A rule-based ad hoc method for selecting a bandwidth in kernel home-range analyses
© Kie; licensee BioMed Central Ltd. 2013
Received: 14 February 2013
Accepted: 29 July 2013
Published: 2 September 2013
An important issue in conducting kernel home-range analyses is the choice of bandwidth or smoothing parameter. To examine the effects of this choice, telemetry data were collected at high sampling rates (843 to 5,069 locations) on 20 North American elk, Cervus elaphus, in northeastern Oregon, USA, during 2000, 2002, and 2003. The elk had their collars replaced annually, hence none were monitored for more than a single year. True home ranges were defined by buffering the actual paths of individuals. Fixed-kernel and adaptive-kernel estimates were then determined with reference bandwidths (h ref ), least-squares cross-validation bandwidths (h lscv ), and rule-based ad hoc bandwidths designed to prevent under-smoothing (h ad hoc ). Both raw data and sub-sampled sparse datasets (1, 2, 4, 6, 12, and 24 locations/elk/day) were used.
With fixed-kernel and adaptive-kernel analyses, reference bandwidths were positively biased (including areas not part of an animal’s home range) but performed better (lower bias, closer match between estimated and true home ranges) with increasing sample size. Least-squares cross-validation bandwidths were positively biased with very small sample sizes, but quickly became negatively biased with increasing sample size, as home-range estimates broke up into disjoint polygons. Ad hoc bandwidths outperformed reference and least-squares cross-validation bandwidths, exhibited only moderate positive bias, were relatively unaffected by sample size, and were characterized by lower Type I errors (falsely including areas not part of the true home range). Ad hoc bandwidths also exhibited lower Type II errors (failure to include portions of the true home range) than did least-squares cross-validation bandwidths, although reference bandwidths resulted in lowest Type II error rates. Auto-correlation indices increased to about 150 to 200 locations per elk, and then stabilized. Bias of fixed-kernel analyses with ad hoc bandwidths was not affected by auto-correlation, but did increase with irregularly shaped home ranges with high fractal dimensions.
The rule-based ad hoc bandwidths, specifically designed to prevent fragmentation of estimated home ranges, outperformed both h ref and h lscv , and gave the smallest value for h consistent with a contiguous home-range estimate. The protocol for choosing the ad hoc bandwidth was shown to be consistent and repeatable.
KeywordsAdaptive kernel Bandwidth Cervus elaphus Fixed kernel Home range North American elk Smoothing parameter
A basic principal in animal ecology is that species, populations, and individuals have finite limits in use of space. Species and populations are delineated by geographical ranges, and individuals are described as having a home range. Burt’s definition of home range is widely used: ‘…that area traversed by the individual in its normal activities of food gathering, mating and caring for young’ . Although plotting animal locations is straightforward, and is subject primarily to measurement errors, estimating the size of the home range is often dependent on a number of assumptions, which are often either not tested or if they are tested, are often determined to be false .
Kernel techniques for estimating the density of a utilization distribution (UD) of a random sample of locations for an individual animal were first proposed by Worton . Kernel analyses are commonly used in statistical density estimation and have the advantage of being non-parametric . They are used not only with single variables, but in bivariate space as well, with the distributions of the x and y coordinates representing animal locations .
Although Worton  used the terms ‘utilization distribution’ and ‘home range’ synonymously, a distinction can be made between the two concepts. Early attempts to quantify the home range of an animal involved drawing polygons around the outermost set of locations. Such techniques result in a contiguous polygon delineating the ‘area traversed by the individual’ , including crucial travel corridors in which an animal spends limited amounts of time, but these fail to portray the intensity of space use within the polygon . Conversely, kernel techniques provide a UD, that is, a three-dimensional probability density map showing which portions of the total home range home are used most frequently . Alternatively, the estimate of the UD can be sliced to reveal a two-dimensional (2D) surface (for example, by taking a 95% volume contour), which is the equivalent of a traditional definition of a home range. Such 2D slices may not be contiguous but rather disjoint, being composed of multiple polygons that more accurately indicate intensity of space use . To capture little-used but important areas such as travel corridors, the 2D slice may be constrained to a single, contiguous polygon .
