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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
�Popul Ecol (2014) 56:463–470
DOI 10.1007/s1014401404318
ORIGINAL ARTICLE
Estimating the abundance of rare and elusive carnivores
from photographicsampling data when the population
size is very small
Brian D. Gerber • Jacob S. Ivan • Kenneth P. Burnham
Received: 18 July 2013 / Accepted: 7 January 2014 / Published online: 25 January 2014
� The Society of Population Ecology and Springer Japan 2014
Abstract Conservation and management agencies
require accurate and precise estimates of abundance when
considering the status of a species and the need for directed
actions. Due to the proliferation of remote sampling cameras, there has been an increase in capture–recapture
studies that estimate the abundance of rare and/or elusive
species using closed capture–recapture estimators (C–R).
However, data from these studies often do not meet necessary statistical assumptions. Common attributes of these
data are (1) infrequent detections, (2) a small number of
individuals detected, (3) long survey durations, and (4)
variability in detection among individuals. We believe
there is a need for guidance when analyzing this type of
sparse data. We highlight statistical limitations of closed
C–R estimators when data are sparse and suggest an
alternative approach over the conventional use of the
Jackknife estimator. Our approach aims to maximize the
probability individuals are detected at least once over the
entire sampling period, thus making the modeling of variability in the detection process irrelevant, estimating
abundance accurately and precisely. We use simulations to
demonstrate when using the unconditionallikelihood M0
(constant detection probability) closed C–R estimator with
profilelikelihood confidence intervals provides reliable
results even when detection varies by individual. If each
individual in the population is detected on average of at
least 2.5 times, abundance estimates are accurate and
precise. When studies sample the same species at multiple
areas or at the same area over time, we suggest sharing
detection information across datasets to increase precision
when estimating abundance. The approach suggested here
should be useful for monitoring small populations of species that are difficult to detect.
Keywords Camera traps � Capture–recapture �
Heterogeneous detection � Small population
Electronic supplementary material The online version of this
article (doi:10.1007/s1014401404318) contains supplementary
material, which is available to authorized users.
B. D. Gerber (&)
Colorado Cooperative Fish and Wildlife Research Unit,
Department of Fish, Wildlife and Conservation Biology,
Colorado State University, Fort Collins, CO 80523, USA
email: bgerber@colostate.edu
J. S. Ivan
Colorado Division of Parks and Wildlife, Wildlife Research
Center, 317 West Prospect Road, Fort Collins, CO 805262097,
USA
K. P. Burnham
Professor Emeritus, Department of Fish, Wildlife, and
Conservation Biology, Colorado State University,
Fort Collins, CO 80523, USA
Introduction
Management and conservation of wildlife requires reliable
information on the population status of species. Knowledge
of abundance is critical to assessing species status and
essential for ecological research (e.g., predator–prey
dynamics, specieshabitat relationships). A common
approach for estimating abundance is to collect data on
detections of uniquely marked individuals that are
(re)captured over two or more sampling occasions (t;
usually days or weeks). Then closed capture–recapture (C–
R) estimators (Borchers et al. 2002) can be fit to the data to
estimate (1) the probability of detecting an individual on a
^
given occasion (^
p) and (2) abundance (N).
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�464
The recent proliferation of remote sampling cameras
(cameratraps) has led to a rapid increase of C–R studies
estimating the abundance of medium to large carnivores
(O’Connell et al. 2010; Foster and Harmsen 2012). Data
common to these studies are often of low information
quantity in that few unique individuals are detected (Mt?1)
and then only a small number of times. Sparse data can
generally be recognized when the capturehistory has many
zeroes. Characteristics of carnivore photographic C–R data
were summarized in Harmsen et al. (2011) and Foster and
Harmsen (2012). These data are typified by (1) infrequent
^ ¼ 0:17,
detections
[p�
SDð^
pÞ ¼ 0:16,
rangeð^
pÞ ¼
0:02�0:79], (2) a small number of individuals detected
� tþ1 ¼ 12:6, SD(Mt?1) = 10.4, range(Mt?1) = 2–65]
[M
due to naturally low densities, (3) long survey durations
(19–65 occasions), and (4) variability in detection among
individuals (heterogeneity). Closed C–R estimators were
developed to estimate abundance when animal populations
are too large for a complete census or when individuals are
difficult to detect. Total sampling duration is assumed to
occur over a small snapshot in time, for example B10 days.
