https://cpw.cvlcollections.org/files/original/fe15b02738bd7e4ebc50ab3b7ba69ba7.pdf 1399cfc8866f7991d2a9ecd24454c8f7 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 simulation to compare methods for estimating 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. 817-826 Published by: Wiley on behalf of the Ecological Society of America Stable URL: https://www.jstor.org/stable/23436295 REFERENCES Linked references are available on JSTOR for this article: https://www.jstor.org/stable/23436295?seq=1&cid=pdfreference#references_tab_contents You may need to log in to JSTOR to access the linked references. 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 17:52:42 UTC All use subject to https://about.jstor.org/terms �Ecology, 94(4), 2013, pp. 817-826 © 2013 by the Ecological Society of America Using simulation to compare methods for estimating density from capture—recapture data Jacob S. Ivan,1'3 Gary C, White,1 and Tanya M. Shenk2'4 1 Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado 80523 USA 2Colorado Division of Wildlife, 317 West Prospect, Fort Collins, Colorado 80526 USA Abstract. Estimation of animal density is fundamental to wildlife research and management, but estimation via mark-recapture is often complicated by lack of geographic closure of study sites. Contemporary methods for estimating density using mark-recapture data include (1) approximating the effective area sampled by an array of detectors based on the mean maximum distance moved (MMDM) by animals during the sampling session, (2) spatially explicit capture-recapture (SECR) methods that formulate the problem hierarchi cally with a process model for animal density and an observation model in which detection probability declines with distance from a detector, and (3) a telemetry estimator (TELEM) that uses auxiliary telemetry information to estimate the proportion of animals on the study site. We used simulation to compare relative performance (percent error) of these methods under all combinations of three levels of detection probability (0.2, 0.4, 0.6), three levels of occasions (5, 7, 10), and three levels of abundance (10, 20, 40 animals). We also tested each estimator using five different models for animal home ranges. TELEM performed best across most combinations of capture probabilities, sampling occasions, true densities, and home range configurations, and performance was unaffected by home range shape. SECR outperformed MMDM estimators in nearly all comparisons and may be preferable to TELEM at low capture probabilities, but performance varied with home range configuration. MMDM estimators exhibited substantial positive bias for most simulations, but performance improved for elongated or infinite home ranges. Key words: closure: density: geographic closure: mean maximum distance moved: simulation: spatially explicit capture-recapture: telemetry; trapping grid. Introduction the super population is larger than the study site itself. Animal density is a fundamental parameter in ecology, and practitioners often estimate density using mark-recapture techniques in conjunction with arrays of This makes abundance estimates difficult to convert to density because the area effectively sampled by the detectors (i.e., the area used by the super population) is unknown. live traps, cameras, hair snags, or other detection , . , . , , ,. Traditionally, the most common strategy for manag devices. A common issue under such a sampling ., . . , , , . . . ' . ing the geographic closure issue has been to attempt framework ,s lack of geographic closure. That is, estimation of the effecti animals move on and off of the study site (e.g., a and then divjde this es convex polygon around the detectors) during the estimate obtained from sampling period. Accordingly, estimates of animal a corrected estimate of abundance obtained from traditional closed-capture distance moved (MMD models (Otis et al. 1978, White et al. 1982, Huggins sampling session, 1989, 1991, Williams et al. 2002) reflect the super averaged across all population of animals that could have used the study 0nce, and the study s site during the sampling period rather than the number one-half this dist of animals within its boundaries (Schwarz and Arnason sampled (Wilson a 1996, Kendall et al. 1997). Intuitively, the area used by More recently, spa (SECR) techniques have been introduced a tive (Efford 2004, Borchers and Efford 2008, Manuscript received 17 January 2012; revised 14 September 2012; accepted 20 September 2012; final version received 15 20096). Like the November 2012. Corresponding Editor: E. G. Cooch. spatial information co 3 Present address: Colorado Parks and Wildlife, Fort each individual. Collins, Colorado 80526 USA. E-mail: Jake.Ivan@state.co.us information to estimate parameters of an observation 4 Present address: National Park Service, Fort Collins, model in which detection probability declines as a Colorado 80525 USA. function of the distance between an animal's home range 817 This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �818 JACOB S. IVAN ET AL. Ecology, Vol. 94, No. 4 center and a given detector. The density of home ran centers is represented as a separate process model an the two sub-models are combined hierarchically such that density is estimated directly given the data (Effo 2004, Borchers and Efford 2008, Royle et al. 2009a, b). Early versions of SECR (inverse prediction) performed well in simulation (Efford 2004) and field experimen (Efford et al. 2005). Currently SECR analyses can accomplished using