The starting point in kernel analyses is to construct a bivariate kernel estimate of a probability density function around each data point (animal location). A standard normal distribution is often used, although kernels can take on other shapes such as triangular, rectangular, or parabolic . The functional shape and width of the kernel is determined by the smoothing parameter or bandwidth, denoted by h. Once probability density functions are in place, a grid structure is placed over the entire field, and volumes under the functions are summed over individual locations.
The choice of a smoothing parameter is a key decision in home-range analyses involving UDs, and the initial value is often obtained from the data themselves, although there is no a priori way to choose the best value for h. Silverman  and Worton  suggested a method of constructing an optimum h for large sample sizes if the data were assumed to be normally distributed. Referred to as h opt (and occasionally, an ad hoc choice of h) by Worton , it is optimal only if the assumption of bivariate normality is met, and will be denoted here as the reference bandwidth h ref . If animal locations are clumped rather than normally distributed, h ref will over-smooth the data, and the estimate of home-range size will be positively biased .
A different approach is to choose a bandwidth that minimizes the least-squares cross-validation score, h lscv [3, 7]. In most instances, h lscv is less than h ref , and is often only a small proportion of the latter. Although mathematically appropriate , h lscv frequently results in under-smoothing, and gives an estimate of the home range that consists of multiple polygons. In extreme instances, such an estimate will generate polygons around each small cluster of points, or even individual points.
A further smoothing issue is whether to use the same h for all points (global bandwidth), resulting in a fixed-kernel analysis, or to allow h to vary as a function of local point densities (local bandwidths), yielding an adaptive-kernel analysis. The local-bandwidth approach allows for larger kernels (greater smoothing) associated with locations, often at the edge of the animal’s distribution, where location-point densities are lower. This approach assigns more uncertainty to sparsely distributed locations near the edge of the home range .
An assumption of both kernel analyses and parametric approaches is that data points are independent. However, animal locations are collected sequentially, and the extent to which the assumption of independence is violated is a function of sampling rate . Sampling rates are rapidly increasing with newer telemetry technologies, such as those based on global positioning systems . Little information is currently available on how auto-correlation interacts with estimation choices in kernel analyses to bias resulting estimates. Moreover, the ability to assess bias and hence performance of different kernel techniques is ultimately dependent on defining the true home range of an animal, an issue that has not received much attention.
Kernel analyses with the reference bandwidth (h ref ) exhibited a consistent positive bias as a function of sampling frequency, although the bias declined somewhat with larger sample sizes (Figure 3). Bias using h ref was generally not affected by choice of fixed versus adaptive kernel, with a significant difference (P = 0.0347) seen only when using raw data (Figure 3). In a manner similar to h ref , h ad hoc resulted in a slight positive bias in the estimation of the size of home range, although the bias was more stable with respect to sampling frequency. Bias using h ad hoc was not affected as a function of fixed versus adaptive kernel (all a priori combinations P>0.10) (Figure 3).
Specific a priori comparisons indicated that the choice of fixed versus adaptive kernel had a significant effect on Type I errors when using h lscv at a sampling frequency of 1 location per day (P<0.0001), h ref at 1, 2, 4, 6, and 12 locations per day (P < 0.0001) and at 24 locations per day (P = 0.0006), and the raw data (P = 0.0226) (Figure 4). No significant differences (P > 0.10) between fixed and adaptive kernels occurred at any sampling frequency when using h ad hoc (Figure 4).
Type II errors (failing to capture area in the estimate that was part of the animal’s home range) were affected by the individual animal (F 19,700 = 24.48, P<0.0001), sampling frequency (F 5,714 = 92.57, P<0.0001), and method of choosing a bandwidth (F 2,717 = 2,050.50, P<0.0001), but less by the choice of fixed or adaptive kernels (F 1,718 = 6.95, P = 0.0086) (Figure 4). When using h lscv , significant differences existed between fixed and adaptive kernels at two (P = 0.0335), four (P = 0.0121), and six (P = 0.0415) locations per day (Figure 4). All other a priori comparisons of fixed versus adaptive kernels within a sampling period or bandwidth selection technique were not significant (all P>0.10) (Figure 4). Type II errors increased sharply with h lscv as the estimates of home range polygons became fragmented, but use of either h ref and h ad hoc resulted in Type II errors that remained relatively stable at less than 200 hectares as a function of sample size (Figure 4).
