Population estimate for the Alberta 3B and 4B Grizzly Bear

2004 Population Inventory and Density Estimates for the Alberta 3B
and 4B Grizzly Bear Management Area
Report Prepared for Alberta Sustainable Resource Development, Fish and Wildlife
Division, May 2005 (with updates November 2005)
John Boulanger1, Gord Stenhouse2, Mike Proctor3, Stefan Himmer4, David
Paetkau5, and 6Jerome Cranston.
1
Integrated Ecological Research, Nelson, B.C. V1L 5T2, [email protected]
2
Sustainable Resource Development, Fish and Wildlife Division, Box 6330, Hinton, AB
T7V 1X7, [email protected]
3
Dept. of Biological Sciences, University of Alberta, Edmonton, AB T6G 2E9,
[email protected]
4
Arctos Wildlife Services, Site 10, Comp. 7, R.R. 1, Crescent Valley, B.C. V0G 1H0,
[email protected]
5
Wildlife Genetics International, Nelson, B.C. V1L 5P9, [email protected]
6 Foothills Model Forest, Box 6330, Hinton, AB T7V 1X7, [email protected]
Alberta 3B and 4B Grizzly Bear Inventory Project
2
1. Abstract
This report provides grizzly bear population and density estimates for the Alberta 3B and
4B grizzly bear management areas. Recent DNA analysis with provincial DNA samples
has shown that these two management areas are in fact one genetically distinct population
unit (Proctor 2004), hence this inventory project was designed to sample a single
population unit. In the spring of 2004 a DNA sampling grid (8820 km2) was sampled.
One hundred and eighty 7x7 km grid cells were sampled with 1 DNA bait site for 4
sampling sessions. Each bait site was moved between sessions to ensure adequate
coverage of cell areas. Thirty-nine bears were captured in bait sites, and 5 other bears
were identified with other forms of sampling. The distribution of bears was clumped
along the west side of the study area with virtually no bears detected in the southeast part
of the sampling grid. Data-based tests and GPS collared bears suggested that bears
traversed across the western border therefore violating the assumption of population
closure. The estimate of the superpopulation of bears (including dependent offspring) in
the grid and surrounding area was 53 (SE=8.3, CI=44 to 80). The average number of
bears on the sampling grid was estimated by multiplying the superpopulation number by
the proportion of GPS bear locations on the sampling grid (0.79) to estimate an average
number of bears on the sampling grid at any one time of 42 (SE=7.3 CI=36 to 55). This
number was divided by the sampling grid area (8820 km2) to derive a density estimate of
4.79 bears per 1000 km2 (SE=8.82 CI=4.10 to 6.28). The levels of precision of this
project were one of the highest attained in any previously conducted DNA census in
British Columbia and Alberta.
2. Introduction
This report provides a population estimate for the grizzly bear DNA mark-recapture
inventory project that occurred in Bear Management Areas 3B and 4B in 2004. Recent
DNA analysis with provincial DNA samples has shown that these two management areas
are in fact one genetically distinct population unit (Proctor 2004). This inventory project
had multiple objectives oriented towards optimization of methodologies for future
inventory projects, with a long-term goal of improving sampling design and reducing
inventory costs. The results of this work will be presented in future documents.
3. Methods
3.1.
Field methods
A DNA sampling area of approximately 8820 km2 was defined within the 3B and 4B
management area. Within this area, a systematic sampling grid with 180 49 km2 grid
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cells was placed. Plains areas towards the eastern boundary were sampled using 17
transect cells to determine bear occurrence in these areas. In the spring of 2004 this area
was sampled. One bait site was placed in each grid cell for 4 14 day sampling sessions.
Site selection was done prior to fieldwork and was based upon grizzly bear resource
selection function models (Nielsen 2004; Nielsen et al. 2002), GPS collar location,
remote sensing-based habitat mapping, and aerial photographs. Each bait site was
moved after each session to ensure equal coverage of each cell. Two strands of barbed
wire were used at bait sites to test the utility of single wire sampling, and maximize bear
capture probabilities.
;
Edson
Hinton
;
Brule
;
Robb
;
Cadomin
;
DNA grid cells
; Towns
GB 3B 4B
Elevation
600 - 1200
1201 - 1600
1601 - 2000
2001 - 2500
2501 - 3300
20
0
20
40 Kilometers
Figure 1: DNA sampling grid layout for the Alberta area 3B and 4B grizzly bear DNA
mark-recapture population inventory project. Each cell was 49 km2.
Transect cells (to the east of the main grid) were also sampled, however, sites were not
moved within each cell. Grizzly bear hair was collected from each bait site using
methods documented in Woods et al. (1999). Applicable samples were then genotyped
using methods documented in Paetkau (2003).
3.2.
