p l (0019) 019 (0020) 013 (0034) p l (0023) 014 (0018) 009 (0030) p l (0024) 010 (0014) 010 (0034) p l (0023) 012 (0016) 012 (0039) p l (0025) 015 (0018) 013 (0036) p l (0022) 015 (0019) 022 (0057) model set focused on more and less general models of parameters for the underlying ecological processes (Table 4). One of these models, (w 97 (.), h(l), h 0 (l), c(l), ɛ(.), ), had a smaller AIC than the top model of the first model set. Under this model, the initial occupancy was modelled as a constant rather than as varying by latitude. However, as noted above, the point estimates of initial occupancy were larger for the central area as predicted (Table 2). Therefore, we have a slight preference for the top model of set 1 (Table 1) and decided to retain it for inference. Inferences from the two models are virtually identical. Discussion We have presented a new model that extends the correlated replicates model of Hines et al. (2010) for single seasons to deal with data from multiple seasons. The modelling leads to more complex expressions than under standard multiseason models (MacKenzie et al. 2003, 2006) for probabilities associated with detection histories (e.g. eqn 5). However, our modelling permits use of those designs that rely on correlated (typically spatial) replicates to draw inferences not only about occupancy but also about probabilities of local extinction and colonization, the vital rates underlying occupancy dynamics. Our use of the single-season model of Hines et al. (2010) as the starting point for this work is based on our familiarity with this model. It should be possible to extend the continuous point process model of Guillera-Arroita et al. (2011) to deal with data from multiple seasons as well. This modelling introduces what we believe to be an improvement to the model of Hines et al. (2010) for single-season data as well. This improvement involves the initial spatial replicate of the correlated sequence and the uncertainty about how to model its local occupancy in the absence of information from a preceding spatial replicate. Rather than using the ad hoc Fig. 2. Annual estimates of occupancy probabilities and approximate 95% confidence intervals of brown-headed nuthatch for BBS routes within three geographic bands: south (Latitude ), central (Latitude ), and north (Latitude ) in the South Atlantic Bird Initiative region in the South Atlantic Coastal Plain, south-eastern United States. Estimates were obtained using 11 years of Breeding Bird Survey data ( ) and multiseason occupancy models that account for spatial correlation among the 50 discrete sampling points along each BBS route (n = 120 routes).

7 Multiseason models for correlated replicates 589 Table 3. Model selection results for brown-headed nuthatch in the SAMBI region of the South Atlantic Coastal Plain. This model set explores alternative (with respect to Table 1) sources of variation in the parameters associated with the sampling process. AIC is Akaike s information criterion, DAIC is the difference between the AIC value of a focal model and the low-aic model in the set, K is the number of model parameters and 2Loglik is the negative of twice the logarithm of the likelihood function evaluated at the maximum likelihood estimates Model AIC DAIC K 2Loglik p(t), p(.) w 97 (l), h(.), h 0 (.), c(l), e(.), p(.), p(.) w 97 (l), h(l + t), h 0 (l + t), c(l), e(.), w 97 (l), h(t), h 0 (t), c(l), e(.), p(l), p(.) p(l 9 t), p(i) w 97 (l), h(l 9 t), h 0 (l 9 t), c(l), e(.), Table 4. Model selection results for brown-headed nuthatch in the SAMBI region of the South Atlantic Coastal Plain. This model set explores alternative (with respect to Table 1) sources of variation in the parameters associated with the underlying ecological process. AIC is Akaike s information criterion, DAIC is the difference between the AIC value of a focal model and the low-aic model in the set, K is the number of model parameters and -2Loglik is the negative of twice the logarithm of the likelihood function evaluated at the maximum likelihood estimates Model AIC DAIC K -2Loglik w 97 (.), h(l), h 0 (l), c(l), e(.), w 97 (l), h(l), h 0 (l), c(.), e(.), w 97 (.), h(l), h 0 (l), c(l), e(t), w 97 (l), h(l), h 0 (l), c(l), e(l + t), w 97 (l), h(l), h 0 (l), c(l + t), e(.), w 97 (.), h(l), h 0 (l), c(t), e(.), w 97 (l), h(l), h 0 (l), c(l 9 t), e(.), w 97 (l), h(l), h 0 (l), c(l), e(l 9 t), w 97 (l), h(l), h 0 (l), c(.), e(l), w 97 (l), h(l), h 0 (l), c(l), e(l), approach of equating the local occupancy of the initial replicate with the equilibrium value of the Markov process (Hines et al. 2010), we directly estimate this probability as a mixture of the probabilities of local occupancy for replicates that are andarenotprecededbyanoccupiedreplicate.simulation results indicate that this approach works well. Although the initial modelling for correlated spatial replicates was motivated by field studies based on surveys of animal sign along trails (Hines et al. 2010; Karanth et al. 2011), we recognized that other kinds of sampling might require such models as well. The North American Breeding Bird Survey uses a design in which bird surveys are conducted at stops along roads, and we suspected that local occupancy might be correlated for adjacent stops. This suspicion was based primarily on the suspected correlation of habitat on successive stops and on basic ideas about bird movement among stops. Results of our modelling of BHNU in the south-eastern United States provided strong evidence of the need for models that do incorporate such spatial correlation. Our inferences do not extend beyond this species, but we believe