https://cpw.cvlcollections.org/files/original/89f0919ec0933d9c32ae03c1d888186b.pdf 14325fda9d1230384277cd42192617fc PDF Text Text The research in this publication was partially or fully funded by Colorado Parks and Wildlife. Dan Prenzlow, Director, Colorado Parks and Wildlife • Parks and Wildlife Commission: Marvin McDaniel, Chair • Carrie Besnette Hauser, Vice-Chair Marie Haskett, Secretary • Taishya Adams • Betsy Blecha • Charles Garcia • Dallas May • Duke Phillips, IV • Luke B. Schafer • James Jay Tutchton • Eden Vardy �Using auxiliary telemetry information to estimate animal density from capture—recapture data Author(s): Jacob S. Ivan, Gary C. White and Tanya M. Shenk Source: Ecology , April 2013, Vol. 94, No. 4 (April 2013), pp. 809-816 Published by: Wiley on behalf of the Ecological Society of America Stable URL: https://www.jstor.org/stable/23436294 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms Ecological Society of America and Wiley are collaborating with JSTOR to digitize, preserve and extend access to Ecology This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms �Ecology, 94(4), 2013, pp. 809-816 © 2013 by the Ecological Society of America Using auxiliary telemetry information to estimate animal density from capture—recapture data Jacob S. Ivan,1'3 Gary C. White,' and Tanya M. Shenk2 Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado 80523 USA 2Colorado Division of Wildlife, Fort Collins, Colorado 80526 USA Abstract. Estimation of animal density is fundamental to ecology, and ecologists often pursue density estimates using grids of detectors (e.g., cameras, live traps, hair snags) to sample animals at a study site. However, under such a framework, reliable estimates can be difficult to obtain because animals move on and off of the site during the sampling session (i.e., the site is not closed geographically). Generally, practitioners address lack of geographic closure by inflating the area sampled by the detectors based on the mean distance individuals moved between trapping events or invoking hierarchical models in which animal density is assumed to be a spatial point process, and detection is modeled as a declining function of distance to a detector. We provide an alternative in which lack of geographic closure is sampled directly using telemetry, and this auxiliary information is used to compute estimates of density based on a modified Huggins closed-capture estimator. Contrary to other approaches, this method is free from assumptions regarding the distribution and movement of animals on the landscape, the stationarity of their home ranges, and biases induced by abnormal movements in response to baited detectors. The estimator is freely available in Program MARK. Key words: density: geographic closure: mark-recapture; telemetry. Introduction deaths, immigration, or emigration during the sampling Estimation of animal density is fundamental to Period (i e" the Population is closed demograph.cally) wildlife ecology. Density is used to evaluate system and that ammals do not move on and off the study slte responses to environmental perturbations and treat- during sampling (i.e., the population is closed geograph ments (Converse et al. 2006, Manning and Edge 2008), it ically [Otis et al. 1978, Williams et al. 2002]). Short can function as a benchmark for listed species recovery sampling sessions appropriately timed with the natural (U.S. Fish and Wildlife Service 1998:319), it can be history of the species of interest can ensure achievement useful for understanding system dynamics (Soulé et al. °f demographic closure. However, outside of a few 2003), or assessing habitat suitability for dependent exceptions (e.g., sampling small islands or ponds), species (Zahratka and Shenk 2008), and it is routinely geographic closure is unlikely (White et al. 1982). used in population monitoring and modeling (Thomp- Without geographic closure, abundance estimates ob son et al. 1998). It is often preferable to estimates of tained through closed-capture models do not reflect the relative density (i.e., abundance) because standardizing number of animals within the boundaries of the study population estimates by area facilitates comparison of she. Rather, they reflect the super population of animals populations across space and time. associated with the site. That is, such estimates reflect all Ecologists often pursue density estimates using grids animals that could have used the site at any time over of detectors (e.g., cameras, traps, hair snags) to sample the course of the sampling period including those both animals at a study site. Under such a sampling inside and outside of the site (Schwarz and Arnason framework, estimates of animal abundance can be 1996, Kendall et al. 1997). Such estimates are difficult to obtained using a variety of closed-capture models (Otis convert to density because the area effectively sampled et al. 1978, White et al. 1982, Huggins 1989, 1991, by the detectors (i.e., the area used by the super Williams et al. 2002). These models assume no births, population) is unknown. Traditionally, the most common strategy for manag ing the geographic closure issue has been to attempt Manuscript received 17 January 2012: revised 14 September