Data summary
Data was summarized in terms of overall frequencies of captures for individuals. For
traditional mark-recapture analysis, multiple captures of an individual are pooled into one
capture per session. Summary statistics were generated for the pool data set. See
Appendix 2 for background information on mark-recapture population estimation.
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Alberta 3B and 4B Grizzly Bear Inventory Project
3.3.
4
Assessment of closure violation
In review, closure violation is caused by bears moving in and out of the study areas
during sampling (White et al. 1982). If closure violation is occurring, mark-recapture
population estimates will pertain to the “superpopulation” of bears in the sampling grid
and surrounding area during the time that sampling was conducted. (Kendall 1999;
White 1996). For estimation of density, and comparison of different areas the average
number of bears on the sampling grid at any one time is most applicable. For this reason,
we used data-based tests for closure violation (Boulanger and McLellan 2001), and used
data from GPS collared bears to scale superpopulation estimates into average population
size estimates (White and Shenk 2001) as discussed later.
The Pradel model in program MARK was used to assess the data set for closure violation
as first described by Boulanger and McLellan (2001). The main premise for this test is
that, if closure violation was occurring, grizzly bears that were near the grid edge ("edge"
grizzly bears) would have lower recapture rates due to a reduced trap encounter rate
compared to grizzly bears farther from the edge ("core" grizzly bears). In addition, if
grizzly bears moved from the grid for the entire sampling period, then edge grizzly bears
would exhibit a lower apparent survival estimate than core grizzly bears. Also, grizzly
bears which immigrated into the grid area during sampling would be more prone to be
captured near the grid edge. The distance from edge of capture was the shortest distance
from the grid edge to the mean location of hair-collection sites where a grizzly bear was
identified during the entire project.
The Pradel (1996) model as incorporated in program MARK (White and Burnham 1999),
which estimates apparent survival (φ), recruitment (f) and recapture probability (p), was
used for this analysis. The estimates for recapture rate are for the exact sampling period,
whereas the estimates for the apparent survival rate (φ) and recruitment (f) correspond to
the interval before the given sampling period.
We assumed that the population of grizzly bears was demographically closed for this
analysis. The duration of sampling was ≈1 month so this assumption was reasonable.
Apparent survival equals true survival (S) (due to mortality) times the fidelity of grizzly
bears (F) to the sampling grid (φ=SF). Because the population was demographically
closed, we assumed that true survival equaled one (S=1) and therefore relative changes in
φ reflect grizzly bear fidelity to the sampling grid rather than actual mortalities, i.e.,
(φ=F). The Pradel recruitment rate estimates the number of new individuals in the
population at time j+1 per individual at time j. We assumed that the number of births
during sampling was minimal and therefore measures of recruitment reflected permanent
immigration or "additions" of grizzly bears into the sampling grid. For the sake of
simplicity we will refer to φ as the rate of “Fidelity” and f as the rate of “Additions” in the
rest of the report.
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Alberta 3B and 4B Grizzly Bear Inventory Project
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As an initial appraisal of population closure we evaluated the goodness of fit of Pradel
models constrained to only allow certain forms of closure violation as first proposed by
Stanley and Burnham (1999). The exact models used in the test of Stanley and Burnham
(1999) were the fully open Jolly Seber model (JS), a recruitment but not mortality model
(NM), a mortality but not recruitment model (NR), and a closed model with no mortality
or recruitment (Mt). We emulated the approach of Stanley and Burnham (1999) by
fixing parameters to appropriately constrain the Pradel model as detailed in Boulanger
and McLellan (2001).
We then used continuous covariates to model the relationship of distance from edge for φ,
f, or p as a logistic function. The potential shapes that the logistic curve, which is used to
model covariates in MARK, could accommodate was restrictive, and therefore logistic
equations with log transformed (+1) (Zar 1996) distance from edge and higher order
polynomial (i.e. dfe2 log (dfe)2+1) distance from edge terms were also considered.
Covariates were standardized in program MARK by the mean and standard deviation of
observed distances (White et al. 2002). A logit link was used for all analyses.
In addition to covariates, both sex and time specific model formulations were considered
in the building of mark-recapture models. The fit of models was evaluated using the
Akaike Information Criterion (AIC) index of model fit. The model with the lowest AICc
score was considered the most parsimonious, thus minimizing estimate bias and
optimizing precision (Burnham and Anderson 1998). The difference in AICc values
between the most supported model and other models (ΔAICc) was also used to evaluate
the fit of models when their AICc scores were close. In general, any model with a ΔAICc
score of less than 2 was worthy of consideration.