inverse prediction (Efford 2004), maximum likelihood (Borchers and Efford 2008) Bayesian (Royle et al. 2009a, b) approaches. Uniform(0,l) and ordering them from smallest to Auxiliary radio-telemetry data has been suggested as largest. We assigned probability of use for home range yet another means to address geographic closure (Boutin cell i (P¡) as P¡ = xí+J - x¡, for i = 2, 3, ..., 15. We 1984, White and Shenk 2001, Ivan et al. 2013). With this similarly computed use for the two remaining cells: Pt = approach, telemetry is used to monitor animals during x\ - 0 and Pl6 = 1 - x15. or immediately after mark -recapture sampling to Once an animal was placed in the simulation arena estimate the proportion of time they spend on the study and assigned its own randomly generated utilization site. These proportions are used to scale the super distribution, we simulated a capture history for that population estimate to only those animals and "frac- individual. For each capture occasion, we compared the tions of animals" that use the site. The corrected super product of the capture probability specified for the population estimate divided by the area of the site simulation and the probability that the animal was on provides an estimate of density accounting for lack of the study site (summation of probability of use across all geographic closure. home range cells that overlapped the grid) to a random We provide a simulation-based comparison to evalu- draw from Uniform(0,l). Products less than the rando ate relative performance of these three classes of draw resulted in a "capture." When captures occurred, estimators under a variety of sampling conditions. We we assigned the location of capture probabilistical hypothesized that (1) the telemetry estimator would based on the utilization distribution for that animal su generally perform best because it makes use of auxiliary that detectors in cells where the animal was more lik information about animal movement on and off the to occur were more likely to capture the animal. We study site, which is generally not used by the other simulated a home range location, utilization distribu estimators (but see Gopalaswamy et al. [2012] and J. A. tion, and capture history in a similar manner (and Royle and R. B. Chandler, unpublished manuscript, for independent of previous animal locations or captu potential advancements using telemetry to help estimate histories) for all animals in the simulation, parameters in SECR models), and (2) in the absence of Each simulation was governed by a specific combin auxiliary information, SECR would perform better than tl0n of capture probability, number of occasions, a MMDM due to a stronger theoretical basis. animals released into a simulation. We considered three We refer based mum use the terms TELEM, MMDM, and SECR to levels for each of these three factors to represent a ran generally to the three classes of density estimators 0f conditions commonly encountered in field resea on auxiliary telemetry information, mean maxi- (capture probability for any single occasion = 0.2, 0.4 distance moved, and spatially explicit capture- and q g. 5^ 7^ and jq occasions; 10, 20, and 40 animals recapture, respectively. Additional modifiers to these reieased into the simulation). We completed 1 terms indicate a specific form of the estimator. For simulations for all 27 possible combinations resultin example, TELEM50 references application of the ¡n 77 000 data sets. Simulations were carried out us telemetry estimator to a case where 50% of the animals SAS 9 2 (SAS Institute, Cary, North Carolina, USA), captured during mark-recapture sampling were also For context! we assumed grid cells were 10 m o sampled via telemetry (e.g., Ivan 2011); 1/2 MMDM sjtjCi resulting in densities of 4-16 animals/ha (10-40 refers to estimates based on approximating effective area animals released int0 2 56_ha arena)> which is consisten sampled as one-half of the mean maximum distance wjth research on voles, mice, and other small rodent moved between detection events (e.g., Wilson and (e g; Hadley and wilson 2004; Tioli et al 2009) Anderson 1985), ML SECR references estimates trom However, the absolute spatial scale of the simulation the maximum likelihood version of SECR (Borchers and inconsequential and does not affect relative performa Efford 2008, Efford 2012). 0f the estimators. For example, we could have assumed Methods 50-m cells resulting in true simulated densities of 0.1-0.6 animals/ha, which corresponds to research on squirrel or Simulation specifications rabbit-sized species (e.g., Zahratka and Shenk 2008, The simulation arena (i.e., simulated landscape) Russell et al. 2010). consisted of a 16 X 16 grid of cells in which we centered Initial simulations were completed with all animals a 10 X 10 grid of detectors (Fig. 1). We defined the 100 assigned a 4 X 4 home range and we assessed This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �April 2013 COMPARING DENSITY ESTIMATION TECHNIQUES 819 Fig. 1. A simulated 10x10 trapping grid centered within a 16 cell X 16 cell simula home ranges were represented by a 4 X 4 square, and utilization of that home range simulations exploring the role of home range shape on estimator performance, we m include (b) 2X8, (c) 16-cell irregular (home range boundary is random for each anim is adjacent to at least one other cell), (d) bivariate normal "circles" (av = cy), an addition to the 4X4. Note that utilization distributions for the 4 X 4, 2 X 8, an multinomial distributions. Here, we