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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 auxiliary telemetry information to estimate animal density from
capture—recapture data
Author(s): Jacob S. Ivan, Gary C. White and Tanya M. Shenk
Source: Ecology , April 2013, Vol. 94, No. 4 (April 2013), pp. 809816
Published by: Wiley on behalf of the Ecological Society of America
Stable URL: https://www.jstor.org/stable/23436294
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�Ecology, 94(4), 2013, pp. 809816
© 2013 by the Ecological Society of America
Using auxiliary telemetry information to estimate animal density
from capture—recapture data
Jacob S. Ivan,1'3 Gary C. White,' and Tanya M. Shenk2
Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado 80523 USA
2Colorado Division of Wildlife, Fort Collins, Colorado 80526 USA
Abstract. Estimation of animal density is fundamental to ecology, and ecologists often
pursue density estimates using grids of detectors (e.g., cameras, live traps, hair snags) to
sample animals at a study site. However, under such a framework, reliable estimates can be
difficult to obtain because animals move on and off of the site during the sampling session
(i.e., the site is not closed geographically). Generally, practitioners address lack of geographic
closure by inflating the area sampled by the detectors based on the mean distance individuals
moved between trapping events or invoking hierarchical models in which animal density is
assumed to be a spatial point process, and detection is modeled as a declining function of
distance to a detector. We provide an alternative in which lack of geographic closure is
sampled directly using telemetry, and this auxiliary information is used to compute estimates
of density based on a modified Huggins closedcapture estimator. Contrary to other
approaches, this method is free from assumptions regarding the distribution and movement of
animals on the landscape, the stationarity of their home ranges, and biases induced by
abnormal movements in response to baited detectors. The estimator is freely available in
Program MARK.
Key words: density: geographic closure: markrecapture; telemetry.
Introduction deaths, immigration, or emigration during the sampling
Estimation of animal density is fundamental to Period (i e" the Population is closed demograph.cally)
wildlife ecology. Density is used to evaluate system and that ammals do not move on and off the study slte
responses to environmental perturbations and treat during sampling (i.e., the population is closed geograph
ments (Converse et al. 2006, Manning and Edge 2008), it ically [Otis et al. 1978, Williams et al. 2002]). Short
can function as a benchmark for listed species recovery sampling sessions appropriately timed with the natural
(U.S. Fish and Wildlife Service 1998:319), it can be history of the species of interest can ensure achievement
useful for understanding system dynamics (Soulé et al. °f demographic closure. However, outside of a few
2003), or assessing habitat suitability for dependent exceptions (e.g., sampling small islands or ponds),
species (Zahratka and Shenk 2008), and it is routinely geographic closure is unlikely (White et al. 1982).
used in population monitoring and modeling (Thomp Without geographic closure, abundance estimates ob
son et al. 1998). It is often preferable to estimates of tained through closedcapture models do not reflect the
relative density (i.e., abundance) because standardizing number of animals within the boundaries of the study
population estimates by area facilitates comparison of she. Rather, they reflect the super population of animals
populations across space and time. associated with the site. That is, such estimates reflect all
Ecologists often pursue density estimates using grids animals that could have used the site at any time over
of detectors (e.g., cameras, traps, hair snags) to sample the course of the sampling period including those both
animals at a study site. Under such a sampling inside and outside of the site (Schwarz and Arnason
framework, estimates of animal abundance can be 1996, Kendall et al. 1997). Such estimates are difficult to
obtained using a variety of closedcapture models (Otis convert to density because the area effectively sampled
et al. 1978, White et al. 1982, Huggins 1989, 1991, by the detectors (i.e., the area used by the super
Williams et al. 2002). These models assume no births, population) is unknown.
Traditionally, the most common strategy for manag
ing the geographic closure issue has been to attempt
Manuscript received 17 January 2012: revised 14 September
estimation
of this effective area, then divide this estimate
2012; accepted 20 September 2012; final version received
7
November 2012. Corresponding Editor: E. G. Cooch. into the super population estimate obtained from closed
3 Present address: Colorado Parks and Wildlife, Fort
Collins, Colorado 80526 USA.
Email:
4
capture models to obtain a corrected estimate of density.
Usually the maximum distance moved by each individ
Jake.Ivan@state.co.us
Present
address:
.