An assumption of the Pradel analysis was that capture, survival, and movements are
independent. In addition, it is assumed that all individuals within a group have similar
apparent survival rates and similar values of other model parameters. If individuals are
not independent, the multinomial variances from the models become inflated or
overdispersed, which causes underestimation of parameter variances and overfitting of
models. Various goodness-of-fit tests are available to test and estimate the degree of
overdispersion in the data set. Because lack of fit in the Pradel models can only be
assessed for the recapture portion of the encounter history, the goodness-of-fit test in
Program RELEASE (Burnham et al. 1987) was used to assess goodness-of-fit. If
overdispersion was detected (as indicated by ĉ > 1), we used QAICc instead of AICc
model selection criterion to select optimal models (Burnham and Anderson 1998, White
et al. 2002).
If a segment of core animal was identified by the Pradel analysis, then population
estimates were calculated for this segment and extrapolated to the entire grid area
(Boulanger and McLellan 2001). This extrapolation was based on the assumption that
differences in population size estimates were due to closure rather than differences in
densities of grizzly bear on the trapping grid. A test of uniform density (Otis et al. 1978)
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Alberta 3B and 4B Grizzly Bear Inventory Project
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was therefore conducted to test whether density was reasonably uniform between core
and extrapolated areas.
3.4.
Closed population estimation model selection
We primarily used the Huggins closed models (Huggins 1991) in program MARK for
estimation model selection and population estimates. Sexes of grizzly bear were entered
as groups in this analysis, testing whether sexes displayed differing forms of capture
probability variation. Models with time, heterogeneity, and behaviour variation were
considered in the analysis. Mean precipitation for each session was considered as a
capture probability covariate to potentially explain temporal variation in capture
probabilities. Mixture model heterogeneity estimators (Pledger 2000) as incorporated in
program MARK (White and Burnham 1999) were used to model heterogeneity variation.
This is modeled by letting capture probabilities come from more than one capture
probability distribution. There are three parameters with the 2-distribution mixture
model. The parameters are the probability that a given capture probability will come
from the first distribution (π), the mean capture probability of the first distribution (θ1),
and the mean capture probability of the second distribution (θ2). The probability that the
capture probability comes from the second distribution is 1- π (Pledger 2000). As with
the Pradel analysis, AICc model selection was used to assess parsimonious estimation
models. In addition, tests for specific forms of capture probability variation in program
CAPTURE (Otis et al. 1978) were used to assess capture probability variation and
compare with MARK model selection results.
3.5.
Population estimates
Population estimates from the program MARK models and program CAPTURE models
were considered. Estimates from all of the Huggins MARK models were modelaveraged, allowing population estimates that were influenced by all the estimation
models considered in the analysis.
3.6.
Simulation tests of estimators
Two simulation methods were used to test estimation models. First, results from past
projects, radio collared studies of grizzly bears, and expert opinion were used to devise
simulation models of likely forms of capture probability variation in bear population.
These models were used to evaluate the precision of estimators across a range of
sampling parameters. Second, direct estimates of capture probability variation obtained
in program MARK were used to parameterize data-based simulations.
As summarized in Table 1, each of these approaches has strengths and weaknesses.
Simulation models based on data-based simulations are limited to detectable forms of
capture probability variation in mark-recapture data sets. Because most data sets are
sparse it is probable that certain forms of heterogeneity variation, such as low capture
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Alberta 3B and 4B Grizzly Bear Inventory Project
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probabilities of cubs of the year, are not detected. Therefore, telemetry and expert based
simulations are useful to allow exploration of non-detected forms of capture probability
variation.
Table 1: Methods used to estimate precision and evaluate estimator performance.
Evaluation method
Strengths
Weaknesses
Makes “expert” assumptions
1) Simulations based on expert Incorporates biologicallyabout main forms of capture
based forms of capture
opinion and telemetry
probability variation in data
probability variation that may
estimates of cub capture
that cannot be directly tested.
be difficult to detect in field
probability
data
2) Data-based simulations
Uses direct estimates of
May not include forms of
variation from data with
capture probability variation
minimal “expert-based”
not detected due to sparse data
assumptions
Each of these methodologies are explained in detail in Appendix 1.
Estimators were evaluated in terms of bias, precision, and confidence interval coverage.
Bias was defined by percent relative bias
ˆ − N)
(N
where N̂ is the estimate population size from each model and N was
N
the true number of bears in the simulation. The optimal level of precision was indexed
by the coefficient of variation (CV) which is
P.R.B. =
ˆ
CV ( Nˆ ) = SE ( N ) ˆ X 100
N
The CV is a convenient way to compare precision of different projects since it expresses
standard error in percentage units of the estimate. The confidence interval width is
approximately ± 1.96 (CV). So if a project has a CV of 10% then the confidence interval
is roughly ± 20% of the estimated population size. Confidence intervals for markrecapture project are seldom symmetrical so this is a rough approximation. Confidence
intervals can be conceptualized as a bell shaped curve with the most probable estimate
being the point estimate and the least probable estimates being on either end of the
confidence interval. Bias and precision were considered simultaneously by the mean
squared error (M.S.E.), which is the absolute value of percent relative bias plus the
coefficient of variation. An estimator with the lowest M.S.E. displays the best balance
between bias and precision. In addition, confidence interval coverage was considered
which is the proportion of times the estimated population size bracketed the true
population value.