Discussion and conclusions
Kernel analyses are widely used in estimating home ranges and UDs of animals, but they have some disadvantages. Choice of initial bandwidth largely determines the resulting estimates of home-range size (Figures 2, 3). A reference bandwidth (h ref ) assumes bivariate normality, although samples of animal locations are frequently not normally distributed. Animals often use space in a clumped or multimodal manner, and h ref , in assuming a unimodal normal distribution, assigns high variance to the data when they are actually distributed more tightly around two or more modes. The result is over-smoothing of data, and an inflated estimate of home-range size. Conversely, a bandwidth that minimizes the least-squares cross-validation score (h lscv ) often under-smoothes location data, and the resulting home-range estimate breaks up into disjointed polygons [5, 6], resulting in negative bias in the estimate of home-range size (Figure 3) and large Type I errors (Figure 4).
Why should an estimate of the home range for an animal be contiguous? One reason is philosophical; such a distribution matches Burt’s definition of home range as ‘that area traversed by the individual in its normal activities of food gathering, mating, and caring for young . Disjoint or separate core areas, such as those defined by a 60% kernel analysis, do not violate this definition, although an estimate of the entire home range that consists of multiple polygons does. For many purposes, such as estimating the intensity of spatial use of habitats , disjoint polygons are appropriate. Consequently, the terms ‘utilization distribution’ and ‘home range’ are not synonymous, with only the former being a legitimate description of disjoint spatial distributions. However, the biggest disadvantage to disjoint home-range polygons resulting from the use of h lscv is that the degree of fragmentation is highly dependent on sample size (Figure 2), which is an undesirable property when analyzing animal location data sampled at high frequencies with new and emerging telemetry technologies [9, 11].
Given the disadvantages of kernel techniques, what available analytical options are essential to minimize bias and error? One issue is that as sampling frequency increases, so does serial auto-correlation. White and Garrott  argued that auto-correlation itself was not as much of an issue as was insuring that the sampling was evenly spread over the time period of interest. De Solla et al.  also recommended maximizing the number of observations using constant time intervals, arguing that such a protocol increases the biological relevance of home-range estimation. In the current study, bias was not influenced by degree of auto-correlation when using h ad hoc choice of bandwidth (Figure 5b). Type I and II errors associated with h ad hoc also appeared to be independent of sampling frequency (Figure 3). Given an appropriate choice of bandwidth such as h ad hoc , auto-correlation is not a concern. However, use of h lscv , is fraught with pitfalls associated with sampling frequency, auto-correlation, bias, and Type I and II errors. The issue is not whether an assumption of independent data has been violated, but rather how robust is a specific choice of bandwidth to such violations. This study indicates that kernel analyses using h ad hoc can be robust under these conditions, and supports previous recommendations [2, 12].
Likewise, the shape of the kernel itself may not be a crucial issue. Wand and Jones  noted that that efficiency of various kernel shapes varied by less than 10%. Most computer programs currently use a standard normal distribution for the kernel probability density function . However, other shapes are possible, including uniform or triangular kernels [4, 13]. Some older programs use a parabola-shaped Epanechnikov kernel to avoid having to evaluate the volume under the extended tails of a bivariate normal distribution . It should be noted that computationally, it is not possible to conduct a strict 100% volume analysis with a standard-normal kernel; the tails of the kernel must be truncated at some point by requesting a volume of less than 100%. In some computer programs, this modification may be done automatically, for example at 99.9%, in a manner not transparent to the user. Although not tested in this study, it has been suggested that choice of kernel shape is not of major concern .
The advantages and disadvantages of using global versus local bandwidths in kernel home-range analyses has been the subject of debate, as has the choice of h . Worton  favored a local bandwidth (adaptive kernel) using h lscv , but also suggested that a global bandwidth (fixed kernel) using h ref also produced valid estimates. Worton  later argued that although the choice of h was very important, the choice of global versus local application of that bandwidth was less so. Seaman and Powell  and Seaman et al.  reported that global use of h lscv resulted in little bias in home-range estimates, but that local-bandwidth approaches overestimated areas of distribution, and thus should not be used. The results from the current study are consistent with Worton ; the choice between global versus local bandwidths is inconsequential in terms of bias (Figure 3), Type I, and Type II errors (Figure 4). Conversely, in this study the use of h lscv resulted in rapidly increasing negative bias and Type I errors in home-range estimates with increasing sample size (Figures 2, 4). Similar concerns have been raised by Hemson et al. .