National
.
Park
Service,
Colorado 80525 USA. events, is averaged across all individuals captured more
809
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Fort
�810 JACOB S. IVAN ET AL. Ecology, Vol. 94, No. 4
process model and the two submodels are combined
hierarchically such that density is estimated directly
given the data (Efford 2004, Borchers and Efford 2008,
Royle et al. 2009a, b). Early versions of SECR (inverse
prediction) performed well in simulation (Efford 2004)
and field experiments (Efford et al. 2005). Currently
SECR analyses can be accomplished using inverse
prediction (Efford 2004), maximum likelihood (Borch
ers and Efford 2008) or Bayesian (Royle et al. 2009a, b)
approaches.
Other than the usual capturerecapture assumptions
concerning demographic closure and accurate reporting
of marks, the main assumptions of the basic SECR
model are (1) home range centers occur randomly on the
landscape at a reasonably constant density (or at least
varying in some way that can be modeled) according to
a Poisson point process or some variant thereof and (2)
individuals occupy stationary, symmetrical activity
ranges during the sampling session. Estimation of the
observation
portionactivity
of the model is based
on informa
Fig. 1. Conceptualization of
animal
ranges
(ovals)
overlaid on a hypothetical
site
(rectangle).
Lack of
tionstudy
from detections.
Thus,
this portion of the model
geographic closure results may
from
some
activity
ranges
be susceptible
to some
of the potential
MMDM being
partially on the site such that animals can move on and off
biases
if
movements
and
detection
are
unnaturally
during a sampling session. The sum of the proportion of each
affecteddivided
by the samplingby
process.
animal on the site (gray areas)
the area of the site
provides an estimate of density
corrected
for
geographic
We formalize
a density estimator
that addresses
closure. Note that the use of
equally
sized,
oval
activity
geographic
closure
by measuring
the process
directly ranges
and a rectangular study site are for illustration only; the method
using auxiliary telemetry data. This method allows home
makes no restrictions about the shape and size of activity ranges
range shifts or irregular movements induced by sam
pling, makes no assumptions about the distribution and
movements of animals on the landscape, and does not
than once (i.e., mean maximum distance moved,
require estimation of effective area sampled. Conceptu
MMDM), and the study site is buffered by this distance
ally, the approach can be traced to the idea of estimating
(or onehalf this distance), to estimate the effective area
fractions of animals or "animal equivalents" within an
sampled (Wilson and Anderson 1985). However, there
area as described by Marten (1972), and later Boutin
are a number of concerns with this strategy, including
(1984). We describe the basic form of the estimator,
(1) movements of animals may be constrained by the
which is similar to that proposed by White and Shenk
detection process itself if sampling occurs in live traps,(2001) and implemented by Grant and Doherty (2007),
(2) movements of animals residing near the edge of the
then extend it to allow for individual covariates and to
or sites.
study site may not be well represented, (3) baited
admit more useful designs. We then apply the estimator
detectors (if used) may induce immigration into the
to an example for illustration.
study site or otherwise bias normal movement patterns,
Methods
(4) movement distances revealed through detection
events are dependent on the number of times an
Estimator
individual is detected, and (5) detector spacing and
Conceptually, we begin with the notion that each
study site size can influence movement estimates
animal occupies some activity range on the landscape
(Parmenter et al. 2003).
during a relatively short sampling session, and these
More recently, spatially explicit capturerecapture activity ranges occur irrespective of the boundaries of a
(SECR) techniques have been introduced as an alterna study site (Fig. 1). During a sampling session, individ
tive (Efford 2004, Borchers and Efford 2008, Royle et al. uals are captured, marked, and released on multiple
20096). Like MMDM, SECR makes use of spatial
occasions. Our method requires tagging at least some of
information contained in the capture history of each the animals subjected to markrecapture sampling with
individual. However, SECR does not attempt to a telemetry device (e.g., radio tag, GPS collar).
estimate effective area sampled. Rather, spatial infor Telemetry tagging can occur prior to or during the
mation is used to estimate parameters of an observation markrecapture process. Locations are then obtained
model in which detection probability declines as afrom tagged animals during and/or immediately follow
function of the distance between an animal's home rangeing the markrecapture sampling session in order to
center and a given detector in the study site. The densityproduce an estimate of the proportion of time that each
of home range centers is represented as a separatetagged animal spent within the study site (Fig. 1).
