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2. We evaluated recently developed spatially explicit capture–recapture (SECR) models using data from a common large carnivore, the American black bear Ursus americanus, obtained by remote sampling of 11 geographically open populations. These models permit direct estimation of population density from C–R data without assuming geographic closure. We compared estimates derived using this approach to those derived using conventional approaches that estimate density as /.
4.Synthesis and applications. We conclude that the higher densities estimated as / compared to estimates from SECR models are consistent with positive bias due to edge effects in the former. Inflated density estimates could lead to management decisions placing threatened or endangered large carnivores at greater risk. Such decisions could be avoided by estimating density by SECR when bias due to geographic closure violation cannot be minimized by study design.
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Effective conservation and management of wildlife populations requires reliable estimates of population density, but the available estimators rely on assumptions that are seldom met in studies of wild populations. Densities of large carnivores are frequently estimated by dividing capture–recapture (C–R) estimates of abundance by the area sampled. The conversion from abundance to density relies on an assumption that is especially troublesome: that the area occupied by the sampled population is well-defined and known.
Large carnivores are difficult to enumerate because they range widely, occur at low densities, exhibit heterogeneous capture probabilities, and are often secretive or elusive (Garshelis 1992; Karanth 1995; Boulanger et al. 2004). This situation has improved through advances in remote identification from photographs or genetic samples (Karanth 1995; Woods et al. 1999) that enable researchers to obtain C–R data quickly while avoiding some of the problems associated with live-capture. However, conventional C–R estimators provide estimates of abundance (), not population density (), and N is a biologically relevant parameter only when the sampled population occupies a known, discrete area (Parmenter et al. 2003). must be divided by the area sampled to obtain the biologically relevant parameter , but for geographically open populations is overestimated relative to the area of the trap array because animals with only part of their home range within the array are available for capture (White et al. 1982). This form of positive bias, termed ‘edge effect’ (Dice 1938), remains a major obstacle to enumeration of large carnivore populations (Karanth et al. 2006; Kendall et al. 2008). To correct for edge effect, a boundary strip of width W can be included in an estimate of the effective trap area (; Dice 1938). W should approximate the distance animals at risk of capture move from the trap array during normal movements (White et al. 1982; Parmenter et al. 2003). However, most trap-revealed movements are underestimates because they are truncated at trap locations. Accurate can be obtained if telemetry data are available for marked animals in the study. Indeed, studies combining C–R with telemetry data for large carnivores demonstrate that where populations are not geographically closed, densities estimated from capture data alone are positively biased (Garshelis 1992; Soisalo & Cavalcanti 2006; Dillon & Kelly 2008). However, the need to instrument large numbers of animals to obtain unbiased from telemetry data prevents many researchers from realizing the benefits of remote sampling in terms of study duration and costs.
Efford (2004) presented a method for estimating population density directly from capture data without assuming geographic closure or estimating the area sampled. His spatially explicit capture–recapture (SECR) approach combines C–R and distance sampling (Burnham, Anderson & Laake 1980) methods to estimate three model parameters: the magnitude of the capture probability function (h0), the spatial extent over which capture probability declines (σ), and population density, defined as the intensity of a spatial point process describing the locations of home range centres (Efford 2004). Model parameters were originally estimated by simulation and inverse prediction (Efford 2004); more flexible, maximum likelihood-based estimators have subsequently been developed (Borchers & Efford 2008). Another approach to SECR models was recently developed using a Bayesian hierarchical framework (Gardner, Royle & Wegan 2009; Royle et al. 2009), but we do not evaluate that approach in this paper. Here, we focus on the SECR approach outlined by Efford (2004) and Borchers & Efford (2008).
As inflated density estimates are especially problematic for carnivores that are at risk, a method that reduces the probability of generating inflated estimates could be useful. We evaluated the utility of the SECR estimator for large carnivores compared to more traditional estimators using data from a common large carnivore, the American black bear Ursus americanus (Pallas 1780). We remotely sampled black bears in 11 geographically open populations using barbed-wire hair corrals (Woods et al. 1999), and identified individuals using molecular methods. Densities were estimated both as / and by SECR. Our objective was to compare density estimators in the context of their assumptions, precision and ability to account for biologically relevant forms of capture heterogeneity to identify a defensible method for estimating large carnivore densities from capture data collected on geographically open populations.
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Molecular data indicated from 9 to 36 (= 21) unique females from 69 to 335 (= 155) accepted genotypes in different WMUs. Numbers of recaptures excluding and including within-occasion recaptures at different traps ranged from 3 to 17 and 4 to 31, and averaged 9·5 and 14·5 respectively.