As such, statistical assumptions of C–R estimators have not
often been considered in the literature when sampling
occasions are spread over long periods of time, which may
be appropriate for longlived species (Karanth and Nichols
2002). It was further assumed that a reasonably large
number of individuals could be detected and that many, but
not all individuals would be detected several times (White
et al. 1982). Thus, the characteristics of common photographicsampling data on rare carnivores do not match well
with the type of data abundance estimation has previously
considered.
We believe there is a need for guidance when estimating
the abundance of low density, longlived species, where
heterogeneity is expected and sampling effort is protracted.
These conditions limit the utility of traditional C–R estimators but these limitations are not clear in the literature.
Clarification is especially important because carnivore C–R
studies often target rare, poorly studied, and/or threatened
species (Jackson et al. 2006; Gerber et al. 2012). To ensure
appropriate conservation actions are implemented, estimating abundance precisely and accurately is critical. The
uncertainty common to estimates from carnivore C–R data
(Foster and Harmsen 2012) may mask the true and possibly
critical status of a population, thus delaying conservation
actions. For large carnivores where the population size is
naturally low, an imprecise abundance estimate is especially problematic, as the uncertainty of only a few individuals is a significant proportion of the population.
Our objectives are fourfold. First, we discuss statistical
limitations relevant to estimating abundance with closed
C–R estimators when detection probability per occasion is
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Popul Ecol (2014) 56:463–470
low and heterogeneous, sampling duration is long, and few
individuals are detected but they are likely to be a large
proportion of the total population (these elements are collectively referred to below as ‘rare carnivore C–R data’).
We also comment on appropriate measures of precision.
Second, we suggest that when a population size is naturally
small (e.g., \20 animals), p is small, and t is long enough
such that most individuals in the population are detected,
using C–R data in a confirmatory census framework is
more appropriate than traditional modeling of the detection
process to estimate abundance. This is in contrast to the
usual approach of using the Jackknife estimator, which is
inappropriate with sparse data (Burnham and Overton
1978; Harmsen et al. 2011; B. D. Gerber, unpublished
data). We provide combinations of p and t useful in planning a C–R study to census a population. We then describe
minimum frequencies of individual detections that will
lead to informative estimates and confirm a complete or
nearcomplete census for a small population size using an
unconditionallikelihood closed C–R estimator with constant detection probability (referred as M0 from here on;
White et al. 1982). Third, using Monte Carlo simulations
we demonstrate the reliability of estimating abundance
with the M0 and quantifying uncertainty with profilelikelihood confidence intervals (PLCI; Venzon and Moolgavkar 1988) when detection is heterogeneous. Fourth, we
discuss sharing information on p to estimate abundance at
multiple sampling areas or at the same area across different
sampling sessions. We emphasize that the suggestions
detailed below should only be applied to the specific type
of data we describe above.
Statistical limitations
For statistical estimation of any parameter, ideal estimators
are unbiased, precise, and produce nominal confidence
interval coverage at the (1  a) level (a = probability of a
Type I error). Most commonly used C–R estimators of
abundance rely on likelihood theory to estimate parameters.
As such, they meet the characteristics of an ideal estimator,
but under asymptotic properties of consistency, normality,
and efficiency, which are generally not applicable when
sample size is small. Thus, commonly used estimators are
not guaranteed to ‘do the right thing’ given typical carnivore
photographicsampling data. White et al. (1982) make this
point clear for C–R studies with a small number of sampling
occasions, ‘‘… if N \ 100 and p \ 0.3, a good capture–
recapture study probably cannot be done.’’
Precision and confidence interval coverage are especially problematic for sparse data. Logbased confidence
intervals are most often used for abundance estimates as
�Popul Ecol (2014) 56:463–470
they can be formulated to ensure that the lower bound is
CMt?1. However, logbased confidence intervals are
known to perform poorly (i.e., they do not include the true
parameter value as intended) with sparse data due to a
poorly estimated variance–covariance matrix and bias in
maximumlikelihood estimates (MLE; Hudson 1971;
Donaldson and Schnabel 1987). An additional problem
occurs when protracted sampling leads to a large proportion of the population being detected, even though per
occasion p may be low. This situation constrains the lower
bound of the abundance estimation to be CMt?1 (except for
conditionallikelihood estimators that derive N^ Huggins
1991), which in turn violates a condition required for largesample likelihood inference to perform well: that the true
parameter value is not on or near a boundary (e.g., it is not
near 0 or 1, or in this case Mt?1 and is therefore ‘‘unconstrained’’ of the parameter space). Under such constraints,
estimates of the variance–covariance matrix are often poor
(i.e., estimates of sampling variance may be 0 or exceedingly large). The boundary issue is compounded when
sample size is small; a circumstance that often leads to
‘‘flat’’ likelihoods (as opposed to highly ‘‘peaked’’ likelihoods with large sample size) which makes it difficult to
find a maximum and increases the chance that the
MLE(s) are on or near a boundary.