Different computer programs have limits on how small h lscv can be as a function of h ref . Home Range Extension (HRE) places a minimum value of h lscv at 0.1025 h ref , a floor unlikely to have a pronounced effect on the calculation of h lscv in this study. However, another commonly used program (Animal Movement Extension, http://alaska.usgs.gov/science/biology/spatial/gistools/index.php/, accessed 29 January 2013) will not allow a value for h lscv of less than 0.9662 h ref , in effect implementing an incorrect definition of h lscv (= 0.9662 h ref ) in many analyses (A. Rodgers, personal communication).
The current study indicates that implementation of h ad hoc , specifically designed to prevent fragmentation of estimated home ranges, in either a global or local context, outperformed both h ref and h lscv . Use of an arbitrary value for h such that h is less than or equal to h ref to improve model fit while preventing fragmentation of home-range estimates has been reported for domestic cattle, Bos taurus (h = 0.8 h ref ) , mule deer, Odocoileus hemionus (h = 0.8 h ref ) , and white-tailed deer, Odocoileus virginianus (h = 0.7 h ref ) . The protocols used in this current study were similar, but rather than select an arbitrary value for the bandwidth, the smallest value for h that was consistent with a contiguous home-range estimate was chosen. These protocols are consistent and repeatable, and have been used in other studies [23, 24].
With emerging telemetry techniques, large numbers of data on animal location can be collected at high sampling frequencies . The technique of plotting the buffered path of an individual [25, 26], similar to that performed in this study to define true home ranges, may provide a useful estimate of the total area used by an animal. However, further research into perceptual ranges of different species [27, 28] will be required refine the distance by which animal paths should be buffered. Conversely, for the foreseeable future, kernel approaches will remain useful for the analysis of spatial use by animals, not only for use with sparse datasets, but most importantly for determining intensity of use within a home range.
This study was conducted at the US Forest Service’s Starkey Experimental Forest and Range (hereafter referred to as ‘Starkey’), located 35 km southwest of La Grande (45°13’N, 118°31’W) in the Blue Mountains of northeastern Oregon, USA (Figure 1). The forest is situated between 1,122 and 1,500 meters in elevation, and supports a mosaic of coniferous forests, grasslands, and riparian areas that typify the summer range for elk in the Blue Mountains . A network of narrow, irregular drainage channels in the project area creates a complex and varied topography [30, 31].
Starkey consists of 10,125 hectares enclosed by a 2.4-m high fence that prevents immigration or emigration of resident elk and other large herbivores . The largest division within Starkey is a main study area if 7,762 hectares, from which data for this research were obtained (Figure 1). Details of the study area and facilities are available elsewhere [29, 32–34].
Determining animal locations
Location data collected for Rocky Mountain elk at Starkey Experimental Forest and Range, Oregon, USA
Elapsed time, minutesa
True home range
31 October to 24 November 2000 (25 days)
2 November to 3 December 2002 (32 days)
4 to 30 November 2003 (27 days)
Protocols were approved by the Institutional Animal Use and Care Committee at Starkey Experimental Forest and Range .
The female elk in this study were (mean ± SD) 6.9 ± 2.85 years of age (range 3 to 14 years). Mean elapsed times between observations were 36.90 ± 5.22 minutes (n = 3 elk) in 2000, 8.67 ± 0.92 minutes (n = 9) in 2002, and 8.95 ± 0.87 minutes (n = 8) in 2003. Numbers of locations per individual ranged from 843 to 1,089 in 2000, and from 3,615 to 5,069 in 2002 to 2003 (Table 1).
Finally, to test the performance of different techniques for estimating home-range size using sparse datasets, data were sub-sampled by choosing at random 1, 2, 4, 6, 12, and 24 locations per elk per day. Techniques for home-range estimation were then applied to each dataset of reduced sampling frequency in addition to raw data. Moreover, bivariate serial auto-correlation and cross-correlation between 2 points, but among 3 or more points in the raw and sparse datasets were estimated with a measure described by Swihart and Slade .