Demographic closure violation was indicated only in WMU 21A (χ2 = 7·973, 3 d.f., P = 0·047); results of component and subcomponent tests showed violation was due to additions and losses between occasions 3 and 4, so we excluded data from the fourth occasion in WMU 21A when estimating N, W and D. We fixed capture probabilities on the fourth occasion in WMU 21A at zero in the C–R analysis, removed data from the fourth occasion in WMU 21A from the SECR analysis, and excluded distances moved between the third and fourth occasion and on the fourth occasion in WMU 21A from .
The top AICc-ranked C–R model included additive effects of t and h in capture probabilities and accounted for 63% of the total AICc weight (Table 1). The second-ranked model included only h in capture probabilities and yielded similar (Tables 1 and 2). Differences in capture probabilities among WMUs or in response to previous capture were not supported (Table 1). Probabilities of initial capture and recapture from the model with b were 0·28 and 0·24 respectively. The mean maximum distance moved across all animals captured more than once was 3152 m (SE 1576); WMU-specific MMDM varied considerably (Table 2).
Table 1. Model selection results for closed capture–recapture models fit to data for female American black bears on 11 study areas in different Wildlife Management Units (WMU) in Ontario, Canada, 2004 or 2005. In model names, ‘t’ denotes time variation, ‘h’ denotes individual heterogeneity, ‘b’ denotes behavioural response to previous capture and ‘WMU’ denotes study area effects
|t + h||17||107·19||0·00||0·626||184·2|
|t + h + WMU||26||110·97||3·78||0·095||169·0|
|h + WMU||23||111·52||4·33||0·072||176·0|
|t + WMU||24||120·01||12·82||0·001||182·3|
Table 2. Numbers of unique females aged >1 year identified by genotyping [M(t+1)], estimates of abundance () and their standard errors (SE), and mean and SE of maximum distances moved between captures for female American black bears in 11 study areas in different Wildlife Management Units (WMU) in Ontario, Canada, 2004 or 2005
|WMU||M(t+1)||Abundance||Maximum distance moved|
|1st-ranked model||2nd-ranked model|
An SECR model with h affecting both h0 and σ ranked first with 99% of the total AICc weight (Table 3). Estimates of h0 under this model were 0·26 (mixture 1) and 0·06 (mixture 2), and of σ were 1360 m (mixture 1) and 7339 m (mixture 2). was higher and less precise when h in h0 and σ were included in the estimating model (Table 4).
Table 3. Model selection results for spatially explicit capture–recapture models fit to data for female American black bears aged >1 year in study areas in different Wildlife Management Units (WMU) in Ontario, Canada, 2004 or 2005. In model names ‘·’ indicates the parameter was held constant, ‘h’ denotes individual heterogeneity, ‘b’ denotes an effect of previous capture, and ‘WMU’ denotes study area effects
Table 4. Densities (; bears km−2) of female American black bears aged >1 year in 11 Wildlife Management Units (WMU) in Ontario, Canada, sampled in 2004–2005 and derived from different estimators. Densities were estimated as / where was calculated using buffer strip widths estimated as half the mean maximum distance moved (MMDM/2) and the mean maximum distance moved between traps (MMDM) within each study area (WMU-specific), and across all animals. Densities were also estimated from null and AICc-selected spatially explicit capture–recapture (SECR) models. The mean coefficient of variation (CV) across WMUs appears in the bottom row
|WMU||/ (WMMDM/2)||/ (WMMDM)||SECR|
|WMU specific||All animals||WMU specific||All animals||h0(·)σ(·)||h0(h)σ(h)|
|Mean CV||0·22|| ||0·21|| ||0·22|| ||0·21|| || ||0·24|| ||0·32|
Figure 2. Median and range of estimated densities of female American black bears aged >1 year across 11 Wildlife Management Units (WMU) in Ontario, Canada, 2004 or 2005, from different estimators. The different estimators were (from left to right): abundance divided by effective trap area with a boundary strip width equal to (1) half the mean maximum distance moved by individuals caught >1 time (MMDM/2) in each WMU, and (2) MMDM/2 across all animals (pooled); abundance divided by effective trap area with a boundary strip equal to (3) MMDM in each WMU, and (4) MMDM across all animals; (5) a null spatially explicit capture–recapture (SECR) model, and (6) the AICc-selected SECR model with individual heterogeneity in detection parameters.