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�April 2013 DENSITY ESTIMATION USING TELEMETRY DATA Sil
Note that the definition of the site is somewhat of animals detected.
arbitrary. However, any workable definition should individualspecific
ensure that animals within the site have a reasonable ing distancechance of detection. We suggest that a "reasonable covariate; compute
chance of detection" can be attained if the site is defined individual, then t
such that detectors are distributed at an adequate and to the edge of the
roughly consistent rate (e.g., four detectors per home edge of the study
range [Otis et al. 1978:76]) within it so that there are no range and a dimi
gaps in sampling effort. We prefer to define the study to animals captur
site as the minimum convex polygon (MCP) encom covariate helps accou
passing the detectors. Such a definition seems natural Next we make a
and trap density inside this polygon would meet our Huggins estimato
criteria, assuming detectors are deployed at a sufficient
r+1
rate. Alternatively, one could define the study site as the _ í Pj_
MCP plus onehalf trap width, or a full trap width, and P*
still claim that trap density is relatively high and
consistent within the site. However, if we defined the where & is the estimated proportion of time animal i
site as the MCP plus two to three trap widths, or if sPent 011 the study site (estimated via telemetry) and
detectors were distributed at a rate of two per home Nss is now the estimated number of animals within the
, c•. , . • , „ boundary of the site. Animals that are detected on the
range over part of the site, but six per home range over / .
,, , „ ... ,, , ,. ■ site during a markrecapture session but never located
other parts, those definitions would likely result in , . . r
, i rr , ., ■, , i j, on the study site via telemetry are assigned p, = 0 and do
unequal sampling effort across the site and may lead to ./ , . J 0 ,
not contribute to the estimate. Those individuals that
bias in the estimator. Note that, because we incorporate
are always located on the study site contribute fully
the proportion of time each individual spends on the site
the estimate and are assigned p, = 1. All other
(i.e., telemetry data), the estimator we describe will be ,. ., , . . . , _ , . ,,
. ,, ... ,, . . , . , , individuals receive a fractional p. and partially contnb
appropriate regardless of how the site is defined, as long
ute.
as it is defined following the guidance we provide here.
Traditional markrecapture estimators can be applied
Finally, we divide NSs by the area of the study site
(e.g., area of the minimum convex polygon encompass
to the markrecapture data collected during the
ing all detectors) in order to obtain an estimate of
sampling session to estimate the super population
density:
(Schwarz and Arnason 1996, Kendall et al. 1997) of
animals that used the site during sampling. Our goal is m,+, ~
to use the telemetry data to estimate the portion of the
super population (i.e., total animal equivalents) that q = — i
occurred within the boundaries of the site to produce an ^
estimate of density corrected for the lack of geographic where D is estimated density (number of animals per
closure. Heuristically then, we start with the super unit area), A is the area of the study site, and p¡, p¡, and
population estimate derived from conventional closed M,+\ are as defined previously,
capture methods then disregard those animals complete
Variance
ly off of the site, partially discount edge animals, and
tally the whole and partial animals left before dividing For the simple case where p is estimated directly from
by the area of the site. In order to work well, the telemetry data as an overall mean across individuals and
telemetry sampling scheme should be appropriate for the no covariates are used to estimate por p*, the estimator
species of interest such that locations can be assumed to reduces to
be representative of how animals use their home range.