Aside from boundary issues, sampling a large portion of
the population can result in nonexistent diagonal elements
of the Fisher information matrix which can lead to poor
estimates of sampling variation or preclude estimates
^ the secondentirely. For the special case where Mt?1 = N,
partial derivatives of the likelihood evaluated at the
MLE(s) are guaranteed to not exist due to lack of curvature
in the likelihood and thus there will be no standard estimate
^ Finally, when a large portion of the population
of Var(N).
is detected, the estimated number of animals that were
never detected (f0) goes to zero. This is problematic when
using lognormal confidence intervals to characterize
uncertainty, as is the standard practice in Programs CAPTURE (Rexstad and Burnham 1991) and MARK (White
and Burnham 1999) because f0 appears in the denominator
of the confidence interval calculation. Thus, precise estimation of N using asymptotic theory, when most of the
population is detected, or when sample size is small, is
either not possible or is unlikely to be at the (1  a) level
using common interval estimators.
Characteristics of typical carnivore C–R data also make it
difficult to address the ubiquitous issue of heterogeneous
detection. Heterogeneity is often regarded as the most dominant form of variation in p for a variety of reasons, including
effects of the sampling layout and its relationship to animal
movement, natural variability among age classes or between
sexes, and intrinsicindividual variation that may be
465
unmeasurable (Noyce et al. 2001; Harmsen et al. 2011; Royle
et al. 2013). Failing to account for existing heterogeneity
when many individuals in the population are not detected can
result in negatively biased estimates of abundance, but adequately accounting for heterogeneity is difficult. With large
samples, one can make use of conditionallikelihood closed
models (Huggins 1991) and incorporate individual covariates
(e.g., sex) to model heterogeneity. Alternatively, one could
use mixture models that assume animals belong to 2 or more
arbitrary groups having distinct capture probabilities (Pledger 2000). However, due to low information quantity in
sparse data, very few parameters may be estimable, thus
estimating multiple detection parameters to account for heterogeneity is often unrealistic. Despite the low quality of data,
many carnivore C–R studies follow recommendations of
foundational carnivore photographicsampling work (Karanth and Nichols 1998, 2002) and estimate abundance using
the Jackknife estimator (Foster and Harmsen 2012). However, the Jackknife estimator also relies on asymptotic and
regularity conditions that are not met with small sample size
and such estimates, especially those of precision should not
be considered reliable (B. D. Gerber, unpublished data).
Furthermore, when most of the population is detected, which
may often be the case, as noted above, the Jackknife estimator
is known to overestimate abundance (Chao and Huggins
2005). For endangered species, this may be of a larger detriment than underestimation.
Recommendations
Design studies to be complete or nearcomplete
censuses
Due to the limitations associated with estimating abundance using rare carnivore C–R data, we suggest it is more
appropriate to refocus the design of such studies to detect
all (or almost all) individuals in the population and provide
confidence of such. By doing so, we can reduce the number
of relevant parameters that need to be estimated to achieve
an accurate and precise abundance estimate. Given the long
duration of many carnivore C–R sampling, even low p can
result in detection of most animals in the population, albeit
with each animal likely observed a small number of times
relative to the total number of occasions. For example, if
p = 0.10 and sampling occurs for 30 occasions (e.g.,
30 days) the probability of detecting an individual at least
once over the total sampling period would be p* = 1 (1  0.10)30 = 0.958. As such, when t gets large, even
when p is very low, Mt?1 converges to N and p* can be
interpreted as the fraction of the population seen over the
entire sampling period. While sampling a large portion of
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Popul Ecol (2014) 56:463–470
Fig. 1 Combinations of
detection probability (p) and
sampling occasions (t), where
C95 % of the true population is
detected (black cells) or not
(grey cells)
the population causes estimation issues as described above,
the need for estimation is secondary for rare carnivore C–R
data as most individuals will be detected given a long
enough sample period.