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For geographically open populations, is poorly defined with respect to the sampled population (White et al. 1982). Studies combining C–R and telemetry data showed that when is estimated as / from capture data alone, is underestimated and overestimated (Garshelis 1992; Soisalo & Cavalcanti 2006; Dillon & Kelly 2008). Estimating and including its area in to correct for edge effect constitutes an ad hoc correction for a violated assumption, which itself relies on additional assumptions about home range sizes, shapes, and degrees of overlap (White et al. 1982; Parmenter et al. 2003; Karanth et al. 2006). Furthermore, it does not account for negative bias in capture probabilities and positive bias in that occur when geographic closure violation causes demographic closure violation because animals are unavailable for capture on a subset of the sampling occasions (Kendall 1999). Densities estimated by SECR here were 20–200% lower than / estimates. We cannot infer bias directly because true densities were unknown; however, the direction of the observed difference is consistent with overestimation of densities estimated as / due to edge effect. We expected severe positive bias due to edge effect in our / estimates because our study areas had high edge: area ratios and were small relative to home ranges of bears. The difference in between spatially explicit and conventional C–R density estimates would probably be less in studies employing large grids of traps. Our trap layout was constrained by the need for vehicle access, and the size of our study areas reflected a trade-off between the size and number of study areas we could achieve with available resources. Elsewhere, trap layouts or study area size may be constrained by costs and logistics (Settlage et al. 2008), trade-offs between potential sources of bias (Boulanger et al. 2004), or the need to set traps along trails used by the study species to maximize capture probabilities (Karanth & Nichols 1998; Ríos-Uzeda, Gómez & Wallace 2007). It is preferable to design field studies to minimize violations of assumptions, but where logistical constraints or characteristics of animals or their habitat preclude using large grids of traps, SECR models are appealing for density estimation because they allow the assumption of geographic closure to be relaxed (Efford 2004; Royle et al. 2009). Prior to the development of SECR models, bias due to geographic closure violation was avoidable only by live-capturing and radio-tagging animals and monitoring their movements (Parmenter et al. 2003; Karanth et al. 2006).
Densities estimated by boundary strip methods were more similar to SECR estimates when was set equal to MMDM. MMDM/2 should theoretically approximate W as half the maximum linear distance of the average home range as recommended by Dice (1938) and was used as the boundary strip width in recent studies of felids and bears (Karanth et al. 2006; Immell & Anthony 2008). However, MMDM, which has no theoretical basis, performed better as an estimator of W in several studies where actual densities or home range sizes were observed (Parmenter et al. 2003; Soisalo & Cavalcanti 2006; Dillon & Kelly 2008). We propose explanations for the apparent superior performance of MMDM as a boundary strip width estimator which nevertheless do not support its general applicability. Home range lengths of voles Microtus montebelli observed from recapture locations were underestimates unless individuals were captured at ≥5 traps (Tanaka 1972). Because large carnivores exist at low densities, researchers may maximize trap spacing to sample more animals over a larger area, exacerbating the underestimation of by truncating measured movements compared to the small mammal studies for which the approach was developed. Further, as defined, MMDM includes zero values when animals are recaptured only at their original capture location (Wilson & Anderson 1985; Karanth & Nichols 1998). When sampling occasions span several days, zero values are likely to reflect failure to detect movement due to imperfect detection and spatially discrete opportunities to detect animals, and contribute to negative bias in . Hence, with wide trap spacing, few recaptures at different traps, and zeros in the data, MMDM may outperform MMDM/2 simply because the theoretically appropriate MMDM/2 more severely underestimates movements during sampling. In our study, most individuals were recaptured at the same trap (27%) or at adjacent traps approximately 2-km apart (38%); only 6% of individuals were captured at traps >6 km apart. MMDM may therefore reflect trap spacing and sampling error rather than approximating mean home range length. Neither MMDM/2 nor MMDM should be expected to approximate well in studies of large carnivores, many of which are characterized by wide trap spacing and few recaptures at different locations. Efford (2004) emphasized that parameter estimates from his SECR models did not depend on trap layout and specified that data from linear arrays were acceptable. Nevertheless, further investigation, including simulations, of effects of trap layout and spacing typical of studies of large carnivores on are warranted.