Mathematically, our approach stems from the Huggins M,+1—
(1989, 1991) closedcapture estimator for abundance: ß _ p
A
1
i
„
Ñsp = V" _ and variance of D can be approximated using the delta
/= i P* method (Seber 1982:7) as
where ÁSP is the estimate of the super population of
~2 (1 p*) Var (p) Var(p*)\
animals that could have used the site during the trapping Var(D) = D I +  2
session, p* is the estimated probability that animal i is \ ^<+l p P* I
captured one or more times during the sampling session
(i.e., if pi is an estimate of the probability animal that i is whSffiJ/') is assumed to be a binomial proportion (i.e.,
detected on any given occasion,p* = 1  (1 — p,)', where t Var(p) = PA 1 — Pi)l(n¡)> 'L_Js__the number of
is the number of occasions), and M,+1 is the total number locations obtained for animal i), Var((* ) follows from
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�812 JACOB S. IVAN ET AL. Ecology. Vol. 94. No. 4
the Huggins estimator (Huggins 1989, 1991), and D and Example
M,+1 are as defined previously. However, we expect the We trapped snowshoe hares {Lepus am
use of individual covanates to be common and desirable. central Colorado during n_l5 August 20
In such cases, the delta method approximation is much ? live_trapping gnd wlth 50.m spacing (I
more complex. We provide guidance for numerically captured 14 adult_sized hares 25 times du
estimating variance in the Appendix. day sampling sessi0n (Colorado State Univ
Assumptions CUC Protocol 06062A03). Of these, eight were
with 28g radio collars (Model TW5SM; BioTrack,
Assumptions for this estimator include the usual Wareham> Dorsek UK) Traps and bajt w
closed markrecapture assumptions (Otis et al. 1978. at the conclusion of mark_reCaPture sam
Wilhams et al. 2002) as well as three additional August Telemetry sampiing commenced
assumptions specific to this method: (1) The animals and occurred dai[y thrQugh ^ Augus
sampled with telemetry are representative of the locations via triangulation at relativel
population of animals that use the study site. (2) (w¡th¡n 250 m)> r£su)ting m accuracy of
Telemetry devices do not affect movements
are no effects of markrecapture sampling
movements. (3) Telemetry location error is
to the size of the study site and assignment
and there were alternately sampled during nlghttim
on an,mal daytime (restmg) hours tQ obtajn a repr
small relative sample of ,ocations We obtained an a
(on/off) of localions/individual during daytime and
locations near the edge of the site is unbiased. indivldual at night. There was significant
Useful variants anc' study site during the telemetry sa
period, and one hare was never located on the site after
It will often be difficult to telemeter all individuals trapping (Fig 9)
detected during a sampling session. In such cases, the Using the estimator described here, as imp
estimator described above can be mod.fied to model use Program MARK; we modeed both
of the study site by individuals that were subjected to intercepts only> as functions of DTE, and
markrecapture sampling but were not telemetered. tions ther£of aiq selected the DXE struct
Specifically one can estimate p, by fitting a logistic parameters (AIC, weight = 0.99). The estim
model to the data from radiotagged animals, then apply (±§E) from tha{ model was j 99 ± J16
this model to untagged animals to estimate their p„ We FaiKng tQ account for ,ack of geographlc
again suggest that minimum DTE from each individu jn a naive density estimate from the n
al's mean trap location could be a useful covariate in modd {M^ Ñ/A = 5 0? hares/ha) that is 15
such a model to account for heterogeneity induced by telemetrycorrected value
location of individuals relative to the study site. Other
models are also possible (including intercept only, which Discussion
simply assigns each individual the mean proportion of The estimator preSented here has advantage
time on site from the telemetered sample) and multiple Qther traditiona, and contemporary density
candidate models can be formulated and compared techniques. First> it does not require esti
using an information theoretic approach such as AICc effe£tiv£ area sampled using ad hoc approache
(Burnham and Anderson 2002). it is free from the assumption that animals
In most cases, we expect that the study site will be stationary home ranges during the sampling p
large enough to encompass numerous activity ranges of ¡t does nQt require spedfic distributional ass
the target species. In these instances, the (DTE)
about home range centers (e.g., that home range centers
covariate is only important when mean capture location foUow a Poisson process) or movements 0f animals (e.g.,
is relatively near the edge of the site. Activity ranges that that ammals mov£ about w¡th¡n a symmetric home
are fully on site expose animals to an equal number of rang£) Notei how£V£r; that we require telemetered
detectors regardless of whether the range is just mside an¡mals tQ be representative of all animals, so we do
the study site or exactly m the middle. Thus, it may also assume all an¡mals come from the same "distribution" in
be useful to consider a threshold model for the som£ s£ns£ Jhird< it allows us£rs t0 mak£ full US£ of
estimation of p, and/or p, such that the DTE covariate anciUary telemetry data that directly describe the process
is only operational up to a point. After that animals are of int£r£St: movement of animals on and off the study
fully on site and the proportion or capture probability is sit£ Finally5 the estimator is intuitive and relatively easy
estimated accordingly. Mathematically, such a model to implement.