Toward this end, we computed p* for plausible combinations of low detection probability (p = 0.01  0.30) and
protracted sampling (t = 1–90; Fig. 1) and identify combinations that result in a nearcensus (p* C 0.95), similar to
White et al. (1982). Given anticipated values for p, researchers
can assess the number of sampling occasions required for a
census. For example, if p = 0.05, 60 occasions would be
required to achieve a nearcomplete census. If pilot data are
available and indicate heterogeneity in detection, then we
recommend the use of the lowest possible or observed p^ in
order to provide a conservative estimate of t required to obtain
a near census. Note that when p* is close to one, there is little
room for variation in detection due to heterogeneity, time, or
other factors and such modeling of this variation becomes
irrelevant in estimating abundance (see simulation below).
Although heterogeneity is expected in many carnivore C–R
datasets, it should not be assumed that a statistical estimator
that incorporates heterogeneity is inherently the most appropriate when the data are sparse. Below we demonstrate when a
simpler model that estimates fewer parameters than any heterogeneity model will perform well with sparse data.
Sample to attain minimum capture frequencies that will
lead to informative estimates
By ‘thinking like an estimator’ we can consider how many
individual detections (capture frequencies) are required to
123
obtain reliable estimates of abundance. Using simulation
we explored the relationship between capture frequencies
and precision as a means for practitioners to discern when a
study can be considered a nearcensus. Specifically, we
started with an initial scenario of t = 20, N = 10, 9 animals were detected once each, and 1 individual was never
detected. For subsequent simulations we added detections
such that for the second scenario 1 individual was caught
twice, others were caught once, and 1 individual was never
detected. For the third scenario 2 individuals were captured
twice, etc. We incremented the scenarios sequentially until
all animals were detected three times each except for the 1
individual that was never detected. We estimated abundance for each of these nineteen scenarios by fitting M0 and
computing PLCI’s. We recommend profilelikelihood
confidence intervals as they have several advantages over
logbased confidence intervals, which are most often used.
First, PLCI do not rely on the assumption of asymptotic
normality and usually provide estimates close to nominal
levels even when sample size is small or estimation is on or
near a boundary (Bates and Watts 1980). Even in the
^ PLCI will still provide
special case where Mt?1 = N,
meaningful confidence bounds. Second, PLCI guarantee
the lower bound will be greater than or equal to Mt?1.
Lastly, profiling the likelihood to estimate confidence
intervals directly uses the information in the data regarding
the parameter estimate(s) and is based on a relatively
robust distributional assumption of the asymptotic v2 distribution (Venzon and Moolgavkar 1988).
When each individual is only caught once, the point
estimate was unreliable and the upper confidence interval
�Popul Ecol (2014) 56:463–470
467
a
b
Fig. 2 Abundance estimates where nine individuals are detected a
varying number of times using the M0 (constant detection probability)
unconditionallikelihood capture–recapture estimator and 95 % profile likelihood confidence intervals. The true population size is 10.
The upper confidence interval is excessively large when the average
capture frequency is \1.4 and is not presented
was positive infinity. However, when most individuals in the
population were captured two or more times, the point estimate was very close to N and the upper confidence interval
was reasonable (Fig. 2). An average capture frequency of 2.5
or more resulted in a nearcomplete census. While we used
t = 20 for this simulation, the number of sampling occasions
is irrelevant to the general patterns in our results. If detection
probability is low, more than 20 sampling occasions will be
necessary to ensure most individuals are detected two or
more times; however, the accuracy and precision will be
similar to the patterns observed here.
Estimating abundance with heterogeneous detections
To investigate the reliability of using M0 and associated 95 %
PLCI under commonly observed heterogeneity in detection,
we simulated poorly informative data as presented in the
Electronic Supplementary Material (ESM). For all simulations, true population size (N) was 10 and t = 10, 20, 30, 40,
50, and 60 occasions. We considered a generic form of heterogeneity where pi is a logitnormal distribution for the ith
individual using the combinations of l = 0.05, 0.1 and
r = 0.1 (Fig. S1 in ESM). Heterogeneity could be due to any
number of reasons, such as the spatial arrangement of the
^ and 95 % profile likelihood
Fig. 3 Expected abundance estimate (N)
confidence intervals from the M0 (constant detection probability)
unconditionallikelihood capture–recapture estimator with simulated
heterogeneous capture–recapture data where the mean detection
probability was a l = 0.05 or b 0.1 and r = 0.1 over sampling
occasions of 10–60. The upper confidence limit was often infinite
when the sampling occasions were 10 and thus is not presented
trapping layout. Our specific choice of variability (r) in
detection was chosen to functionally capture p’s often
observed in carnivore studies (Harmsen et al. 2011). For each
individual i we randomly drew a single pi from the specified
distribution. We then created a toccasion capture history for
individual i by inserting a ‘1’ or ‘0’ for each occasion
according to a Bernoulli process with probability pi. For each
combination of N and t, we simulated 1000 iterations and
^
investigated empirical distributions, bias, and precision of N.
