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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, ViceChair
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. 817826
Published by: Wiley on behalf of the Ecological Society of America
Stable URL: https://www.jstor.org/stable/23436295
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�Ecology, 94(4), 2013, pp. 817826
© 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 markrecapture is often complicated by lack of geographic
closure of study sites. Contemporary methods for estimating density using markrecapture
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 capturerecapture (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 capturerecapture: 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
markrecapture 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 closedcapture 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 onehalf 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.
Email: 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
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�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 submodels 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 radiotelemetry 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 simulationbased 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 markrecapture 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 416 animals/ha (1040
refers to estimates based on approximating effective area animals released int0 2 56_ha arena)> which is consisten
sampled as onehalf 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 50m cells resulting in true simulated densities of 0.10.6
animals/ha, which corresponds to research on squirrel or
Simulation specifications rabbitsized 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
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�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) 16cell 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 16cell 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
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�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
markrecapture 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 closedcapture 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 100m 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 nonzero 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 —3040 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
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�April 2013 COMPARING DENSITY ESTIMATION TECHNIQUES 821
cumulative plot of PE for a perfect estimator would be
errorfree 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" midsections 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 submodels for a detection
unique parameterlevel 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). Twentyseven 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. A1A3).
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
parameterlevel 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
A1A3, leftmost 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
Fiftythree 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).
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�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 4353%
. __ 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 8290% 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
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�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 HorvitzThompson 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 traplevel 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 traplevel process, spacing of dete
markrecapture 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)
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�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 hardedged, 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 livetrapping studies. We did not
identified, measured, combined appropriately into a simulate special cases where detectors such as cameras
model, and estimated well using capturerecapture 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 markrecapture 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 wellsampled. 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
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�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 76:133140.
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
47:725732.
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 94:809816.
capturerecapture
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 capturerecapture data with
working suggesting p = 0.2 would correspond
well with
Pollock's robust design.
Ecology 78:563578.
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 7135.
figures that adequately bracket anticipated
conditions,
Parmenter, R. R., et al. 2003. Smallmammal density estima
and assessing estimator performance
generally. Making
tion: a field comparison of gridbased vs. webbased density
use of ancillary data is likely to estimators.
be helpful
in Monographs
many
Ecological
73:126.
situations and we urge consideration
Royle, J.
ofA.,
such
R. B. Chandler,
methods
K. D. Gazenski, and T. A.
Graves. 2013. Spatial capturerecapture models for jointly
where appropriate.
Acknowledgments
estimating population density and landscape connectivity.
Ecology 94:287294.
Royle, J. A., and B. Gardner. 2009. Hierarchical spatial
We thank P. Lukacs for assistance in formulating the
capturerecapture models for estimating density from trap
telemetry estimator. M. Efford and J. A. Royle provided
ping arrays. Pages 163190 in A. O'Connell, J. D. Nichols,
insightful discussions regarding density estimation in general
and spatially explicit capturerecapture 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 capturerecapture models. Ecology
90:32333244.
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.
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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
E094071A1).
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�
https://cpw.cvlcollections.org/files/original/55692878020c22a1c331fa91a5fd4e5f.pdf
eed59445b09cac79bf8bb8943107b1ed
PDF Text
Text
Ecological Archives E094071A1
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/10.1890/120102.1
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 10m trap spacing, 0.16 animals/ha for 50m 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 10m trap spacing, 0.3 animals/ha for 50m 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 10m trap spacing, 0.6 animals/ha for 50m 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 E094071]
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Using simulation to compare methods for estimating density from capture–recapture data
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<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>
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Ivan J. S., G. C. White, and T. M. Shenk. 2013. Using simulation to compare methods for estimating density from capturerecapture data. Ecology 94:817–826. <a href="https://doi.org/10.1890/120102.1" target="_blank" rel="noreferrer noopener">https://doi.org/10.1890/120102.1</a>
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Ivan, Jacob S.
White, Gary C.
Shenk, Tanya M.
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Closure
Density
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Mean maximum distance moved
Simulation
Spatially explicit capturerecapture
Telemetry
Trapping grid
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10 pages
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20130401
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Ecology
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