it wise for testing purposes to include some models for correlated replicates in model sets used for other occupancy analyses based on BBS data. As noted above, there are multiple hypotheses about processes underlying a Markov structure (h, h 0 ) for local occupancy, and one of these involves correlations between habitats of adjacent survey stops. A preferable approach to dealing with this hypothesis would entail use of stop-specific habitat covariates for modelling either the detection parameters in standard occupancy models or the h and h 0 in our models. Such modelling would lead to greater ecological interest in these local occupancy parameters and their interpretation. In the absence of detailed habitat data, it is also possible that temporal variation in the local occupancy parameters (h t, h t 0 ) could be used to test ideas about changes in preferred habitat over time. For example, if correlated habitat along routes was the primary process responsible for spatial correlation in local occupancy, then a substantive decrease in suitable habitat throughout a study region would lead to the expectation of increases in h t and decreases in h t 0 through time, a hypothesis that could be tested using our models. We could also compute equilibrium local occupancy, h t *, the probability that a randomly selected stop will be locally occupied, as a statistic reflecting general habitat suitability along the route: h t ¼ h t h t þð1 h 0 t Þ : Again, we would prefer to test such ideas more directly using habitat data (e.g. MacKenzie et al. 2011), but simply note that our new models might be useful in the absence of habitat data. Ouranalysespermittedustodrawinferencesaboutvariation in vital rates associated with latitude and position within a species range. Specifically, we predicted that the latitudinal band in the centre of the species range would be characterized by higher abundances of BHNU within occupied areas, higher

8 590 J. E. Hines et al. occupancies, higher rates of colonization and lower rates of local extinction. The first prediction about abundances cannot be addressed with our analyses, but the other predictions were addressed. The probability that a sample unit (BBS route) was occupied by BHNU during the initial year of the analyses was estimated to be larger for the central area (066) than for the northern and southern areas (046) (Table 2). Subsequent occupancies were similarly larger for the central area (Fig. 2). Colonization probabilities were larger in the central area than in either the northern or southern areas (Table 2), as predicted. This result supports the mechanistic ideaofmorepotentialcol- onists in regions with higher occupancy (Hanski 1999; Bled, Royle & Cam 2011; Yackulic et al. 2012). However, our predictions of increased extinction probabilities in the northern and southern areas were not corroborated. In retrospect, we note that previous studies that have supported this prediction (e.g. Doherty, Boulinier & Nichols 2003; Karanth et al. 2006) defined edge areas that were not as wide as our geographic bands, but we do not know whether this difference is responsible for the differing results. In summary, our look at occupancy dynamics of BHNU within three geographic bands leads to the conclusion that occupancy of sample units in the latitudinal centre of the study area was higher than that of units in more northern and southern areas. Colonization probabilities of unoccupied sample units in this central area were higher than for unoccupied units in northern and southern units. We found no evidence of variation among the three geographic bands in local extinction. These findings about occupancy and colonization are consistent with basic ideas about greater occupancy providing more sources of colonists and leading to higher colonization, and about higher colonization leading to higher equilibrium occupancy (w ¼ c c þ e Acknowledgements, e.g. MacKenzie et al. 2006). We thank the USGS Patuxent Wildlife Research Center, North Carolina Cooperative Fish and Wildlife Research Unit and the Department of Applied Ecology at North Carolina State University for their support (Research Work Order 194). We also thank Steve Williams for his assistance in obtaining Breeding Bird Survey data and map of the South Atlantic Coastal Plain. We thank Krishna Pacifici, John Sauer and Sophie Veran for useful discussions. We also thank the associate editor and two anonymous reviewers for helpful comments and suggestions. Any use of trade, product or firms names is for descriptive purposes only and does not imply endorsement by the US government. 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9 Multiseason models for correlated replicates 591 Slater, G.L., Lloyd, J.D., Withgott, J.H. & Smith, K.G. (2013) Brown-headed Nuthatch (Sitta pusilla). The Birds of North America Online (ed. A. Poole). Cornell Lab of Ornithology, Ithaca; Retrieved from the Birds of North America Online: Watson, C. & McWilliams, K. (2005) The South Atlantic Migratory Bird Initiative- An integrated approach to conservation of All Birds Across All Habitats. USDA Forest Service General Technical Report. Yackulic, C.B., Reid, J., Davis, R., Hines, J.E., Nichols, J.D. & Forsman, E. (2012) Neighborhood and habitat effects on vital rates: expansion of the barred owl in the Oregon Coast Ranges. Ecology, 93, Supporting Information Additional Supporting Information may be found in the online version of this article. Appendix S1. Simulation study assessing bias and mean squared error of model estimators. Received 4 January 2014; accepted 10 March 2014 Handling Editor: Olivier Gimenez