To test the accuracy of location data obtained from individual elk, each year a radio collar was placed at a known location and its position monitored regularly, along with the study animals. Based on approximately 3,000 locations determined each year for the fixed collars, the estimated error (mean ± SD) was 35.3 ± 35.9 m, comparing favorably with a previous estimate of 52.8 ± 5.87 m (mean ± SE) .
Analyses of home ranges
The true home range of an animal was defined by assuming first that the actual path followed by an individual was a straight line between each pair of successive locations. Elapsed times between locations were relatively short, particularly in 2002 and 2003, hence this is likely to be an accurate portrayal. Again, using a perception threshold of 183 m, the path of each elk was buffered by that amount, and then any lacuna within the resulting polygon were removed  to arrive at the true home range. In addition, the fractal dimension of each home range was estimated to give a measure of the irregularity of its shape.
HRE  for ArcView (ESRI, Redlands, CA, USA) was used to estimate elk home ranges. The 95% volumetric kernel analyses  were calculated using a variety of techniques, including both a global bandwidth (fixed kernel) and local bandwidth (adaptive kernel), all with a default resolution (70 × 70 cell grid) option in HRE . Three different methods were used in choosing an initial bandwidth. The first was to use the reference bandwidth, h ref ; the second was to use the bandwidth that minimized the cross-validation score, h lscv ; and the third was based on an ad hoc approach.
Silverman stated that ‘a natural method for choosing a smoothing parameter is to plot out several curves and choose the estimate that is most in accordance with one’s prior ideas about the density’ . In the current study, the goal was to delineate a single, contiguous polygon representing a complete home range as described by Burt . Therefore, the reference bandwidth (h ref ) was sequentially reduced in 0.10 increments (0.9 h ref , 0.8 h ref , 0.7 h ref , …0.1 h ref ). This rule-based h ad hoc was the smallest increment of h ref that: 1) resulted in a contiguous rather than disjoint 95% kernel home-range polygon, and 2) contained no lacuna within the home range. When sequentially reducing h ref , lacuna occasionally appeared that subsequently disappeared at successively smaller values of h ref . However, Once an estimate of the home range fractured into two or more polygons, the process of searching for h ad hoc was halted. In most instances, h lscv < h ad hoc < h ref , although h ad hoc < h lscv < h ref was considered. Conversely, we did not allow h ad hoc to be greater than h ref when the estimate of the home range was fragmented at h ref , but accepted the fragmented estimate instead. Note that the definition of h ad hoc used in the current study should not be confused with the discussion of h ref as an ad hoc choice by Worton . This ad hoc choice of a bandwidth has previously been used to delineate home ranges in coyotes, Canis latrans , and in pronghorns, Antilocapra americana .
Finally, the various estimates of elk home ranges were compared with what were previously defined as true home ranges. Differences in size between the estimates and the true home range (% bias) and Type I (area included as part of the estimate, which was not part of the true home range) and Type II errors (area within the true home range, which was not included within the estimate), were examined. For kernel analyses, statistical tests were conducted with a general linear model in SAS software (SAS Institute, Cary, NC, USA)  with main factors including individual animal (n = 20), initial bandwidth (n =2: global, local), bandwidth selection technique (n = 3: h ref , h lscv , h ad hoc ), and sampling frequency (n = 6: 1, 2, 4, 6, 12, and 24 locations per day plus raw data), along with interactions between the main factors. Total sample size was thus 720 records (20 × 2 × 3 × 6). Bias was transformed with a square-root arc sin function to ensure additivity of treatment effects , and specific a priori comparisons were made with least-squares means . The relationship between percentage bias of the various home-range estimates as functions of degree of auto-correlation between among of individual elk and the fractal dimension of the true home range was examined.
Long range navigation
Home Range Extension
I thank Alan A. Ager for assistance with preparing raw data files for ArcView GIS, and R. T. Bowyer, A.R. Rodgers, and J. K. Young for comments on previous versions of this manuscript.