estimation of this effective area, then divide this estimate 2012; accepted 20 September 2012; final version received 7 November 2012. Corresponding Editor: E. G. Cooch. into the super population estimate obtained from closed 3 Present address: Colorado Parks and Wildlife, Fort Collins, Colorado 80526 USA. E-mail: 4 capture models to obtain a corrected estimate of density. Usually the maximum distance moved by each individ Jake.Ivan@state.co.us Present address: . National . Park Service, Colorado 80525 USA. events, is averaged across all individuals captured more 809 This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms Fort �810 JACOB S. IVAN ET AL. Ecology, Vol. 94, No. 4 process model and the two sub-models are combined hierarchically such that density is estimated directly given the data (Efford 2004, Borchers and Efford 2008, Royle et al. 2009a, b). Early versions of SECR (inverse prediction) performed well in simulation (Efford 2004) and field experiments (Efford et al. 2005). Currently SECR analyses can be accomplished using inverse prediction (Efford 2004), maximum likelihood (Borch ers and Efford 2008) or Bayesian (Royle et al. 2009a, b) approaches. Other than the usual capture-recapture assumptions concerning demographic closure and accurate reporting of marks, the main assumptions of the basic SECR model are (1) home range centers occur randomly on the landscape at a reasonably constant density (or at least varying in some way that can be modeled) according to a Poisson point process or some variant thereof and (2) individuals occupy stationary, symmetrical activity ranges during the sampling session. Estimation of the observation portionactivity of the model is based on informa Fig. 1. Conceptualization of animal ranges (ovals) overlaid on a hypothetical site (rectangle). Lack of tionstudy from detections. Thus, this portion of the model geographic closure results may from some activity ranges be susceptible to some of the potential MMDM being partially on the site such that animals can move on and off biases if movements and detection are unnaturally during a sampling session. The sum of the proportion of each affecteddivided by the samplingby process. animal on the site (gray areas) the area of the site provides an estimate of density corrected for geographic We formalize a density estimator that addresses closure. Note that the use of equally sized, oval activity geographic closure by measuring the process directly ranges and a rectangular study site are for illustration only; the method using auxiliary telemetry data. This method allows home makes no restrictions about the shape and size of activity ranges range shifts or irregular movements induced by sam pling, makes no assumptions about the distribution and movements of animals on the landscape, and does not than once (i.e., mean maximum distance moved, require estimation of effective area sampled. Conceptu MMDM), and the study site is buffered by this distance ally, the approach can be traced to the idea of estimating (or one-half this distance), to estimate the effective area fractions of animals or "animal equivalents" within an sampled (Wilson and Anderson 1985). However, there area as described by Marten (1972), and later Boutin are a number of concerns with this strategy, including (1984). We describe the basic form of the estimator, (1) movements of animals may be constrained by the which is similar to that proposed by White and Shenk detection process itself if sampling occurs in live traps,(2001) and implemented by Grant and Doherty (2007), (2) movements of animals residing near the edge of the then extend it to allow for individual covariates and to or sites. study site may not be well represented, (3) baited admit more useful designs. We then apply the estimator detectors (if used) may induce immigration into the to an example for illustration. study site or otherwise bias normal movement patterns, Methods (4) movement distances revealed through detection events are dependent on the number of times an Estimator individual is detected, and (5) detector spacing and Conceptually, we begin with the notion that each study site size can influence movement estimates animal occupies some activity range on the landscape (Parmenter et al. 2003). during a relatively short sampling session, and these More recently, spatially explicit capture-recapture activity ranges occur irrespective of the boundaries of a (SECR) techniques have been introduced as an alterna study site (Fig. 1). During a sampling session, individ tive (Efford 2004, Borchers and Efford 2008, Royle et al. uals are captured, marked, and released on multiple 20096). Like MMDM, SECR makes use of spatial occasions. Our method requires tagging at least some of information contained in the capture history of each the animals subjected to mark-recapture sampling with individual. However, SECR does not attempt to a telemetry device (e.g., radio tag, GPS collar). estimate effective area sampled. Rather, spatial infor Telemetry tagging can occur prior to or during the mation is used to estimate parameters of an observation mark-recapture process. Locations are then obtained model in which detection probability declines as afrom tagged