We found that the bias was minimal for both l = 0.05 and
l = 0.1 when t C 20 (Fig. 3; empirical distributions are presented in Fig. S2 in ESM). However, precision was relatively
poor until t C 30 for l = 0.1 or t C 50 for l = 0.05. The
upper 95 % PLCI was often infinite at t = 10 for both detection distributions. In general, when expected p* was[0.6, this
approach estimated abundance accurately and with reasonable
precision. In other words, despite considerable heterogeneity
in detection, this simple model estimated abundance well. The
failure rate in estimating abundance for both scenarios was
B5 % after t = 30 (Fig. S3 in ESM). Models that require
estimation of additional parameters over M0 will have worse
precision, due to the limited information in the data.
Sharing information on p to estimate N
Sample sizes in many carnivore C–R studies are simply not
large enough to use closed C–R estimators as they were
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intended, modeling p to estimate N. This sentiment is stated
plainly by White et al. (1982), ‘‘experiments in which Mt?1
is on the order of 10 or 20 animals simply do not provide
enough information for any procedure to perform well’’.
However, in many studies, the target species is sampled at
more than one area or across multiple sessions at the same
area. While information regarding the detection process
may be sparse with one dataset, information can be shared
across datasets when estimating abundance for each area
(White 2005). This can be accomplished by specifying an
appropriate linear model for detection, as is possible in
several software packages (e.g., Program MARK). Care
must be taken to sample each area similarly such that
sharing information about detection is reasonable. An
informationtheoretic approach (Burnham and Anderson
2002) can be used to evaluate whether sharing information
is supported by the data. One simulation found this to be a
useful approach when a population at a single sampling
area was small (N = 20; Conn et al. 2006). When sample
sizes of combined datasets are [20 individuals and sharing
detection information is considered reasonable a priori,
modeling the detection process in a shared framework is
appropriate. This approach has the potential to allow for
added model complexity to explain realistic variation in
detection, better identify appropriate models through model
selection, and likely produce more reliable abundance
estimates (Boulanger et al. 2002; Bowden et al. 2003;
White 2005). In other words, sharing information may
release an analysis from the bounds of poor data and allow
additional parameters to be estimated. Although, when
most animals are detected, sharing information may not be
particularly useful, as point estimates will be similar,
regardless of variation in detection.
Discussion
We emphasize again that the approach described here
should only be considered for the unique data typical of
photographicsampling: (1) infrequent detections, (2) a
small number of individuals detected, (3) long survey
durations, and (4) variability in detection among individuals. With better data, more sophisticated approaches are
warranted. Given the type of data described here, the goal
of carnivore C–R studies of low density populations should
be to make modeling the detection process irrelevant by
maximizing p* and detecting all or almost all of the population. Even with heterogeneous detection among individuals, as long as p* C 0.6, the M0 estimator with PLCIs
can be used to accurately estimate abundance and quantify
uncertainty. When a species is sampled multiple times at an
area or at multiple areas using the same methodology, we
encourage researchers to combine these data and determine
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Popul Ecol (2014) 56:463–470
whether information about the detection process can be
shared.
M0 and PLCIs can be computed by many software
programs, including Program MARK (White and Burnham
1999), Program DENSITY (Efford et al. 2004), or through
the R programming language in packages RMark (Laake
2013) and secr (Efford 2011). Since PLCI’s require
parameters are in a likelihood framework, they cannot be
used with conditionallikelihood models or with the Jackknife estimator.
We recommend that C–R studies of very small populations consider capture frequencies as sampling proceeds
and provide such frequencies in publications, so readers
can (1) examine how often individuals are detected and
qualitatively assess the likelihood of a census, (2) determine how reliable estimates are likely to be, and (3) consider the degree of heterogeneity and whether it is
important in the estimation of abundance. This will allow
reviewers to examine when a heterogeneity model may or
may not be useful.
Results shown here suggest that detecting most of the
population two or three times will provide minimal
expected bias and acceptable levels of precision that will
be useful to conservation and management agencies. A
common suggestion in photographicsampling studies to
achieve appropriate frequencies of detection is to ensure
multiple camera sites are within each potential home range
of individuals within the study area (Karanth and Nichols
2002). However, this may be impractical due to restrictions
on the number of cameras available. An alternative design
is to move cameras during sampling to maximize the
chance of placing multiple cameras within each individual’s home range, thereby increasing an individual’s
probability of being detected at least once. A goal of every
study should be to ensure no individual in the population
has a zero probability of detection; one way to do this is to
move cameras. When animals have zero probability of
detection due to the sampling technique or layout, estimating their numbers is equivalent to asking the question,
how many invisible animals are there? There is of course
no statistical solution to mitigate this bias.
