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Keywords:

  • American black bear;
  • carnivore;
  • density estimation;
  • edge effect;
  • geographic closure;
  • spatially explicit capture–recapture;
  • Ursus americanus

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

1. Population density is a critical ecological parameter informing effective wildlife management and conservation decisions. Density is often estimated by dividing capture–recapture (C–R) estimates of abundance (inline image) by size of the study area, but this relies on the assumption of geographic closure – a situation rarely achieved in studies of large carnivores. For geographically open populations inline image is overestimated relative to the size of the study area because animals with only part of their home range on the study area are available for capture. This bias (‘edge effect’) is more severe when animals such as large carnivores range widely. To compensate for edge effect, a boundary strip around the trap array is commonly included when estimating the effective trap area (inline image). Various methods for estimating the width of the boundary strip are proposed, but inline image/inline image estimates of large carnivore density are generally mistrusted unless concurrent telemetry data are available to defineinline image. Remote sampling by cameras or hair snags may reduce study costs and duration, yet without telemetry data inflated density estimates remain problematic.

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 inline image/inline image.

3. Spatially explicit C–R estimates were 20–200% lower than densities estimated as inline image/inline image. AICc supported individual heterogeneity in capture probabilities and home range sizes. Variable home range size could not be accounted for when estimating density as inline image/inline image.

4.Synthesis and applications. We conclude that the higher densities estimated as inline image/inline image 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.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

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 (inline image), not population density (inline image), and N is a biologically relevant parameter only when the sampled population occupies a known, discrete area (Parmenter et al. 2003). inline image must be divided by the area sampled to obtain the biologically relevant parameter inline image, but for geographically open populations inline image 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 (inline image; 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 inline image 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 inline image 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 inline image/inline image 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.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Study area and sampling

We sampled black bears in 11 different Wildlife Management Units (WMU) within the Boreal Forest (Rowe 1972) of Ontario, Canada (Fig. 1). All study areas were comprised of forest stands of varied ages and were characterized by low levels of human disturbance (see Obbard & Howe 2008 for a description of black bear habitat in the boreal forest of Ontario).

image

Figure 1.  Wildlife Management Units in Ontario, Canada, as defined by the Ontario Ministry of Natural Resources. Numbered Wildlife Management Units are those where black bears were sampled in 2004 or 2005. Inset map shows the trap layout in Unit 15B with a 2·5-km concave buffer around traps.

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We sampled bears at 20–25 barbed-wire corrals spaced 2 km apart along secondary roads in each study area in 2004 or 2005. Sampling routes had high edge: area ratios (Fig. 1 inset) so positive bias in inline image due to edge effect was potentially severe (White et al. 1982). We baited traps with three partially opened tins of sardines suspended from a board nailed 2·5 m up a tree >2 m from any point on the wire. Hair samples were collected on four occasions over 47 days from late May through mid-July. On each occasion traps were baited for 1 week, after which samples were collected and baits removed. Hair samples were air-dried in paper envelopes and stored at room temperature. Sampling occasions were separated by a 1-week interval with no bait present, after which any snagged hairs were burned off when traps were re-baited. We deemed our study areas too small to measure movements of male bears because the size of their home ranges could approach that of our trap arrays (Alt et al. 1980; Koehler & Pierce 2003), so we analysed data from females only. We assumed the height of the wire (50 cm) excluded cubs and yearlings from the sample.

DNA analyses

We selected 15–20 hairs with visible roots from each sample when possible (samples with <5 hairs were not analysed). Samples were suspended in 250 μL of 1× lysis buffer, treated with 15 units of proteinase K (>600 U mL−1; Qiagen, Inc., Mississauga, ON, Canada), and incubated at 37 °C for 12 h. A paramagnetic bead automated DNA extraction (Promega MagneSil ONE; Promega Corporation, Madison, WI, USA) protocol (Cullingham, Smeeton & White 2007) was implemented using an Evolution P3 (Perkin Elmer, Waltham, MA, USA) liquid handler, eluting in a final volume of 70 μL.