can be represented as A disadvantage is that purchasing and tracking
1 u ~\ a , 5 i ■ ti, Tw telemetry tags, in addition to the required mark
logit(p.) = ßn + ß, (min(ß7, DTE )
recapture sampling, may be cost prohibitive when the
where ß0 and ßi are the usual intercept and slope only goal is density estimation. However, p
estimates from a logistic model using distance to edge as routinely use telemetry for home ran
a covariate, and ß2 is the threshold parameter. quantification of habitat use, and/or survival
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�April 2013 DENSITY ESTIMATION USING TELEMETRY DATA 813
Fio. 2. Telemetry locations obtained on eight snowshoe hares (Lepus american
rectangle) in central Colorado. August 2006. Different symbols indicate unique i
day and night over a 13day period immediately following trapping.
at the same time they obtain density estimates. In fact, equal and the
many authors (e.g., Di Betti et al. 2006, Soisalo and traps (e.g., at lea
Cavalcanti 2006, Dillon and Kelly 2008, Balme et al. to occur within
2009, Sharma et al. 2010) have used location data to interest. Also, d
correct markrecapture density estimates by first calcu that research
lating average home range size for animals in the study, determine
then applying a buffer to their study site based on this common acro
home range estimate. We suggest that a better use of on, or whethe
these location data to correct density estimates would be whether each
the approach outlined here. GPS technology provides an and Doherty
excellent opportunity to implement this method as can be realiz
potentially more locations can be collected with better sites.
precision but without the additional cost of physically In our examp
following the telemetry devices after deployment. individual to est
We presented an example from a single study site laid manee impro
out in a regular grid, but other designs are possible, asymptote. How
Traps can be set in virtually any configuration as long as manee of th
the distance between neighboring traps remains roughly error among
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�814 JACOB S. IVAN ET AL. Ecology, Vol. 94. No. 4
individuals. Thus, error can be minimized more effi H
ciently by sacrificing number of locations/individual
favor of deploying telemetry devices on more individuals
(Ivan et al. 2013). For example, the estimator is like
be less biased when 100% of the animals in the markrecapture sample have telemetry devices and only five
locations are obtained per animal than when 25% are
telemetered and 20 locations per animal are obtained. w
One unique assumption of this estimator is that t
movements of telemetered animals during and/or We illustrated the method using data from a
immediately after markrecapture sampling are reflec traditional livetrapping study on small mammals,
tive of ordinary presampling movements. The use of However, the method is applicable to any study in
bait or other attractants during markrecapture can which markrecapture data and telemetry data are
cause violations of this assumption and deserves special available simultaneously. Such work has been completed
consideration. First, under such conditions, we recom on large carnivores (Miller et al. 1997, Soisalo and
mend discarding from the analysis any telemetry Cavalcanti 2006, Sharma et al. 2010), birds (Powell et al.
locations obtained during the markrecapture session 2000, Bächler and Schaub 2007), reptiles (McMaster
as it is likely that baited detectors influenced animal and Downs 2006, Grant and Doherty 2007), amphibians
movement. Second, it is imperative to remove all bait (Peterman et al. 2008), fish (Naughton et al. 2009), and
from the site at the end of the markrecapture session so even invertebrates (Robinson et al. 2000). Furthermore,
there is no unnatural attractant left to influence the estimator can accommodate any means (in addition
movements after the session. Third, we suggest that it to telemetry) by which p¡ can be estimated, such as
may be appropriate to wait one to two days post mark tallying the proportion of tracks an individual left on the
recapture before collecting location data to allow study site (Marten 1972, Lernen and Freeman 1985).
animals to revert to their normal activity patterns. Thus, the method is versatile and could be implemented
Telemetry sampling should, however, be completed for a variety of taxa.