smoothed probability of use across cells w comparison to the other home range configurations. performance across a range of capture probabilities, cells were adjace occasions, and true densities (see below). We then cells as before. For bivariate normal home ranges, we assessed the influence of home range shape by holding assigned a such that the 95% home range encompassed capture probability, occasions, and true density at an area equal to 16 cells. Thus, for this comparison, intermediate levels (p = 0.4, seven occasions, 20 animals home ranges varied only in shape, not size. Our released into the simulation) while varying home range simulated capture process required computation of the shape and use from regular to irregular. Specifically, we probability of use within each cell on the landscape. For completed batches of simulations in which each animal the simulations involving bivariate normal home ranges, was assigned a circular, bivariate normal home range we accomplished this using the probbnrm function in (av = av, Fig. Id [where ax and crv are standard SAS, which integrates the bivariate normal probability deviation in the .v and y direction]), a bivariate normal density "southwest" of a point given the mean (home ellipse (cyA. < a,., Fig. le), a 4 X 4 home range utilization assigned randomly as described above la), a 2 X 8 home range with randomly assigned utilization (Fig. lb), and a 16-cell irregular home with range center) and SD specified in the simulation. By (Fig. computing this quantity for the "northeast" corner of each cell, then subtracting off the quantity computed at range the other three corners, and adding back in the piece boundary with randomly assigned utilization (Fig. lc). that was subtracted twice, we derived the probability of The latter home range was created by selecting a single use for each cell. We ran 1000 simulations for each of the cell at random within the simulation arena, adding an five home range shapes. adjacent cell at random, adding a third cell adjacent to Analysis of simulated data sets either of the previous two, and so on, until the home range consisted of 16 cells. This procedure could have We analyzed each data set using full MMDM, 1/2 produced any shape under the constraint that the 16 MMDM (Wilson and Anderson 1985), ML SECR This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �820 JACOB S. IVAN ET AL. Ecology, Vol. 94. No. 4 (Borchers and Efford 2008), and 12 forms of TELEM (Ivan et al. 2013): four levels representing the percent of captured animals that were telemetered (25%, 50%, 75%, 100%) and three levels of telemetry sampling (5, and 20 locations obtained per individual via telemetry), To simulate collection of locations for each telemeter individual, we assumed the number of locations on th sampling grid was ~Binomial (N, p¡) where N = number of locations specified in the simulation and p¡ = est proportion of home range on the study site for anim This arrangement does not allow for telemetry error, but if telemetry error in the field is unbiased with respe to on/off grid, such a model is adequate. encompassed by the study site plus a buffer. ML SECR We did not build time or behavioral effects into the aims to estimate the rate at which home range centers mark-recapture portion of the simulations, nor did we occur on the landscape. TELEM tries to estimate the simulate heterogeneity among individuals except for that number of animal equivalents (whole and partial induced by location of home ranges relative to the study animals) within the study site. Thus, any single site. Therefore, we employed basic forms of each definition of truth based on one estimator could stack estimator to produce density estimates from each the deck against the other two. To be fair, we defined simulation. Specifically we used a conditional likelihood truth for MMDM estimators as the number of animals (Huggins 1989, 1991), null closed-capture model (i.e., released into the arena, divided by the area of the arena, model M0; Otis et al. 1978) to estimate abundance (N) For ML SECR, we defined it as the number of home under the MMDM approaches. For the observation range centers within the study site (which we tallied portion of the ML SECR model, we specified a constant using a weighted average of the 16 cells comprising the half normal detection function for each trap (go(') cr(-)), home range for each animal), divided by the area of the using the conditional likelihood formulation for multi- study site. For TELEM, we tallied the proportion of catch traps, with the default 100-m integration buffer each animal on the study site, summed these proportions around traps. Choice of integration buffer is user across all animals, then divided by the area of the site, defined and should be large enough to include all animals with a non-zero probability of detection Assessment of performance (Borchers and Efford 2008). Given that our simulation Our use of realistic parameter inputs complicated specifications required all animal home range centers to summarization as some combinations of parameters fall within 30 m of the study site (or about 150 m from resulted in no point estimates (and/or no SEs) or the farthest detector), we felt the default buffer was unrealistically large point estimates (and/or unrealisti conservative, as it would encompass all animals with any cally large SEs) due to numerical problems with probability of being "captured" by any detector at the optimizing the likelihood function, or chance construc site. However, we