We amplified 15 microsatellite loci using three multiplex polymerase chain reactions. Loci included G1A, G1D, G10B, G10L, G10C, G10J, G10P, G10X, G10U, G10M (Paetkau & Strobeck 1994; Paetkau et al. 1995); G10H, UarMU59, UarMU05, UarMU50 (Taberlet et al. 1997) and Msut-6 (Kitahara et al. 2000). To identify gender, we amplified a region of the Amelogenin gene (Ennis & Gallagher 1994). One primer of each pair was synthesized with a fluorescent dye group, HEX, 6-FAM, or NED for subsequent detection and analysis on a MegaBACE 1000 capillary sequencer (GE Healthcare, Baie d'Urfe, QC, Canada).

To identify individuals, we initially compared genetic profiles at five microsatellite loci and the Amelogenin locus. Samples that did not amplify at ≥four of six loci were removed from subsequent analyses; samples profiled to ≥four loci were grouped using genecap software (Wilberg & Dreher 2004). Subsequently, when possible, two or three representative samples from each group of identical profiles were profiled with the remaining 10 loci. Samples with a mixture of more than one individual profile were removed from further analysis. genecap was then used to confirm unique individuals from the 15 microsatellite loci and the gender locus. We considered pairs of samples that could not be distinguished from one another after taking two potential allelic dropouts into account to have originated from one bear, and reanalysed samples at specific loci to verify profiles that differed by two alleles that could not be explained by allelic dropout. Finally, we considered that profiles were from unique individuals only when they differed from other profiles by at least two of 16 loci. Positive and negative controls were run at all stages (extraction, amplification and visualization on the genetic analyser), and unknown samples were evaluated in light of results of the two positive controls (2·5 ng and 250 pg of DNA).

Data analysis

We tested the assumption of demographic closure in C–R data using tests described by Stanley & Burnham (1999). We fit eight closed-capture models with all additive combinations of time variation (t), individual heterogeneity (h) and WMU (as a grouping variable) on capture probabilities in Program mark v. 5.1 (White 2008). Two-point mixture distributions were used to model h (Pledger 2000). Traps were baited, so we also fit a model with a behavioural response to previous capture (b) and compared capture and recapture probabilities to check for trap response. We assessed support for different forms of capture heterogeneity and selected models for abundance estimation using Akaike’s Information Criterion adjusted for small sample size (AICc; Hurvitch & Tsai 1989).

We estimated effective trap area as the area of concave buffers of width W around all traps on each study area, and estimated W as the mean of maximum distances moved by all individuals captured more than once (MMDM), and half of MMDM (MMDM/2), on each study area and from pooled data. We calculated inline image in each WMU by dividing inline image from the selected C–R model byinline image calculated using each of the four boundary strip widths. Naïve standard errors for inline image were calculated by dividing standard errors ofinline imagebyinline image.

We also estimated density by maximizing the full SECR likelihood for proximity detectors (Efford, Borchers & Byrom 2009) using density software (v. 4.3; Efford 2008). For numerical integration, the likelihood function was evaluated at 4096 evenly distributed points within a rectangular area extending 10 km around traps. We assumed a half normal spatial capture probability function, and that home range centre locations were Poisson distributed. We initially fit nine candidate models with three forms of variation in h0 (constant, with h, and varying among WMUs) crossed with the same forms of variation in σ. We then fit three additional models with the best-supported model of σ variation and additional forms of h0 variation and compared AICc values across all fitted models. Two-point mixture distributions were used to model h in h0 and σ.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Molecular data indicated from 9 to 36 (inline image= 21) unique females from 69 to 335 (inline image= 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., = 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 inline image.