animals during and/or immediately follow function of the distance between an animal's home rangeing the mark-recapture sampling session in order to center and a given detector in the study site. The densityproduce an estimate of the proportion of time that each of home range centers is represented as a separatetagged animal spent within the study site (Fig. 1). This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms �April 2013 DENSITY ESTIMATION USING TELEMETRY DATA Sil Note that the definition of the site is somewhat of animals detected. arbitrary. However, any workable definition should individual-specific ensure that animals within the site have a reasonable ing distancechance of detection. We suggest that a "reasonable covariate; compute chance of detection" can be attained if the site is defined individual, then t such that detectors are distributed at an adequate and to the edge of the roughly consistent rate (e.g., four detectors per home edge of the study range [Otis et al. 1978:76]) within it so that there are no range and a dimi gaps in sampling effort. We prefer to define the study to animals captur site as the minimum convex polygon (MCP) encom- covariate helps accou passing the detectors. Such a definition seems natural Next we make a and trap density inside this polygon would meet our Huggins estimato criteria, assuming detectors are deployed at a sufficient r+1 rate. Alternatively, one could define the study site as the _ í Pj_ MCP plus one-half trap width, or a full trap width, and P* still claim that trap density is relatively high and consistent within the site. However, if we defined the where & is the estimated proportion of time animal i site as the MCP plus two to three trap widths, or if sPent 011 the study site (estimated via telemetry) and detectors were distributed at a rate of two per home Nss is now the estimated number of animals within the , c•. , . • , „ boundary of the site. Animals that are detected on the range over part of the site, but six per home range over / . ,, , „ ... ,, , ,. ■ site during a mark-recapture session but never located other parts, those definitions would likely result in , . . r , i- rr , ., ■, , i j, on the study site via telemetry are assigned p, = 0 and do unequal sampling effort across the site and may lead to ./ , . J 0 , not contribute to the estimate. Those individuals that bias in the estimator. Note that, because we incorporate are always located on the study site contribute fully the proportion of time each individual spends on the site the estimate and are assigned p, = 1. All other (i.e., telemetry data), the estimator we describe will be ,. ., , . . . , _ , . ,, . ,, ... ,, . . , . , , individuals receive a fractional p.- and partially contnb appropriate regardless of how the site is defined, as long ute. as it is defined following the guidance we provide here. Traditional mark-recapture estimators can be applied Finally, we divide NSs by the area of the study site (e.g., area of the minimum convex polygon encompass to the mark-recapture data collected during the ing all detectors) in order to obtain an estimate of sampling session to estimate the super population density: (Schwarz and Arnason 1996, Kendall et al. 1997) of animals that used the site during sampling. Our goal is m,+, ~ to use the telemetry data to estimate the portion of the super population (i.e., total animal equivalents) that q = — i occurred within the boundaries of the site to produce an ^ estimate of density corrected for the lack of geographic where D is estimated density (number of animals per closure. Heuristically then, we start with the super unit area), A is the area of the study site, and p¡, p¡, and population estimate derived from conventional closed- M,+\ are as defined previously, capture methods then disregard those animals complete Variance ly off of the site, partially discount edge animals, and tally the whole and partial animals left before dividing For the simple case where p is estimated directly from by the area of the site. In order to work well, the telemetry data as an overall mean across individuals and telemetry sampling scheme should be appropriate for the no covariates are used to estimate por p*, the estimator species of interest such that locations can be assumed to reduces to be representative of how animals use their home range. Mathematically, our approach stems from the Huggins M,+1— (1989, 1991) closed-capture estimator for abundance: ß _ p A 1 i „ Ñsp = V" _ and variance of D can be approximated using the delta /= i P* method (Seber 1982:7) as where ÁSP is the estimate of the super population of ~2 (1 -p*) Var (p) Var(p*)\ animals that could have used the site during the trapping Var(D) = D -I + - 2 session, p* is the estimated probability that animal i is \ ^<+l p P* I captured one or more times during the sampling session (i.e., if pi is an estimate of the probability animal that i is whSffiJ/') is assumed to be a binomial proportion (i.e., detected on any given occasion,p* = 1 - (1 — p,)', where t Var(p) = PA 1 — Pi)l(n¡)> 'L_Js__the number of is the number of occasions), and M,+1 is the total number locations obtained for animal i), Var((* ) follows from This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms �812 JACOB S. IVAN ET AL. Ecology. Vol. 94. No. 4 the Huggins estimator (Huggins 1989, 1991), and D and Example M,+1 are as defined previously. However, we expect the We trapped snowshoe hares {Lepus am use of individual covanates to be common and desirable. central Colorado during n_l5 August 20 In such cases, the delta method approximation is much ? live_trapping gnd wlth 50.m spacing (I more complex. We provide guidance for numerically captured 14 adult_sized hares 25 times du estimating variance in the Appendix. day sampling sessi0n (Colorado State Univ Assumptions CUC Protocol 06-062A-03). Of these, eight were with 28-g radio collars (Model TW5SM; BioTrack, Assumptions for this estimator include the usual Wareham> Dorsek UK) Traps and bajt w closed mark-recapture assumptions (Otis et al. 1978. at the conclusion of mark_reCaPture sam Wilhams et al. 2002) as well as three additional August Telemetry sampiing commenced assumptions specific to this method: (1) The animals and occurred dai[y thrQugh ^ Augus sampled with telemetry are representative of the locations via triangulation at relativel population of animals that use the study site. (2) (w¡th¡n 250 m)> r£su)ting m accuracy of Telemetry devices do not affect movements are no effects of mark-recapture sampling movements. (3) Telemetry location error is to the size of the study site and assignment and there were alternately sampled during nlghttim on an,mal daytime (restmg) hours tQ obtajn a repr small relative sample of ,ocations We obtained an a (on/off) of localions/individual during daytime and locations near the edge of the site is unbiased. indivldual at night. There was significant Useful variants anc' study site during the telemetry sa period, and one hare was never located on the site after It will often be difficult to telemeter all individuals trapping (Fig 9) detected during a sampling session. In such cases, the Using the estimator described here, as imp estimator described above can be mod.fied to model use Program MARK; we mode|ed both of the study site by individuals that were subjected to intercepts only> as functions of DTE, and mark-recapture sampling but were not telemetered. tions ther£of aiq selected the DXE struct Specifically one can estimate p, by fitting a logistic parameters (AIC, weight = 0.99). The estim model to the data from radio-tagged animals, then apply (±§E) from tha{ model was j 99 ± J16 this model to untagged animals to estimate their p„ We FaiKng tQ account for ,ack of geographlc again suggest that minimum DTE from each individu- jn a nai-ve density estimate from the n al's mean trap location could be a useful covariate in modd {M^ Ñ/A = 5 0? hares/ha) that is 15 such a model to account for heterogeneity induced by telemetry-corrected value location of individuals relative to the study site. Other models are also possible (including intercept only, which Discussion simply assigns each individual the mean proportion of The estimator preSented here has advantage time on site from the telemetered sample) and multiple Qther traditiona, and contemporary density candidate models can be formulated and compared techniques. First> it does not require esti using an information theoretic approach such as AICc effe£tiv£ area sampled using ad hoc approache (Burnham and Anderson 2002). it is free from the assumption that animals In most cases, we expect that the study site will be stationary home ranges during the sampling p large enough to encompass numerous activity ranges of ¡t does nQt require spedfic distributional ass the target species. In these instances, the (DTE) about home range centers (e.g., that home range centers covariate is only important when mean capture location foUow a Poisson process) or movements 0f animals (e.g., is relatively near the edge of the site. Activity ranges that that ammals mov£ about w¡th¡n a symmetric home are fully on site expose animals to an equal number of rang£) Notei how£V£r; that we require telemetered detectors regardless of whether the range is just mside an¡mals tQ be representative of all animals, so we do the study site or exactly m the middle. Thus, it may also assume all an¡mals come from the same "distribution" in be useful to consider a threshold model for the som£ s£ns£ Jhird< it allows us£rs t0 mak£ full US£ of estimation of p, and/or p, such that the DTE covariate anciUary telemetry data that directly describe the process is only operational up to a point. After that animals are of int£r£St: movement of animals on and off the study fully on site and the proportion or capture probability is sit£ Finally5 the estimator is intuitive and relatively easy estimated accordingly. Mathematically, such a model to implement. can be represented as A disadvantage is that purchasing and tracking 1 -u ~\ a , 5 i ■ ti, T-w telemetry tags, in addition to the required mark logit(p.) = ßn + ß, (min(ß7, DTE ) recapture sampling, may be cost prohibitive when the where ß0 and ßi are the usual intercept and slope only goal is density estimation. However, p estimates from a logistic model using distance to edge as routinely use telemetry for home ran a covariate, and ß2 is the threshold parameter. quantification of habitat