- Burt WH: Territoriality and home range concepts as applied to mammals. J Mammal 1943, 24: 346–352. 10.2307/1374834View ArticleGoogle Scholar
- White GC, Garrott RA: Analysis of Wildlife Radio-Tracking Data. San Diego, California: Academic; 1990.Google Scholar
- Worton BJ: Kernel methods for estimating the utilization distribution in home-range studies. Ecology 1989, 70: 164–168. 10.2307/1938423View ArticleGoogle Scholar
- Silverman BW: Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability. London: Chapman and Hall; 1986.View ArticleGoogle Scholar
- Kie JG, Matthiopoulos J, Fieberg J, Powell RA, Cagnacci F, Mitchell MS, Gaillard J-M, Moorcroft PR: The home-range concept: are traditional estimators still relevant with modern telemetry technology? Phil Trans Royal Soc B 2010, 365: 2221–2231. 10.1098/rstb.2010.0093View ArticleGoogle Scholar
- Powell RA: Animal home ranges and territories and home range estimators. In Research Technologies in Animal Ecology – Controversies and Consequences. Edited by: Boitani L, Fuller TK. New York: Columbia University Press; 2000:65–110.Google Scholar
- Gitzen RA, Millspaugh JJ: Comparison of least-squares cross-validation bandwidth options for kernel home-range estimation. Wildl Soc Bull 2003, 31: 823–831.Google Scholar
- Swihart RK, Slade NA: Influence of sampling interval on estimates of home-range size. J Wildl Manage 1985, 49: 1019–1025. 10.2307/3801388View ArticleGoogle Scholar
- Tomkiewicz SM, Fuller MR, Kie JG, Bates KK: Global positioning system and associated technologies in animal behaviour and ecological research. Phil Trans Royal Soc B 2010, 365: 2163–2176. 10.1098/rstb.2010.0090View ArticleGoogle Scholar
- Marzluff JM, Millspaugh JJ, Hurvitz P, Handcock MS: Relating resources to a probabilistic measure of space use: forest fragments and Steller’s jays. Ecology 2004, 85: 1411–1427. 10.1890/03-0114View ArticleGoogle Scholar
- Rodgers AR: Recent telemetry technology. In Radio Tracking and Animal Populations. Edited by: Millspaugh JJ, Marzluff JM. San Diego, California, USA: Academic; 2001:79–121.View ArticleGoogle Scholar
- De Solla SR, Bonduriansky R, Brooks RJ: Eliminating autocorrelation reduces biological relevance of home range estimates. J Animal Ecol 1999, 68: 221–234. 10.1046/j.1365-2656.1999.00279.xView ArticleGoogle Scholar
- Wand MP, Jones MC: Kernel Smoothing. Monographs on Statistics and Applied Probability 60. London: Chapman and Hall; 1995.Google Scholar
- Rodgers AR, Carr AP: HRE: the Home Range Extension for ArcView. Thunder Bay, Ontario: Centre for Northern Forest Ecosystem Research, Ontario Ministry of Natural Resources; 1998.Google Scholar
- Kie JG, Baldwin JA, Evans CJ: CALHOME: a program for estimating animal home ranges. Wildl Soc Bull 1996, 24: 342–344.Google Scholar
- Worton BJ: Using Monte Carlo simulation to evaluate kernel-based home range estimators. J Wildl Manage 1995, 59: 794–800. 10.2307/3801959View ArticleGoogle Scholar
- Seaman DE, Powell RA: An evaluation of the accuracy of kernel density estimators for home range analysis. Ecology 1996, 77: 2075–2085. 10.2307/2265701View ArticleGoogle Scholar
- Seaman DE, Millspaugh JJ, Kernohan BJ, Brundige GC, Raedeke KJ, Gitzen RA: Effects of sample size on kernel home range estimates. J Wildl Manage 1999, 63: 739–747. 10.2307/3802664View ArticleGoogle Scholar
- Hemson G, Johnson P, South A, Kenward R, Ripley R, Macdonald D: Are kernels the mustard? Data from global positioning system (GPS) collars suggests problems for kernel homerange analyses with least-squares cross-validation. J Animal Ecol 2005, 74: 455–463. 10.1111/j.1365-2656.2005.00944.xView ArticleGoogle Scholar
- Kie JG, Boroski BB: Cattle distribution, habitats, and diets in the Sierra Nevada of California. J Range Manage 1996, 49: 482–488. 10.2307/4002286View ArticleGoogle Scholar