To maximize p* and reduce potential bias of ‘‘invisible’’
individuals by moving cameras, one could use a block
design where a grid of cameras placed close together is
moved jointly to survey a larger area in total (Karanth and
Nichols 2002). Alternatively, each camera site could be
independently moved within the survey area after a given
criterion is met. For example, when a camera site detects
individuals at least twice within a given time frame (e.g.,
one month), it would be advantageous to move the site in
an attempt to increase the detection of others not observed
yet. After detecting the same individual three times, there
is little benefit in further detections. Lastly, if photographs
�Popul Ecol (2014) 56:463–470
cannot be reviewed and individuals identified within the
sampling period, camera sites could be deployed throughout the study area for a set time period (e.g., one month)
and then all simultaneously redeployed within the same
area for the same amount of time.
In this paper, we have assumed that the closure
assumption of C–R models is met, such that there are no
geographic or demographic changes in the population
during sampling. While we have demonstrated the usefulness of protracted sampling to maximize p*, which makes
modeling detection irrelevant and thus helps overcome the
issues of small sample size to estimate abundance accurately and precisely, we do not encourage researchers to do
so at the cost of violating this assumption. Because carnivores are typically longlived (relative to the study period)
it has been common to assume populations are closed to
permanent demographic changes over a period of a month
or even more (Karanth and Nichols 2002). However, a
consequence of a long sampling duration is that highly
mobile animals may temporarily move on and off of the
study area. Minor temporary emigration (e.g., absence
from the study area on only a few occasions) over a long
sampling duration may be inconsequential (although it
does depress p and induce heterogeneity). However, considerable movement in and out of the study area will
introduce bias to estimating abundance and/or necessitate a
redefinition of the population inference to the super population of animals that could have used the area over the
course of sampling (Kendall 1999).
Recently, much focus has been on the use of spatiallyexplicit C–R models to estimate animal density and/or
abundance (Efford and Fewster 2013; Royle et al. 2013).
However, to do so requires estimation of more parameters
(i.e., a spatial component), which generally requires better
data than considered here and an additional set of
assumptions (Efford and Fewster 2013; Ivan et al. 2013).
Given the estimation challenges we’ve noted here with
sparse data, estimating additional parameters to obtain
density estimates is seemingly unrealistic without ancillary
information (e.g., more data or prior information). We
suggest that if the goal is to monitor a small population in a
relatively well defined area in which N is expected to be
less than 20 individuals and p is expected to be B0.1, a
confirmatory census approach as outlined here may be
reasonable. It’s interesting to note that simulations using
spatiallyexplicit models found that when the population
size is much larger than considered in this paper, the M0
model performs well (low bias and high precision) when
the detection process is assumed to follow a strict circular
bivariate Gaussian kernel and even when the sampling
occurs over patchy habitat, as long as the suitable habitat of
the target animal is appropriately covered (Efford and
469
Fewster 2013). In these simulations, the Jackknife estimator was found to be unreliable.
Throughout we have focused on carnivore photographicsampling studies, but our suggestions pertain more
generally when the population is very small and most
individuals in the population are detected. For instance,
when the sampling methodology is highly efficient, such as
when scat dogs are used to collect DNA (Wasser et al.
2004) most of the individuals may be detected and p* may
be near one. Also, estimating species richness may result in
sample sizes on the order of those discussed here (Walther
and Morand 1998).
Lastly, researchers should always keep ecology in mind
when estimating abundance. C–R estimators and the
mathematics that underlie them are ignorant of biology—
that piece will always be the responsibility of the
investigator.
Acknowledgments We thank S. Karpanty and D. Catlin for early
discussions on this topic, G. White for taking the time to answer
questions throughout the development of this research, and Sunarto
for inspiring this work. We are grateful to reviewers that helped
clarify this manuscript.
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�
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Electronic Supplementary Information (ESM)
Estimating the abundance of rare and elusive carnivores from photographicsampling data when
the population size is very small
Brian D. Gerber1, Jacob S. Ivan2, and Kenneth P. Burnham3
1
Colorado Cooperative Fish and Wildlife Research Unit, Department of Fish, Wildlife and
Conservation Biology, Colorado State University, Fort Collins, CO 80523, USA.