within a reasonable time to avoid biasing estimates of Density is a fundamental parameter used in a variety
Pi due to seasonal movements, migration, or dispersal. of contexts in the field of ecology. Here we described an
In summary, our experience suggests that several intuitively appealing method for estimating animal
animals may make extensive movements to take density that can be applied in a variety of situations,
advantage of baited detectors and become part of the The approach accounts for lack of geographic closure
markrecapture data set, yet they do not spend any time using ancillary information not typically used by other
on the study site once bait is removed. Following the estimators. Auxiliary information usually improves
guidelines outlined here, telemetry information should parameter estimation (e.g., Burnham 1993, Barker
result in p = 0 for these individuals, and they will not 1997, Powell et al. 2000, Gopalaswamy et al. 2012),
contribute to the density estimate, which is appropriate, and we expect that to be the case here, especially given
Another unique assumption of this estimator is that that we employ auxiliary information that directly
the sample of telemetered animals should represent the measures animal movement, which is the root of the
population of animals that use the study site. For geographic closure issue. Further, unlike other contem
traditional livetrapping studies in which telemetry porary estimators, our approach makes no assumptions
devices are deployed on a subset of animals during the about distribution of animals, stationarity of home
markrecapture sampling, one can help meet this ranges, movement patterns, or estimation of effective
assumption via sound design. We recommend checking area trapped. It can be easily implemented in the field,
traps from a random starting point on each occasion, especially within studies where telemetry is already used
This strategy should help equalize the probability of for other purposes. The method can be fully implement
capturing and telemetering edge vs. interior animals. In ed in Program MARK ("Density with Telemetry" data
addition, we suggest retaining some telemetry devices for type [White and Burnham 1999]) to obtain estimates and
deployment during the latter portion of a sampling associated sampling variances,
session to facilitate the inclusion of trapshy individuals
in the radiotagged sample in addition to traphappy
Acknowledgments
individuals that are captured early and often. Despite ,We thank. P' Lukacs í°r ¡n formulating the
, , r ■ , i i , , i r telemetry estimator. M. Efford and J. A. Royle provided
these design features, it is plausible that the sample of insightful discussions regarding d
telemetrytagged individuals could be biased toward and spatially explicit capturerecapt
those with a higher proportion of their range on the support was provided by the Co
study site as these individuals stand a better chance of Research Unit. Funding was
, , j Division of Wildlife. We thank K. Wilson, P. Doherty, V.
capture on any occasion. This phenomenon would „ ■. ,, .„j j • , , , .
^ r Dreitz, M. Alldredge, two anonymous reviewers, and students
inflate the numerator (p) of the estimator, resulting in in the Wagar
a positively biased density estimate (Efford 2004). drafts of this
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�April 2013 DENSITY ESTIMATION USING TELEMETRY DATA 815
McMaster, M. K., and C. T. Downs. 2006. Population
structure and density of leopard tortoises (Geoche lone
Bächler, E., and M. Schaub. 2007. The effects of permanent
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Supplemental Material
Appendix
Estimate of Var(D) when individualspecific covariates are used to estimate D (Ecological Archives E094070A1).
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�
https://cpw.cvlcollections.org/files/original/d5c31e04eed2965109b9be9bdd754ac0.pdf
057968fc4210e14954df6bf5e0a2281f
PDF Text
Text
Appendix A. Estimate of Var
when individualspecific covariates are used to estimate .
Similar to the equations described in the text, the basic forms of the estimator and its
variance without individualspecific covariates are:
·
Var
where
̂
·
1
Var ⁄ ̂
⁄ ̂
Var
is estimated density (number of animals per unit area),
animals detected during a sampling session,
is the total number of
is the estimated proportion of time an animal spent
on the study site, ̂ is the probability an animal is detected one or more times during the
sampling session (i.e., if ̂ is the estimated probability an animal is detected on any given
occasion, ̂
1
1
̂ , where is the number of occasions), and
is the area of the study
site .
However, we expect use of individualspecific covariates to be both helpful and common.
To accommodate individual covariates the estimator becomes:
̂
where
is the estimated proportion of time animal spent on the study site, ̂ is the
probability animal is captured one or more times during the sampling session, and other
parameters are as defined above (i.e.,
and ̂ can now be different for each individual).
Analytical solutions for variance of this individualspecific expression are complex.
Approximate variance can be estimated more simply using the delta method with numerical
�methods to approximate partial derivatives. Thus, to accommodate individualspecific
covariates, make the following 3 substitutions for the appropriate quantities in Var
1) Assuming
is a binomial random variable, Var
1
is
above:
, where
is the
number of animals actually using the site and at risk of capture (i.e., the superpopulation).