use, and/or survival This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms �April 2013 DENSITY ESTIMATION USING TELEMETRY DATA 813 Fio. 2. Telemetry locations obtained on eight snowshoe hares (Lepus american rectangle) in central Colorado. August 2006. Different symbols indicate unique i day and night over a 13-day period immediately following trapping. at the same time they obtain density estimates. In fact, equal and the many authors (e.g., Di Betti et al. 2006, Soisalo and traps (e.g., at lea Cavalcanti 2006, Dillon and Kelly 2008, Balme et al. to occur within 2009, Sharma et al. 2010) have used location data to interest. Also, d correct mark-recapture density estimates by first calcu- that research lating average home range size for animals in the study, determine then applying a buffer to their study site based on this common acro home range estimate. We suggest that a better use of on, or whethe these location data to correct density estimates would be whether each the approach outlined here. GPS technology provides an and Doherty excellent opportunity to implement this method as can be realiz potentially more locations can be collected with better sites. precision but without the additional cost of physically In our examp following the telemetry devices after deployment. individual to est We presented an example from a single study site laid manee impro out in a regular grid, but other designs are possible, asymptote. How Traps can be set in virtually any configuration as long as manee of th the distance between neighboring traps remains roughly error among This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms �814 JACOB S. IVAN ET AL. Ecology, Vol. 94. No. 4 individuals. Thus, error can be minimized more effi- H ciently by sacrificing number of locations/individual favor of deploying telemetry devices on more individuals (Ivan et al. 2013). For example, the estimator is like be less biased when 100% of the animals in the markrecapture sample have telemetry devices and only five locations are obtained per animal than when 25% are telemetered and 20 locations per animal are obtained. w One unique assumption of this estimator is that t movements of telemetered animals during and/or We illustrated the method using data from a immediately after mark-recapture sampling are reflec- traditional live-trapping study on small mammals, tive of ordinary pre-sampling movements. The use of However, the method is applicable to any study in bait or other attractants during mark-recapture can which mark-recapture data and telemetry data are cause violations of this assumption and deserves special available simultaneously. Such work has been completed consideration. First, under such conditions, we recom- on large carnivores (Miller et al. 1997, Soisalo and mend discarding from the analysis any telemetry Cavalcanti 2006, Sharma et al. 2010), birds (Powell et al. locations obtained during the mark-recapture session 2000, Bächler and Schaub 2007), reptiles (McMaster as it is likely that baited detectors influenced animal and Downs 2006, Grant and Doherty 2007), amphibians movement. Second, it is imperative to remove all bait (Peterman et al. 2008), fish (Naughton et al. 2009), and from the site at the end of the mark-recapture session so even invertebrates (Robinson et al. 2000). Furthermore, there is no unnatural attractant left to influence the estimator can accommodate any means (in addition movements after the session. Third, we suggest that it to telemetry) by which p¡ can be estimated, such as may be appropriate to wait one to two days post mark- tallying the proportion of tracks an individual left on the recapture before collecting location data to allow study site (Marten 1972, Lernen and Freeman 1985). animals to revert to their normal activity patterns. Thus, the method is versatile and could be implemented Telemetry sampling should, however, be completed for a variety of taxa. within a reasonable time to avoid biasing estimates of Density is a fundamental parameter used in a variety Pi due to seasonal movements, migration, or dispersal. of contexts in the field of ecology. Here we described an In summary, our experience suggests that several intuitively appealing method for estimating animal animals may make extensive movements to take density that can be applied in a variety of situations, advantage of baited detectors and become part of the The approach accounts for lack of geographic closure mark-recapture data set, yet they do not spend any time using ancillary information not typically used by other on the study site once bait is removed. Following the estimators. Auxiliary information usually improves guidelines outlined here, telemetry information should parameter estimation (e.g., Burnham 1993, Barker result in p = 0 for these individuals, and they will not 1997, Powell et al. 2000, Gopalaswamy et al. 2012), contribute to the density estimate, which is appropriate, and we expect that to be the case here, especially given Another unique assumption of this estimator is that that we employ auxiliary information that directly the sample of telemetered animals should represent the measures animal movement, which is the root of the population of animals that