2
Colorado Division of Parks and Wildlife, Wildlife Research Center, 317 West Prospect Road,
Fort Collins, Colorado 805262097, USA
3
Professor Emeritus, Department of Fish, Wildlife, and Conservation Biology, Colorado State
University, Fort Collins, CO 80523, USA.
Corresponding Author:
Brian D. Gerber, Colorado Cooperative Fish and Wildlife Research Unit Department of Fish,
Wildlife and Conservation Biology, Colorado State University, Fort Collins, CO 80523, USA.
bgerber@colostate.edu
1
�R code for simulations
#####################################
#Simulate heterogeneous capturerecapture
#data to examine how many individuals will
#likely be detected and secondly, to
#estimate abundance using the M0unconditional
#likelihood CR estimator and quantify uncertainty
#with profilelikelihood confidence intervals
#############################################
#Author: Brian D. Gerber
#Contact: bgerber@colostate.edu
#Last Modified: 8/22/2012
############################################
###########################
#Load Functions and Libraries
###########################
library(RMark)
library(gregmisc)
expit = function (logit){
exp(logit)/(1+exp(logit))
}
logit = function (expit){
log(expit/(1expit))
}
###########################
#Setup Variables
###########################
J=30 #J will be the number of sampling occasions, identified as t in manuscript
N=10 #True abundance animals that have some chance of capture
sets=100 #Number of datasets to simulate
l=0
num_capt=rep(0,sets) #number of animals detected M(t+1)
M0_N=M0_se=M0_lcl=M0_ucl=rep(0,sets) # Store abundance estimates
#Set the heterogeneity for detection probability logit(p) as
mu=0.1
sigma=0.1
##############################################################
#Plot heterogeneity of p
2
�##############################################################
plot(1, type="n", axes=T, xlab="", ylab="", xlim=c(0,1), ylim=c(0,4))
curve(dnorm(x,mu,sigma),col=2,lwd=2,add=TRUE)
#######################
#Create heterogeneity
#CR Data and start simulation
#######################
while (l<sets){
if(l%%10==0)cat("Simulation # ",l,"\n"); flush.console()
mat=data.frame(matrix(0,N,J))
for (q in 1:N){
detection=expit(rnorm(1,logit(mu),sigma))
for (r in 1:J){
sample_unif=runif(1,0,1)
if(sample_unif<=detection){
mat[q,r]=1}
else{mat[q,r]=0}
}
}
#Drop the capture histories with no detections
Rcapture_input=as.matrix(mat[rowSums(mat) != 0, , drop=FALSE])
#Create input for RMARK
mark_input=mat[rowSums(mat) != 0, , drop=FALSE]
mark_input=as.data.frame(apply(mark_input,1,paste,collapse=""))
mark_input < data.frame(lapply(mark_input, as.character), stringsAsFactors=FALSE)
colnames(mark_input)= c("ch")
num_capt_temp=dim(Rcapture_input)[1]
#With low p and N, sum CH will have zero animals
#or some CH will be too small, such as 2 animals, which
#can't be used to estimate N
if(num_capt_temp<=2){
next}
else{
l=l+1
num_capt[l]=dim(Rcapture_input)[1] #this is the number of animals caught
3
�#Define Null model
pdotshared=list(formula=~1,share=TRUE)
#Run model through RMARK M0unconditional with profile likelihood CI.