- Kie JG, Bowyer RT, Boroski BB, Nicholson MC, Loft ER: Landscape heterogeneity at differing scales: effects on spatial distribution of mule deer. Ecology 2002, 83: 530–544. 10.1890/0012-9658(2002)083[0530:LHADSE]2.0.CO;2View ArticleGoogle Scholar
- Bertrand MR, DeNicola AJ, Beissinger SR, Swihart RK: Effects of parturition on home ranges and social affiliations of female white-tailed deer. J Wildl Manage 1996, 60: 899–909. 10.2307/3802391View ArticleGoogle Scholar
- Berger KM, Gese EM: Does interference competition with wolves limit the distribution and abundance of coyotes? J Animal Ecol 2007, 76: 1075–1085. 10.1111/j.1365-2656.2007.01287.xView ArticleGoogle Scholar
- Jacques CN, Jenks JA, Klaver RW: Seasonal movements and home-range use by female pronghorns in sagebrush-steppe communities of western South Dakota. J Mammal 2009, 90: 433–441. 10.1644/07-MAMM-A-395.1View ArticleGoogle Scholar
- Ostro LET, Young TP, Silver SC, Koontz FW: A geographic information system method for estimating home range size. J Wildl Manage 1999, 63: 748–755. 10.2307/3802665View ArticleGoogle Scholar
- Pulliainen E: Use of the home range by pine martens ( Martes martes L.). Acta Zool Fennica 1984, 171: 271–274.Google Scholar
- Mech SG, Zollner PA: Using body size to predict perceptual range. Oikos 2002, 98: 47–52. 10.1034/j.1600-0706.2002.980105.xView ArticleGoogle Scholar
- Zollner PA: Comparing the landscape level perceptual abilities of forest sciurids in fragmented agricultural landscapes. Landscape Ecol 2000, 15: 523–533. 10.1023/A:1008154532512View ArticleGoogle Scholar
- Rowland MM, Bryant LD, Johnson BK, Noyes JH, Wisdom MJ, Thomas JW: The Starkey project: history, facilities, and data collection methods for ungulate research. US Forest Serv 1997. General Technical Report PNW-GTR-396Google Scholar
- Ager AA, Johnson BK, Kern JW, Kie JG: Daily and seasonal movements and habitat use by female Rocky Mountain elk and mule deer. J Mammal 2003, 84: 1076–1088. 10.1644/BBa-020View ArticleGoogle Scholar
- Kie JG, Ager AA, Bowyer RT: Landscape-level movements of North American elk ( Cervus elaphus ): effects of habitat patch structure and topography. Landscape Ecol 2005, 20: 289–300. 10.1007/s10980-005-3165-3View ArticleGoogle Scholar
- Rowland MM, Coe PK, Stussy RJ, Ager AA, Cimon NJ, Johnson BK, Wisdom MJ: The Starkey habitat database for ungulate research: construction, documentation, and use. US Forest Serv 1998. General Technical Report PNW-GTR-430Google Scholar
- Stewart KM, Bowyer RT, Kie JG, Cimon NJ, Johnson BK: Temporospatial distributions of elk, mule deer, and cattle: resource partitioning and competitive displacement. J Mammal 2002, 83: 229–244. 10.1644/1545-1542(2002)083<0229:TDOEMD>2.0.CO;2View ArticleGoogle Scholar
- Stewart KM, Bowyer RT, Kie JG, Dick BL, Ben-David M: Niche partitioning among elk, mule deer, and cattle: do stable isotopes reflect dietary niche? Ecoscience 2003, 10: 297–302.Google Scholar
- Weber KT, Burcham M, Marcum CL: Assessing independence of animal locations with association matrices. J Range Manage 2001, 54: 21–24. 10.2307/4003522View ArticleGoogle Scholar
- Reynolds HG: Use of a ponderosa pine forest in Arizona by deer, elk, and cattle. US Forest Serv 1966. Research Note RM-63Google Scholar
- Wisdom MJ, Cook JG, Rowland MM, Noyes JH: Protocols for care and handling of deer and elk at the Starkey Experimental Forest and Range. US Forest Serv 1993. General Technical Report PNW-GTR-311Google Scholar
- Findholt SL, Johnson BK, Bryant LD, Thomas JW: Corrections for position bias of a LORAN-C radio-telemetry system using DGPS. Northwest Sci 1996, 70: 273–280.Google Scholar
- SAS Institute: SAS/STAT User's Guide, version 8. Cary, North Carolina: SAS Institute, Inc.; 1999.Google Scholar
- Gilbert N: Biometrical Interpretation. Oxford: Clarendon Press; 1973.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.