use the study site. For geographic closure issue. Further, unlike other contem traditional live-trapping studies in which telemetry porary estimators, our approach makes no assumptions devices are deployed on a subset of animals during the about distribution of animals, stationarity of home mark-recapture sampling, one can help meet this ranges, movement patterns, or estimation of effective assumption via sound design. We recommend checking area trapped. It can be easily implemented in the field, traps from a random starting point on each occasion, especially within studies where telemetry is already used This strategy should help equalize the probability of for other purposes. The method can be fully implement capturing and telemetering edge vs. interior animals. In ed in Program MARK ("Density with Telemetry" data addition, we suggest retaining some telemetry devices for type [White and Burnham 1999]) to obtain estimates and deployment during the latter portion of a sampling associated sampling variances, session to facilitate the inclusion of trap-shy individuals in the radio-tagged sample in addition to trap-happy Acknowledgments individuals that are captured early and often. Despite ,We thank. P' Lukacs í°r ¡n formulating the , , r ■ , -i i , , i r telemetry estimator. M. Efford and J. A. Royle provided these design features, it is plausible that the sample of insightful discussions regarding d telemetry-tagged individuals could be biased toward and spatially explicit capture-recapt those with a higher proportion of their range on the support was provided by the Co study site as these individuals stand a better chance of Research Unit. Funding was , , j Division of Wildlife. We thank K. Wilson, P. Doherty, V. capture on any occasion. This phenomenon would „ ■. ,, .„j j • , , , . ^ r Dreitz, M. Alldredge, two anonymous reviewers, and students inflate the numerator (p) of the estimator, resulting in in the Wagar a positively biased density estimate (Efford 2004). drafts of this This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms �April 2013 DENSITY ESTIMATION USING TELEMETRY DATA 815 McMaster, M. K., and C. T. Downs. 2006. Population structure and density of leopard tortoises (Geoche lone Bächler, E., and M. Schaub. 2007. The effects of permanent Literature Cited partialis) on farmland in the Nama-Karoo. Journal of local emigration and encounter technique on stopover Herpetology 40:495-502. duration estimates as revealed by telemetry and mark Miller, S. D., et al. 1997. Brown and black bear density recapture. Condor 109:142-154. estimation in Alaska using radiotelemetry and replicated Balme, G. A., L. T. B. Hunter, and R. Slotow. 2009. 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Radio tracking and animal of Mammalogy 66:13-21. populations. Academic Press, San Diego, California, USA. Zahratka, J. L., and T. M. Shenk. 2008. Population estimates of Williams, B. K.. J. D. Nichols, and M. J. Conroy. 2002. Analysis and management of animal populations. Academic snowshoe hares in the southern Rocky Mountains. Journal of Press, New York, New York, USA. Wildlife Management 72:906-912. Supplemental Material Appendix Estimate of Var(D) when individual-specific covariates are used to estimate D (Ecological Archives E094-070-A1). This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 21:35:55 UTC All use subject to https://about.jstor.org/terms � https://cpw.cvlcollections.org/files/original/d5c31e04eed2965109b9be9bdd754ac0.pdf 057968fc4210e14954df6bf5e0a2281f PDF Text Text Appendix A. Estimate of Var when individual-specific covariates are used to estimate . Similar to the equations described in the text, the basic forms of the estimator and its variance without individual-specific covariates are: · Var where ̂ · 1 Var ⁄ ̂ ⁄ ̂ Var is estimated density (number of animals per unit area), animals detected during a sampling session, is the total number of is the estimated proportion of time an animal spent on the study site, ̂ is the probability an animal is detected one or more times during the sampling session (i.e., if ̂ is the estimated probability an animal is detected on any given occasion, ̂ 1 1 ̂ , where is the number of occasions), and is the area of the study site . However, we expect use of individual-specific covariates to be both helpful and common. To accommodate individual covariates the estimator becomes: ̂ where is the estimated proportion of time animal spent on the study site, ̂ is the probability animal is captured one or more times during the sampling session, and other parameters are as defined above (i.e., and ̂ can now be different for each individual). Analytical solutions for variance of this individual-specific expression are complex. Approximate variance can be estimated more simply using the delta method with numerical �methods to approximate partial derivatives. Thus, to accommodate individual-specific covariates, make the following 3 substitutions for the appropriate quantities in Var 1) Assuming is a binomial random variable, Var 1 is above: , where is the number of animals actually using the site and at risk of capture (i.e., the superpopulation). The number of individuals captured ( ) is an estimate for Var 1 , therefore: ̂ However, if ̂ is to be specific to each individual, the above expression is not appropriate. Instead: Var 1 ̂ 2) Use the delta method approximation (Seber 1982, p. 7) to compute Var ⁄ ̂ : a) Compute a (1 ) vector of approximate partial derivatives for each of the in the density expression (i.e., all parameters used to estimate both parameters [ ,  , …  ] and α , α , … α ]). To accomplish this, successively change (one at a time) each parameter ( or α ) in the density expression ( ∑ by  or α (where  is ⁄ ̂ ⁄ very small) and calculate the difference quotient (e.g., [    / ) for the density expression with each change. The value of the difference quotient at each step is a numerically derived estimate of the partial derivative for that parameter and becomes an element in the vector .    … �parameters ( ,  , …  ) appear first b) Assuming the partial derivatives for the estimated in the vector, the variance-covariance matrix for the density expression consists of the parameters in the upper-left quadrant and variance-covariance matrix of the estimated parameters (α , α , … α ) in the lower- the variance-covariance matrix of the estimated right quadrant. Remaining quadrants are populated by zeros. The estimated variancecovariance matrix for the parameters is can be obtained from the statistical package used to compute the logistic regression. The parameters for come from abundance- estimation software such as Program MARK (White and Burnham 1999). Var Cov Cov , Cov , , Var Cov , Cov , 0 0 0 Cov , 0 0 0 0 0 0 Var Cov 0 0 0 Var 0 0 0 Cov , 0 0 0 Cov , , Var Cov , Cov , Cov , Var c) Multiply the vector of partial derivatives by the variance-covariance matrix then by the transpose of the vector of partials, such that: Var ⁄ ̂ 3) Because · and ̂ are specific to individuals: ⁄ ̂ The final expression for Var( Var where · ⁄ ̂ when using individual-specific covariates: ∑ 1 ̂ · ∑ · ̂ is the form of the estimator that allows for individual covariates. �Note that the variance expression derived here is fully implemented in Program MARK (White and Burnham 1999). LITERATURE CITED Seber, G. A. F. 1982. The estimation of animal abundance and related parameters. 2nd edition. Charles Griffin & Company LTD, London, UK. White, G. C. and K. P. Burnham. 1999. Program MARK: survival estimation from populations of marked animals. Bird Study 46 Supplement:120–138. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Journal Articles Description An account of the resource CPW peer-reviewed journal publications Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Using auxiliary telemetry information to estimate animal density from capture–recapture data Description An account of the resource <span>Estimation of animal density is fundamental to ecology, and ecologists often pursue density estimates using grids of detectors (e.g., cameras, live traps, hair snags) to sample animals at a study site. However, under such a framework, reliable estimates can be difficult to obtain because animals move on and off of the site during the sampling session (i.e., the site is not closed geographically). Generally, practitioners address lack of geographic closure by inflating the area sampled by the detectors based on the mean distance individuals moved between trapping events or invoking hierarchical models in which animal density is assumed to be a spatial point process, and detection is modeled as a declining function of distance to a detector. We provide an alternative in which lack of geographic closure is sampled directly using telemetry, and this auxiliary information is used to compute estimates of density based on a modified Huggins closed-capture estimator. Contrary to other approaches, this method is free from assumptions regarding the distribution and movement of animals on the landscape, the stationarity of their home ranges, and biases induced by abnormal movements in response to baited detectors. The estimator is freely available in Program MARK.</span> Bibliographic Citation A bibliographic reference for the resource. Recommended practice is to include sufficient bibliographic detail to identify the resource as unambiguously as possible. Ivan, J. S., G. C. White, and T. M. Shenk. 2013. Using auxiliary telemetry information to estimate animal density from capture-recapture data. Ecology 94:809–816. <a href="https://doi.org/10.1890/12-0101.1" target="_blank" rel="noreferrer noopener">https://doi.org/10.1890/12-0101.1</a> Creator An entity primarily responsible for making the resource Ivan, Jacob S. White, Gary C. Shenk, Tanya M. Subject The topic of the resource Density Geographic closure Mark-recapture Telemetry Extent The size or duration of the resource. 9 pages Date Created Date of creation of the resource. 2013-04-01 Rights Information about rights held in and over the resource <a href="http://rightsstatements.org/vocab/InC-NC/1.0/" target="_blank" rel="noreferrer noopener">In Copyright - Non-Commercial Use Permitted</a> Format The file format, physical medium, or dimensions of the resource application/pdf Language A language of the resource English Is Part Of A related resource in which the described resource is physically or logically included. Ecology Type The nature or genre of the resource Article