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 inline image (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
ModelNo. parametersAICcΔAICcwiDeviance
t + h17107·190·000·626184·2
h14109·432·240·205192·7
t + h + WMU26110·973·780·095169·0
h + WMU23111·524·330·072176·0
t + WMU24120·0112·820·001182·3
t15121·7414·550·000202·9
WMU21121·7714·580·000190·4
Null12123·5716·380·000211·0
b13125·2018·010·000210·5
Table 2.   Numbers of unique females aged >1 year identified by genotyping [M(t+1)], estimates of abundance (inline image) 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
WMUM(t+1)AbundanceMaximum distance moved
1st-ranked model2nd-ranked model
inline imageSEinline imageSEMeanSE
5265010·05010·225261301
6285310·65410·838101043
816307·0307·12388566
9A326111·86112·038941700
15A18347·6347·72680328
15B14266·4276·544451478
21A24479·3479·52442740
24265010·05010·238482238
2614266·4276·551052155
279174·8174·8187881
3117327·3327·42262852

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). inline image 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
ModelNo. parametersAICcΔAICcwiDeviance
h0(h)σ(h)162318·0100·9931868·9
h0(WMU)σ(h)252328·1410·130·0061857·1
h0(·)σ(h)152332·7414·730·0011885·9
h0(h+WMU)σ(h)262347·6629·650·0001872·4
h0(h)σ(WMU)252394·5976·580·0001923·5
h0(WMU)σ(WMU)332398·8880·870·0001905·3
h0(h)σ(·)152400·1882·170·0001953·4
h0(·)σ(·)132410·6292·610·0001968·4
h0(b)σ(h)162411·8693·850·0001962·7
h0(·)σ(WMU)232413·3695·350·0001947·3
h0(b+h)σ(h)172415·9797·960·0001964·5
h0(WMU)σ(·)232416·3598·340·0001950·3
Table 4.   Densities (inline image; 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 inline image/inline image where inline image 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
WMUinline image/inline image (WMMDM/2)inline image/inline image (WMMDM)SECR
WMU specificAll animalsWMU specificAll animalsh0(·)σ(·)h0(h)σ(h)
inline imageSEinline imageSEinline imageSEinline imageSEinline imageSEinline imageSE
50·4670·0930·3570·0710·2070·0410·1620·0320·0710·0150·1270·038
60·3100·0620·3900·0780·1410·0280·1750·0350·0780·0160·1370·04
80·3090·0720·2190·0510·1360·0320·1000·0230·0440·0120·0800·027
9A0·3410·0660·4390·0850·1560·0300·1970·0380·0880·0170·1520·043
15A0·2960·0660·2430·0540·1320·0290·1100·0250·0490·0120·0880·029
15B0·1490·0370·2240·0550·0680·0170·1000·0250·0470·0130·0840·030
21A0·4800·0950·3510·0690·2140·0420·1620·0320·0800·0180·1470·045
240·3310·0660·4200·0840·1490·0300·1870·0370·0840·0180·1470·045
260·1000·0250·1760·0430·0460·0110·0790·0190·0360·0100·0620·022
270·2660·0750·1260·0360·1010·0290·0560·0160·0250·0090·0440·018
310·4380·1000·2810·0640·1860·0420·1290·0290·0600·0160·1060·035
Mean CV0·22 0·21 0·22 0·21  0·24 0·32

Densities calculated using MMDM/2 as the boundary strip width were more than double those using MMDM (Fig. 2). Calculating W from pooled rather than WMU-specific data reduced variation in inline image among WMUs but had only a small effect on the central tendency (Fig. 2). inline image from the AICc-selected SECR model was >200% lower, on average, than inline image estimated as inline image/inline image with inline image equal to MMDM/2, and >20% lower than inline image estimated as inline image/inline image with inline image equal to MMDM (Table 4; Fig. 2). Densities estimated by SECR were apparently less precise than those estimated as inline image/inline image (Table 4), but standard errors around inline image/inline image estimates underestimated the actual uncertainty in inline image.

image

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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Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

We advocate using SECR models to estimate densities of large carnivores from capture data collected on geographically open populations for three reasons: (1) they did not rely on the assumption of geographic closure (Efford 2004), (2) they accounted for biologically relevant forms of heterogeneity in both capture probabilities and home range sizes and (3) the variance of inline image included additional forms of uncertainty and process variation compared to where inline image was estimated as inline image/inline image.