The number of individuals captured (
) is an estimate for
Var
1
, therefore:
̂
However, if ̂ is to be specific to each individual, the above expression is not appropriate.
Instead:
Var
1
̂
2) Use the delta method approximation (Seber 1982, p. 7) to compute Var ⁄ ̂ :
a) Compute a (1
) vector of approximate partial derivatives for each of the
in the density expression (i.e., all parameters used to estimate both
parameters
[ , , … ] and
α , α , … α ]). To accomplish this, successively change (one at a time) each parameter
( or α ) in the density expression (
∑
by or α (where is
⁄ ̂ ⁄
very small) and calculate the difference quotient (e.g., [
/ ) for
the density expression with each change. The value of the difference quotient at each
step is a numerically derived estimate of the partial derivative for that parameter and
becomes an element in the vector .
…
�parameters ( , , … ) appear first
b) Assuming the partial derivatives for the estimated
in the vector, the variancecovariance matrix for the density expression consists of the
parameters in the upperleft quadrant and
variancecovariance matrix of the estimated
parameters (α , α , … α ) in the lower
the variancecovariance matrix of the estimated
right quadrant. Remaining quadrants are populated by zeros. The estimated variancecovariance matrix for the
parameters is can be obtained from the statistical package
used to compute the logistic regression. The parameters for
come from abundance
estimation software such as Program MARK (White and Burnham 1999).
Var
Cov
Cov
,
Cov
,
,
Var
Cov
,
Cov
,
0
0
0
Cov
,
0
0
0
0
0
0
Var
Cov
0
0
0
Var
0
0
0
Cov
,
0
0
0
Cov
,
,
Var
Cov
,
Cov
,
Cov
,
Var
c) Multiply the vector of partial derivatives by the variancecovariance matrix then by the
transpose of the vector of partials, such that:
Var ⁄ ̂
3) Because
·
and ̂ are specific to individuals:
⁄ ̂
The final expression for Var(
Var
where
·
⁄ ̂
when using individualspecific covariates:
∑
1
̂
·
∑
·
̂
is the form of the estimator that allows for individual covariates.
�Note that the variance expression derived here is fully implemented in Program MARK (White
and Burnham 1999).
LITERATURE CITED
Seber, G. A. F. 1982. The estimation of animal abundance and related parameters. 2nd edition.
Charles Griffin & Company LTD, London, UK.
White, G. C. and K. P. Burnham. 1999. Program MARK: survival estimation from populations
of marked animals. Bird Study 46 Supplement:120–138.
�
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Using auxiliary telemetry information to estimate animal density from capture–recapture data
Description
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<span>Estimation of animal density is fundamental to ecology, and ecologists often pursue density estimates using grids of detectors (e.g., cameras, live traps, hair snags) to sample animals at a study site. However, under such a framework, reliable estimates can be difficult to obtain because animals move on and off of the site during the sampling session (i.e., the site is not closed geographically). Generally, practitioners address lack of geographic closure by inflating the area sampled by the detectors based on the mean distance individuals moved between trapping events or invoking hierarchical models in which animal density is assumed to be a spatial point process, and detection is modeled as a declining function of distance to a detector. We provide an alternative in which lack of geographic closure is sampled directly using telemetry, and this auxiliary information is used to compute estimates of density based on a modified Huggins closedcapture estimator. Contrary to other approaches, this method is free from assumptions regarding the distribution and movement of animals on the landscape, the stationarity of their home ranges, and biases induced by abnormal movements in response to baited detectors. The estimator is freely available in Program MARK.</span>
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Ivan, J. S., G. C. White, and T. M. Shenk. 2013. Using auxiliary telemetry information to estimate animal density from capturerecapture data. Ecology 94:809–816. <a href="https://doi.org/10.1890/120101.1" target="_blank" rel="noreferrer noopener">https://doi.org/10.1890/120101.1</a>
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Ivan, Jacob S.
White, Gary C.
Shenk, Tanya M.
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Density
Geographic closure
Markrecapture
Telemetry
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9 pages
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20130401
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English
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Ecology
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Article