The SECR models we evaluated avoided the assumption of geographic closure but relied on other assumptions about home ranges, which previous work suggests may not severely bias if violated. For example, SECR models assumed home ranges were circular, but violating this assumption probably affects only the variance of (Efford 2004). Secondly, we assumed capture probabilities decreased with distance according to a half normal distribution and did not compare the fit of other detection functions. We considered this a reasonable assumption because occupied habitat extended well beyond the area of integration so capture probability would not have declined abruptly at any specific distance from the home range centre location. Further, sampling was complete before bears began summer foraging excursions outside their breeding ranges so bears outside the area of integration would have had negligible probabilities of capture. Other detection functions might be more appropriate for species with different movement patterns. For example, the negative exponential model could be used for animals that spend most of their time near a den or nest, or a threshold response applied to species with well-defined territories. In any case, densities estimated from SECR models were robust to the choice of detection function (Efford et al. 2009). Thirdly, we assumed that home range centre locations were randomly distributed. Home range locations are likely to be non-random in heterogeneous habitats like the boreal forest. However, because black bears exhibit mutual avoidance within overlapping home ranges (Schenk, Obbard & Kovacs 1998; Samson & Huot 2001) rather than spacing themselves evenly, randomly distributed home range centre locations may be a reasonable approximation. Finally, we assumed study populations were completely geographically open, and that all bears were able to traverse study area boundaries as there were no significant geographic barriers to movements within our areas of integration. However, where animal movements are constrained, for example by topography, water bodies, or fragmented habitat, areas of integration could include habitat not available to the sampled population, potentially causing underestimation of density. Because discrete approximations of areas of integration can be defined explicitly in SECR models, this assumption can be relaxed where necessary (Borchers & Efford 2008; Royle et al. 2009).
The second reason we preferred SECR was that it accounted for relevant forms of detection heterogeneity. SECR models account for spatially induced individual heterogeneity due to variable exposure to traps, while allowing for additional h in both the magnitude and spatial extent of the detection function (Borchers & Efford 2008). Exposure to traps was probably variable among individuals in our study because most animals were exposed to few traps and had home ranges that included areas outside the trap array. By treating this source of h explicitly, SECR reduces reliance on statistical approaches to accounting for h in C–R data (see Royle et al. 2009:125). We included models with additional h in detection parameters among candidate SECR models because bears exhibit heterogeneous probabilities of capture beyond what can be explained by variable exposure to traps (Noyce, Garshelis & Coy 2001; Boulanger et al. 2004), and home range sizes of female black bears may vary with local differences in habitat quality (Koehler & Pierce 2003), or with age and encumbrance status (Alt et al. 1980; Wooding & Hardisky 1994). Heterogeneous home range sizes cannot be accommodated by conventional density estimators. Generally, where density was estimated as /, N estimators included h in capture probabilities, but was calculated by averaging movements or home range sizes across all individuals (Karanth et al. 2006; Immell & Anthony 2008). Royle et al. (2009) and Gardner et al. (2009) presented density estimates for tigers Panthera tigris and American black bears, respectively, from Bayesian analyses of hierarchical SECR models, but did not evaluate models with h in detection parameters. In a reanalysis of their black bear data, Gardner et al. (in press) observed differences in σ between sexes, but did not evaluate models with h within sexes. Our results, with individual heterogeneity in both h0 and σ strongly supported by AICc, and higher density estimates from the SECR model with h, suggest that candidate models with h in detection parameters should be evaluated even when h induced by variable exposure to traps is treated explicitly using telemetry data or a spatial model.
Behavioural responses to previous capture affected densities estimated by SECR (Borchers & Efford 2008; Gardner et al. in press), but were not supported by AICc in our analyses. The small food reward and the 1-week interval between sampling occasions probably reduced behavioural responses to capture in our study. Time variation was apparent in our C–R data, but had negligible effects on . Local weather data had weak and inconsistent relationships with occasion-specific capture probabilities (M. E. Obbard, unpublished data), and models with sampling-occasion-specific detection parameters were not implemented in density, so we did not evaluate time variation in SECR detection parameters.
For black bears in Ontario, the higher density estimates obtained where geographic closure was assumed could translate into higher, potentially unsustainable harvest levels. Harvest levels could be reduced if population declines were observed or other data indicated that mortality rates were unsustainable, although populations could take many years to recover from overharvest (Miller 1990). The consequences of management decisions based on inflated density estimates for large carnivore populations in other systems could be even more severe. Negative consequences could include local extirpation or underestimation of the minimum reserve size necessary to support a viable population. We recommend using the SECR approach to estimate densities of large carnivores when bias due to the violation of geographic closure cannot be minimized by study design because inflated estimates could lead to management decisions that place threatened or endangered populations at greater risk.