also tested the suggest.buffer function tion of very poor data sets. Such results were observed within the software (which returned values of —30-40 for each class of estimator, estimators did not always fail m) and we provided a "mask" to limit integration to the in concert, and no estimator routinely produced bounds of the simulation arena (Efford 2012). Neither estimates when others did not. Thus, usual data approach changed results perceptibly (median difference summaries involving measures of central tendency, in density estimates compared to the default buffer was dispersion, mean squared error, or evaluation of linear <0.6% for n = 1000 simulations of regular, bivariate, models relating performance to input parameters (i.e., normal home ranges and n = 1000 irregular home ANOVA or AIC) were not possible without censoring ranges), so we proceeded with the default buffer for all unrealistic results. However, censoring could not be simulations. For the process portion of ML SECR, we accomplished objectively as the distribution of estimates specified that density of home range centers followed a blended smoothly from those that seemed reasonable to homogeneous Poisson distribution. For versions of the those that were unreasonable, with no natural break telemetry estimator in which <100% of captured along which to delineate the two. Relative performance animals were telemetered, we used a single individual ranking of the estimators (as measured by central covariate (distance to the edge of the study site from the tendency, dispersion, and so on), depended on where mean capture location of individual i) to model we chose to censor. To maintain objectivity, we opted to detection probability (p¡) and "proportion on grid" retain all results and present them graphically as follows. {pi) for individuals captured but not telemetered (Ivan et We calculated percent error (PE) for each simulation al. 2012). This represents the minimum model likely to ( PE = [(D - D)/D] X 100, where Z) = the appropriate true be implemented in practice when more animals are density), ordered the simulations by PE in ascending captured than can be telemetered. Estimates for full fashion, then plotted these values against their percentile MMDM, 1/2 MMDM, and TELEM were computed forming a cumulative distribution plot (Fig. 2). The This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �April 2013 COMPARING DENSITY ESTIMATION TECHNIQUES 821 cumulative plot of PE for a perfect estimator would be error-free for each simulation and its curve would never deviate from zero. Intuitively, then, curves that ap proach zero "quickly" and remain near it for the greatest percentage of simulations represent the most desirable estimators. Two curves that track each other nearly perfectly indicate two estimators performing similarly; two curves that separate quickly indicate disparate performance. Better estimators have curves with rela -20 tively "flat" mid-sections and produce unreasonable -40 results in only a small percentage of simulations in their tails. To facilitate comparisons, we arbitrarily quantified -60 the flatness of curves by calculating the percentage of simulations in which PE was within ±20%. For initial Telem25 Telem50 Telem75 TelemlOO 1/2MMDM -80 Full MMDM ML SECR assessments of overall performance, all simulations from -100 ' all combinations of sampling parameters were included O^O 0.1 0.2 0.3 0.4 0^5 0.6 0.7 0.8 0.9 in the curve for each type of estimator (i.e., each curve Percentile of ordered results represented 27 000 data sets). We used similar plots to assess estimator performance given various home range Fig. 2. Cumulative percentage error (PE = [(D - D)/D] X configurations described in Simulation specifications. 100, where D = true density) for seven density estimators confronted with the same simulated data sets. Estimators Interactions and sensitivity included those that attempted to address the issue of geographic closure using auxiliary telemetry information (TELEM) col To determine how interactions between parameterslected on 25%, 50%, 75%, or 100% of captured animals, those that attempted to address the issue by adjusting the area of the and parameter levels influenced estimator performance, study site (full and 1/2 MMDM), and one that formulated the we also created cumulative distribution plots for each problem hierarchically by fitting sub-models for a detection unique parameter-level combination (e.g., animals function and spatial point process (ML SECR). Each curve released into the simulation, low; occasions, medium; represents estimates from 27 000 data sets: 1000 simulated for capture probability, high). Twenty-seven such combieach combination of capture probability (0.2, 0.4, 0.6), occasions (5, 7, 10), and true density (10, 20, 40 animals nations were possible based on the initial simulation released into arena). PE was calculated for each simulation, specification (i.e., holding home range constant). then values were ordered smallest to largest and plotted against their percentile. While all estimates were used to create plots, we Estimator sensitivity to a particular parameter can also be assessed using these plots by focusing on the only show those in the range of ±100% error to facilitate comparisons among estimators (i.e., extremely large errors are relative change in performance as the level of a given off the plot). sampling parameter changes. For example, if for a given estimator, the three curves representing three levels of a given parameter are nearly