m0=try(mark(mark_input,model="Closed",profile.int =
TRUE,adjust=FALSE,model.parameters=list(p=pdotshared), brief=FALSE,output=FALSE,
delete=TRUE, invisible=TRUE), silent=TRUE)
#
#
#Save estimates in simulation
M0_N[l]=as.numeric(m0$results$real[2,1]) #Abundance only
M0_se[l]=as.numeric(m0$results$real[2,2]) #Abundance SE
M0_lcl[l]=as.numeric(m0$results$real[2,3]) #LCL
M0_ucl[l]=as.numeric(m0$results$real[2,4]) #UCL
if(!is.na(M0_se[l])){if(M0_se[l]>N*inflation){M0_lcl[l]=M0_ucl[l]=NA
M0_se[l]= NA}}
}
}
########################################################
M0_N=as.numeric(M0_N)
M0_se=as.numeric(M0_se)
M0_lcl=as.numeric(M0_lcl)
M0_ucl=as.numeric(M0_ucl)
M0_se[M0_se=="NaN"]=NA
M0=cbind(M0_N,M0_se,M0_lcl,M0_ucl)
#Display frequency of captures by individual
rowSums(Rcapture_input)
mean(rowSums(Rcapture_input)) #Mean frequency of detection
# Plot the histogram of individuals detected
hist(num_capt, xlim=c(0,(N+10)), freq=FALSE)
abline(v=N, lwd=2, col=2) #Plots truth
# Plot proportion of population detected
hist((num_capt/N), xlim=c(0,1), freq=FALSE)
#Plot histogram of abundance estimates
hist(M0_N, breaks=10, main="Abundance Estimate", xlim=c(0,(N+10)),freq=FALSE)
abline(v=mean(M0_N), col=1, lwd=2) #Expected value of Nhat
abline(v=N, col=2, lwd=2) #Ntruth
#Expected_bias
mean(M0_N)N
4
�#Plot all simulations abundance estimates and profile condidence intervals
win.graph()
plotCI(seq(1,sets,1), M0_N, pch=16,ui=M0_ucl, li=M0_lcl,xlim=c(1,sets), ylim=c(0,50), gap=0,
xlab="Simulation #",ylab=expression(hat(N)))
abline(h=N, lwd=2, col=2)
5
�Fig. S1 Distributions of detection probability used to model individual heterogeneity, where
logit(pi) = N(expit(µ), σ) and σ = 0.1.
6
�A
7
�B
Fig. S2 Empirical distributions of abundance estimates ( ̂ ) using the M0 (null, constant p) closedcapture recapture estimator and simulated heterogeneous data where A) logit(pi) = N(expit(µ =
0.05), σ = 0.1) and B) logit(pi) = N(expit(µ = 0.10), σ = 0.1) over sampling occasions (t) of 1060; the dotted line at N = 10 indicates true abundance and the solid line is E[N].
8
�Fig. S3 Percentage of failed point or standard error estimates from simulated heterogeneous
capturerecapture data where logit(pi) = N(expit(µ), σ) where µ = 0.05 or 0.1 and σ = 0.1 over
sampling occasions of 1060.
9
�Fig. S4 The expected number of unique individuals detected (Mt+1) from simulated
heterogeneous capturerecapture data where logit(pi) = N(expit(µ), σ) where µ = 0.05 or 0.1 and
σ = 0.1 over sampling occasions of 1060.
10
�
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Estimating the abundance of rare and elusive carnivores from photographicsampling data when the population size is very small
Description
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<span>Conservation and management agencies require accurate and precise estimates of abundance when considering the status of a species and the need for directed actions. Due to the proliferation of remote sampling cameras, there has been an increase in capture–recapture studies that estimate the abundance of rare and/or elusive species using closed capture–recapture estimators (C–R). However, data from these studies often do not meet necessary statistical assumptions. Common attributes of these data are (1) infrequent detections, (2) a small number of individuals detected, (3) long survey durations, and (4) variability in detection among individuals. We believe there is a need for guidance when analyzing this type of sparse data. We highlight statistical limitations of closed C–R estimators when data are sparse and suggest an alternative approach over the conventional use of the Jackknife estimator. Our approach aims to maximize the probability individuals are detected at least once over the entire sampling period, thus making the modeling of variability in the detection process irrelevant, estimating abundance accurately and precisely. We use simulations to demonstrate when using the unconditionallikelihood </span><i>M</i><sub>0</sub><span> (constant detection probability) closed C–R estimator with profilelikelihood confidence intervals provides reliable results even when detection varies by individual. If each individual in the population is detected on average of at least 2.5 times, abundance estimates are accurate and precise. When studies sample the same species at multiple areas or at the same area over time, we suggest sharing detection information across datasets to increase precision when estimating abundance. The approach suggested here should be useful for monitoring small populations of species that are difficult to detect.</span>
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Gerber, B. D., J. S. Ivan, and K. P. Burnham. 2014. Estimating the abundance of rare and elusive carnivores from photographicsampling data when the population size is very small. Population Ecology 56:463470. <a href="https://doi.org/10.1007/s1014401404318" target="_blank" rel="noreferrer noopener">https://doi.org/10.1007/s1014401404318</a>
Creator
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Gerber, Brian D.
Ivan, Jacob S.
Burnham, Kenneth P.
Subject
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Camera traps
Capture–recapture
Heterogeneous detection
Small population
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8 pages
Date Created
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20140125
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<a href="http://rightsstatements.org/vocab/InCNC/1.0/" target="_blank" rel="noreferrer noopener">In Copyright  NonCommercial Use Permitted</a>
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application/pdf
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English
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Population Ecology
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Article