For geographically open populations, inline image is poorly defined with respect to the sampled population (White et al. 1982). Studies combining C–R and telemetry data showed that when inline image is estimated as inline image/inline image from capture data alone, inline image is underestimated and inline image overestimated (Garshelis 1992; Soisalo & Cavalcanti 2006; Dillon & Kelly 2008). Estimating inline image and including its area in inline image 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 inline image 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 inline image/inline image 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 inline image/inline image due to edge effect. We expected severe positive bias due to edge effect in our inline image/inline image estimates because our study areas had high edge: area ratios and were small relative to home ranges of bears. The difference in inline image 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 inline image 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 inline image 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 inline image. 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 inline image 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 inline image 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 inline image if violated. For example, SECR models assumed home ranges were circular, but violating this assumption probably affects only the variance of inline image (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 inline image/inline image, N estimators included h in capture probabilities, but inline image 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 inline image. 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.

Finally, we preferred SECR because the variance of inline image is estimated directly from the fitted spatial model (Efford et al. 2009) thereby reflecting all forms of uncertainty and process variation included in the model. Where density is estimated as inline image/inline image, naïve standard errors for inline image are obtained by dividing the SE of inline image by a point estimate of inline image. This assumes inline image is measured without error and overstates the precision of inline image. Naïve standard errors for inline image are currently presented only when the sampling variance of inline image is unknown (Kawanishi & Sunquist 2004; Dobey et al. 2005). More frequently, inline image is estimated from samples of individual movements so the variance of inline image may be calculated using the delta method to include uncertainty in both inline image and inline image (Seber 1982). However, published formulae for the variance of inline image are available only for grids (Wilson & Anderson 1985) and approximately circular study areas (Karanth & Nichols 1998). Approximating our sampled areas as grids or circles was inappropriate, so we presented naïve standard errors for inline image estimated as inline image/inline image. Furthermore, under the assumption of Poisson distributed home range centres the sampling variance of inline image is not conditional on the number of individuals within the area of integration, which is appropriate when sampled populations are geographically open. This variance is always larger than when home range centre locations are assumed to be binomially distributed because the variance of inline image is then conditional on the area of integration (Borchers & Efford 2008). This aspect of the sampling variance of inline image is ignored when density is estimated as inline image/inline image under the assumption of geographic closure. The lower precision of our SECR estimates is therefore misleading, because sources of uncertainty included in the variance of SECR inline image did not contribute to the variance of inline image estimated as inline image/inline image.

Spatially explicit capture–recapture is an emerging analytical tool that is theoretically superior to inline image/inline image approaches to estimating animal densities when study populations are geographically open (Efford 2004; Royle et al. 2009). Our results are consistent with previous studies that showed densities estimated from capture data collected on geographically open populations using models that rely on the assumption of geographic closure were higher than where models allowed the assumption to be relaxed or telemetry data were available to correct for edge effect (Soisalo & Cavalcanti 2006; Kendall et al. 2008; Gardner et al. 2009). Although we cannot infer bias directly, we prefer SECR estimates over inline image/inline image estimates because the latter were subject to biases associated with violation of the assumption of geographic closure, which were likely to be severe with our sampling design. Our results also highlight the need to evaluate SECR models with different forms of detection heterogeneity, as has become standard practice in C–R analyses. Specifically, models with non-spatial individual heterogeneity in detection parameters should be evaluated. The improving availability of free, open-source software for fitting SECR models (Efford 2009; Royle et al. 2009) will allow a wider range of models to be explored, further enhancing the utility of the method to wildlife managers.

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.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

We thank the many biologists, technicians and field assistants with the Ontario Ministry of Natural Resources who conducted hair sampling, and K. Wozney, S. Coulson, and D. Abdelhakim for assistance with DNA analyses. The Applied Research and Development Branch, Ontario Ministry of Natural Resources provided funding. M. Efford provided advice on the theory and implementation of SECR. J. Laake provided advice and supplemental code for processing data and fitting C–R models using RMark, and E. Cooch provided advice about closed-population models on the Program MARK online forum (http://www.phidot.org). Comments from J. Bowman, J. Nocera, J. Boulanger, B. Gardner, and an anonymous reviewer helped us improve upon earlier drafts.

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  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
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