identical, then the estimator is insensitive to changes in that parameter (at Interactions of sampling parameters TELEM estimators performed best for 20 of the 27 least over the range of simulated values). If the curve parameter combinations (Appendix: Figs. A1-A3). MMDM estimators returned highly positive errors changes drastically as the parameter level is incre under all scenarios. No estimator performed well given mented, then the estimator is sensitive to that low levels of each sampling parameter (Fig. Al). parameter. After creating the 27 plots of unique Estimators were most sensitive to capture probability parameter-level combinations, and assessing model and large gains in performance occurred with each sensitivity, we assessed TELEM sensitivity to the estimator as capture probability was increased from 0.2 number of telemetry locations (levels 5, 10, 20) using to 0.4 (Figs. A1 A3; performance increases most from a similar strategy. left panels to right panels). Relatively smaller gains were Results realized as capture probability was increased further to 0.6. When capture probability was low (p = 0.2; Figs. Overall performance A1-A3, left-most panels), ML SECR usually outper With respect to PE, 1/2 MMDM performed formed poorly, as estimators, but TELEM performed other over 80% of estimates had PE >50% (Fig.similarly 2). Fullif occasions were high. When capture proba MMDM performed better (curve stayed closer to 0 bility was intermediate (p = 0.4) or high {p — 0.6), longer), but was inferior to ML SECR and TELEM. TELEM estimators generally performed best (Figs. Al TELEM 100 and TELEM75 performed best using the A3) regardless of the number of animals or occasions PE metric; ~75% of estimates were within ±20 PE. simulated. TELEM estimators were relatively insensitive Fifty-three percent of ML SECR estimates were withinto number of locations obtained per individual (Fig. ±20 PE. All methods produced more positive than A4). Overall ML SECR performed most consistently and was least sensitive to changes in parameter levels. negative errors (Fig. 2). This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �822 JACOB S. IVAN ET AL. Ecology, Vol. 94, No. 4 b) a) BVN circle y I, c) 4 X 4 j\ BVN J ellipse d) 2 x 8 J 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 /i e) Irregular Percentile of ordered results ■ — Telem75 1/2MMDM Full MMDM MLSECR i i i i i i 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Percentile of ordered results Fig. 3. Cumulative percentage error for four density estimators confronted with simulated data sets in which capture probability, occasions, and number of animals released into the simulation were fixed to intermediate levels (0.4, 7, and 20, respectively) and home range size varied from regular (bivariate [BVN] normal circle) to highly irregular (16 cells allowed to take any shape in which each cell is adjacent to > 1 other cell). See Fig. 1 for a description of home range types. Influence of home range irregular (Fig. 3b, d, e; PE was within ±20% in 43-53% . __ r j ■ of simulations). Performance of Full MMDM estima With respect to PE, TELEM75 performed best across . ; , , _ . , . . , , . tors improved dramatically given bivanate normal the range of simulated home range shapes and its . , . . , , . . . ., B or circles, or irregular home ranges compared to the 4 X performance was unaffected by home range configura- 4 configuration (Fig tion (Fig. 3; PE was ±20% in 82-90% of simulations for each shape). ML SECR performed well for circular and Discussion square home ranges (Fig. 3a, c; PE was within ±20% in Our simulations represent the first tests of the —60% of simulations), but was more likely to produce telemetry estimator across a wide range of conditions, negative errors when home ranges were elongated or It is also the first comparison among the three This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �April 2013 COMPARING DENSITY ESTIMATION TECHNIQUES 823 contemporary estimators that use primarily mark- at least once du recapture data to estimate density from arrays of area of the grid), detectors. As such we provide an overview of the (Efford 2004). How relative merits of each approach based on these tion of their ra simulations as well as other characteristic of the higher capture pro methods. denominator of the estimator as well, potentially We found variations of TELEM performed best canceling out some of th across most combinations of capture probabilities, should be perform sampling occasions, and true densities. Its performance suggest that design was not affected by the home range model assumed to this problem to som govern movement of animals on the landscape. That TELEM (and MMD TELEM performed well was not surprising given it uses distributed at a rela auxiliary information that provides a direct measure of (approximately fou the process leading to lack of geographic closure - study area. SECR does movement of individuals on and off of the study site. In cases where This information is generally not used by other TELEM is otherwise n estimators (but see Gopalaswamy et al. [2012] and SECR (or Bayesian J. A. Royle and R. B. Chandler, unpublished manuscript, similarly) as it ou for recent advancements in combining telemetry with in nearly all compar SECR). Although we did not explicitly test this background. When c phenomenon, we also expect TELEM to outperform poorly estimated, other estimators in sampling situations that involve TELEM as well. Othe baited detectors. In that case, it is plausible, if not likely, following: altern that animal movements will be influenced by bait and account for differen thus will not be representative of usual movement can occur (e.g., "m patterns. Because the only spatial information typically observations" in w available to SECR and MMDM is that derived from the only one location vs detectors, information upon which those models are observations" in which built could be biased. TELEM allows use of spatial times at multiple locatio information collected outside of the period when bait Gardner 2009]). was used, thus providing opportunity to minimize this modeling the detec bias. Finally, TELEM makes no assumption about heterogeneity among distribution of animals on the landscape. Here we provide a coarser treatm simulated this phenomenon in random fashion, but the practitioners may estimator should be able to handle any sort of be used in the SECR approach. SECR also has the distribution (i.e., territoriality). added potential advantage in that density can be Disadvantages of the TELEM approach include modeled separately for different portions of the stud additional costs associated with obtaining telemetry area or directly as a function of covariates (Borcher data. This may preclude its use in many situations. Also, Efford 2008). TELEM and MMDM report a sin we found an increased probability of TELEM returning density estimate for the study site, so if the site c positive errors at low capture probability. This is likely multiple habitat types, an average density acros due to its construction as a Horvitz-Thompson estima- types is reported. Similarly, with the SECR appro tor, with capture probability (p) in the denominator of capture probability can be modeled as a functi the expression. When this denominator becomes very trap-level covariates if desired whereas TELEM a small and/or is estimated poorly, density estimates are MMDM model detection at a more coarse grid- inflated. Another concern with TELEM arises during (Borchers and Efford 2008). Because SECR the special case in which animals are telemetered during detection as a trap-level process, spacing of dete mark-recapture sampling (rather than having been may be less important than for TELEM or MM telemetered prior). In this instance it is possible that approaches as well. individuals with a greater proportion of their home The standard SECR implementation makes range on the study site would be more likely to be assumption that exposure to capture is governed b captured and telemetered than animals with a home the distance from the home range center to a giv range near the periphery (Efford 2004). This phenom- detector and that this function is monotonicall enon would inflate the numerator of the TELEM decreasing (Borchers and Efford 2008, Royle a estimator (Ivan et al. 2013): D = (Y^=i(Pí/Pí))IA, Young 2008). This, in turn, forces a model in w where Mt+l is the number of animals captured, p¡ animal movements are symmetric, circular probabil estimated proportion of home range on grid for animal distributions (Royle et al. 2013). This home range i, pf is the estimated probability animal i was captured is a special case of the Jennrich and Turner (1969) This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �824 JACOB S. IVAN ET AL. Ecology, Vol. 94, No. 4 range estimator, which has seldom been found to b realistic (White and Garrott 1990). Further, in practice it is usually assumed that all animals have the same fi home range size, which is restrictive, although with w enough data, one can allow home ranges to vary by group (e.g., sex) or individual covariate. Efford (200 noted that SECR may not perform well given elongat home ranges. Our simulations substantiate this claim, We found that SECR performed well given circular or square home ranges, but we observed substantial man negative errors when we confronted it with elongated or irregular home ranges. We suggest animal movement are likely governed by a variety of factors in addition distance from a home range center including habit preference, patchiness of the landscape, gradients in elevation or moisture, interspecific interactions, intraspecific interactions, and individual behaviors. Practitioners should proceed with caution given systems in which heterogeneous habitats and/or animal behavior could produce irregular home ranges, or at least W recognize that estimates may be lower than true density, Fortunately, this type of error would be viewed as fo conservative in most applications. For both harvested i and rare species, it may be better from a management perspective to operate with information that errs in t direction, than to assume that density is higher than really is. Furthermore, the error we observed here w far less than that observed for MMDM, and the ma estimator was more robust than might be expected be given this assumption. estimates of the effective area sampled and decreased Royle et al. (2012) and J. A. Royle and R. B. estimates of density. To the degree that one Chandler (unpublished manuscript) noted the same assume infinite home ranges (or some other deficiencies in the standard SECR model in their shape) are adequate representations of a s simulation work. They suggested the negative bias was Full MMDM may perform adequate due to a failure to account for various sources of hard-edged, compact, home ranges may heterogeneity in space use, similar to how a null closed- abstraction of animal movements over s capture model (A/0) underestimates abundance when sessions as the daily routine of an individ heterogeneity is present (Otis et al. 1978). They also confined to a fairly discrete area, produced novel ways of dealing with this issue via Regardless of the estimator, performan inclusion of a least cost surface or resource selection sensitive to changes in capture probability. Thus, function into the observation portion of the model, practically speaking, investment in that aspect of a These are significant advancements, but require that project is likely to pay the largest dividends. However, important variables (landscape, inter- or intraspecific our simulations were based on only a few design points interactions, behavior) governing movement can be typical of traditional live-trapping studies. We did not identified, measured, combined appropriately into a simulate special cases where detectors such as cameras model, and estimated well using capture-recapture data or hair snags are deployed for >10 occasions. We expect alone or in conjunction with auxiliary telemetry data, a large number of occasions to better aid in overcoming The small number of detections per animal commonly potentially low capture probability and/or animal observed in mark-recapture data sets typically precludes density, but we cannot say how this might affect any but the simplest models, so that results are very performance of the estimators relative to each other, model specific with little opportunity to evaluate more TELEM estimators were relatively insensitive to number realistic alternative models. The basic TELEM estimator of locations, and we suggest that if this method is is less concerned with why or how animals move; it only employed and design tradeoffs are necessary, it is better requires that the process is well-sampled. Even for more to telemeter more individuals (e.g., move from TEL complicated formulations (i.e., when <100% of cap- EM25 to TELEM50), than it is to collect more locations tured animals are telemetered), it only requires that the per individual. binary outcome of being on or off the study site can be In addition to consideration of the merits of each predicted adequately using one or more covariates (e.g., estimator discussed above, we urge ecologists to consult This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms �April 2013 COMPARING DENSITY ESTIMATION TECHNIQUES 825 Huggins, R. M. 1989. On thestudy statistical analysis of capture the Appendix when designing a density estimation recapture experiments. Biometrika 76:133-140. to gain perspective regarding estimator performance Huggins, R. M. 1991. Some practical aspects of a conditional given anticipated field conditions. Note however, that likelihood approach to capture experiments. Biometrics matching estimates of detection probability from pilot 47:725-732. work to the true detection probability parameters Ivan, J. S. 2011. Density, demography, and seasonal movement of snowshoe hares central Colorado. Dissertation. depicted in the figures is nuanced. Estimates ofindetection Colorado State University, Fort Collins, Colorado, USA. probability from pilot work are the product of true Ivan, J. S., G. C. White, and T. M. Shenk. 2013. Using auxiliary detection probability (i.e., the probability of detecting telemetry information to estimate animal density from an animal given that it is available—the number that 94:809-816. capture-recapture data. Ecology Jennrich, in R. I., and F. B. Turner. 1969. Measurement of non went into the simulations and appears the figures), circular homean range. Journal is of Theoretical Biology 22:227 and availability (i.e., the probability that animal 237. actually available for detection). For example, if animals Kendall, W. L., J. D. Nichols, and J. E. Hines. 1997. Estimating are thought to be available 50% of the time, then pilot temporary emigration using capture-recapture data with working suggesting p = 0.2 would correspond well with Pollock's robust design. Ecology 78:563-578. Otis, D. L., K. P. Burnham, C. White, and D. R. Anderson. lines for p = 0.4 in the Appendix. We suggest thatG.the 1978. Statistical inference from capture data on closed animal greatest utility can be gained by focusing on two to three populations. Wildlife Monographs 7-135. figures that adequately bracket anticipated conditions, Parmenter, R. R., et al. 2003. Small-mammal density estima and assessing estimator performance generally. Making tion: a field comparison of grid-based vs. web-based density use of ancillary data is likely to estimators. be helpful in Monographs many Ecological 73:1-26. situations and we urge consideration Royle, J. ofA., such R. B. Chandler, methods K. D. Gazenski, and T. A. Graves. 2013. Spatial capture-recapture models for jointly where appropriate. Acknowledgments estimating population density and landscape connectivity. Ecology 94:287-294. Royle, J. A., and B. Gardner. 2009. Hierarchical spatial We thank P. Lukacs for assistance in formulating the capture-recapture models for estimating density from trap telemetry estimator. M. Efford and J. A. Royle provided ping arrays. Pages 163-190 in A. O'Connell, J. D. Nichols, insightful discussions regarding density estimation in general and spatially explicit capture-recapture in particular. and We U. K. Karanth, editors. Camera traps in animal ecology: methods and analyses. Springer, New York, New York, particularly appreciate M. Efford's assistance with initial USA. implementation of the R package seer. Logistical support was Royle, J. A., K. U. Karanth, A. M. Gopalaswamy, and N. S. provided by the Colorado Cooperative Wildlife Research Unit. Kumar. 2009a. Bayesian inference in camera trapping studies Funding was provided by the Colorado Division of Wildlife. We thank M. Efford, k. Wilson, P. Doherty, E. Bergman,for M. a class of spatial capture-recapture models. Ecology 90:3233-3244. Rice, Larkin Powell, the "Wagar 113 Superpopulation," and an Royle, anonymous reviewer for useful comments on previous drafts of J. A., J. D. Nichols, K. U. Karanth, and A. M. Gopalaswamy. 2009b. A hierarchical model for estimating this manuscript. density in camera-trap studies. Journal of Applied Ecology Literature Cited 46:118-127. Royle, J. A., and K. V. Young. 2008. A hierarchical model for spatial capture-recapture data. Ecology 89:2281-2289. maximum likelihood methods for capture-recapture studies. Russell, R. E., J. F. 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Supplemental Material Appendix Relative performance of estimators across all combinations of capture probability, occasions, and abundance that were evaluated via simulation, and evaluation of number of locations on performance of the TELEM estimator (Ecological Archives E094-071-A1). This content downloaded from 174.205.238.193 on Thu, 21 Oct 2021 17:52:42 UTC All use subject to https://about.jstor.org/terms � https://cpw.cvlcollections.org/files/original/55692878020c22a1c331fa91a5fd4e5f.pdf eed59445b09cac79bf8bb8943107b1ed PDF Text Text Ecological Archives E094-071-A1 Jacob S. Ivan, Gary C. White, Tanya M. Shenk. 2013. Using simulation to compare methods for estimating density from capture–recapture data. Ecology 94:817–826. http://dx.doi.org/10.1890/12-0102.1 Appendix A. Relative performance of estimators across all combinations of capture probability, occasions, and abundance that were evaluated via simulation, and evaluation of number of locations on performance of the TELEM estimator. Fig. A1. Cumulative percent error (PE = ( - D / D × 100%), where D = true density) for simulated data sets in which 10 animals were released into each simulation (density = 4 animals/ha for 10-m trap spacing, 0.16 animals/ha for 50-m spacing) for all levels of capture probability (0.2, 0.4, 0.6) and all levels of sampling occasions (5, 7, 10). �Fig. A2. Cumulative percent error (PE = ( - D / D × 100%), where D = true density) for simulated data sets in which 20 animals were released into each simulation (density = 8 animals/ha for 10-m trap spacing, 0.3 animals/ha for 50-m spacing) for all levels of capture probability (0.2, 0.4, 0.6) and all levels of sampling occasions (5, 7, 10). �Fig. A3. Cumulative percent error (PE = ( - D / D × 100%), where D = true density) for simulated data sets in which 40 animals were released into each simulation (density = 16 animals/ha for 10-m trap spacing, 0.6 animals/ha for 50-m spacing) for all levels of capture probability (0.2, 0.4, 0.6) and all levels of sampling occasions (5, 7, 10). �Fig. A4. Cumulative percent error (PE = ( - D / D × 100%), where D = true density) for TELEM estimator using 5, 10, and 20 locations obtained per individual. All combinations of occasions, number of animals, and detection probability are pooled in this figure. [Back to E094-071] � 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 simulation to compare methods for estimating density from capture–recapture data Description An account of the resource <span>Estimation of animal density is fundamental to wildlife research and management, but estimation via mark–recapture is often complicated by lack of geographic closure of study sites. Contemporary methods for estimating density using mark–recapture data include (1) approximating the effective area sampled by an array of detectors based on the mean maximum distance moved (MMDM) by animals during the sampling session, (2) spatially explicit capture–recapture (SECR) methods that formulate the problem hierarchically with a process model for animal density and an observation model in which detection probability declines with distance from a detector, and (3) a telemetry estimator (TELEM) that uses auxiliary telemetry information to estimate the proportion of animals on the study site. We used simulation to compare relative performance (percent error) of these methods under all combinations of three levels of detection probability (0.2, 0.4, 0.6), three levels of occasions (5, 7, 10), and three levels of abundance (10, 20, 40 animals). We also tested each estimator using five different models for animal home ranges. TELEM performed best across most combinations of capture probabilities, sampling occasions, true densities, and home range configurations, and performance was unaffected by home range shape. SECR outperformed MMDM estimators in nearly all comparisons and may be preferable to TELEM at low capture probabilities, but performance varied with home range configuration. MMDM estimators exhibited substantial positive bias for most simulations, but performance improved for elongated or infinite home ranges.</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 simulation to compare methods for estimating density from capture-recapture data. Ecology 94:817–826. <a href="https://doi.org/10.1890/12-0102.1" target="_blank" rel="noreferrer noopener">https://doi.org/10.1890/12-0102.1</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 Closure Density Geographic closure Mean maximum distance moved Simulation Spatially explicit capture-recapture Telemetry Trapping grid Extent The size or duration of the resource. 10 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