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Alberta 3B and 4B Grizzly Bear Inventory Project
3.7.
8
Average population size on study area grid and density estimates
We estimated average number of bears on the sampling grid using two methods. If
assumptions were met we used the core-extrapolation method of Boulanger and
McLellan (1999) as discussed previously. In addition, superpopulation estimates were
multiplied by the proportion of sampling occasions that GPS collared bears were on the
sampling grid (White and Shenk 2001) to obtain estimates of the average number of bears
on the sampling grid. We estimated proportion of locations for GPS bears during the time
of DNA sampling for 2003 and 2004 when a suitable spatial distribution of GPS bears
were present on the 3B and 4B sampling area. Bears were included if any of their
locations traversed the grid during sampling in either 2003 or 2004. Proportion for bears
that occurred on the grid in 2003 and 2004 were averaged to avoid psuedoreplication.
This resulted in a sample size of 23 GPS bears for estimation of proportion of time on
sampling grid. Both the CAPTURE and MARK superpopulation estimate and the
proportion of radio marked bears on the grid estimate have error. Therefore, we used the
delta method (Seber 1982) to estimate combined variances under the assumption that
correlation between population estimates and the proportion of time on the grid was zero.
We calculated log-based confidence-intervals for the average number of bears on the
sampling grid estimates using formulas presented in White et al. (2002).
4. Results
4.1.
Data summary
Thirty-nine bears (18 males and 21 females) were identified with the inventory sites. An
additional 5 bears were identified using other sites (Figure 2). The distribution of
captures was clumped in the western mountainous regions with few captures on the
eastern plains area. Only 1 bear was captured in the transects.
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Alberta 3B and 4B Grizzly Bear Inventory Project
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Elevation
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Figure 2: Distribution of captures at bait sites for the Alberta 3B4B inventory projects.
Sites are colored by the number of unique bear captures that occurred at the site over the
course of the project.
Mean capture locations of DNA bears also suggest that most bears were found on the
western edge of the study area (Figure 3). Male and female bears were intermixed
evenly.
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
10
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Grizzly bear mean DNA capture location
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Elevation
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1601 - 2000
2001 - 2500
2501 - 3300
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Figure 3: Mean capture locations for individual bears in the Alberta 3B and 4B inventory
project.
Summary statistics for mark-recapture modeling suggest that there was temporal
variation in capture rates for bears as summarized by animals caught each session
(Table 2). Male captures decreased with session whereas female captures increased and
then decreased in latter sessions. The number of newly caught (genotyped) individuals
decreased with session for both sexes suggesting that sampling was effective in capturing
the majority of bears in the area. Capture frequencies suggested higher recapture rates
with more bears being captured 2 or more times than bears captured once.
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Alberta 3B and 4B Grizzly Bear Inventory Project
11
Table 2: Summary statistics for the DNA data set.
Statistic
males
Animals caught n(j)
Total individuals caught M(j)
Newly caught u(j)
Frequencies f(j)
females
Animals caught n(j)
Total individuals caught M(j)
Newly caught u(j)
Frequencies f(j)
Pooled
Animals caught n(j)
Total individuals caught M(j)
Newly caught u(j)
Frequencies f(j)
4.2.
Session
1
2
3
4
13
0
13
8
9
13
3
6
8
16
1
3
3
17
1
1
7
0
7
10
14
7
10
6
10
17
1
4
7
18
3
1
20
0
20
18
23
20
13
12
18
33
2
7
10
35
4
2
total
18
21
39
Tests for closure violation
The Program RELEASE goodness-of-fit test for model {φ[sex x time] p[sex x time]}
suggested minimal overdispersion of multinomial likelihoods or ĉ = 1 (χ2 = 1.07, df=7,
p=0.99) and therefore AICc was used for model selection. The Pradel model test for
closure violation suggested that a significant degree of emigration and immigration of
bears occurred during the course of sampling (Table 3). For example, a model that
assumed the population was closed (φ (1) p(.) f(0)) was substantially less supported by
the data then a model that suggested constant emigration rates between sexes and sexspecific immigration rates (φ (.) p(T2) f(sex)). Capture probability was best described by
a quadratic term p(T2) which allowed capture probabilities to initially be low then
increase and then decrease. Models that constrained demography to vary as a function of
distance of mean capture from grid edge were not well supported by the data. This may
have been due to the clumping of bear distribution along the western border of the
sampling grid (Figure 3).
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July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
12
Table 3: Pradel model selection. Akaike Information Criteria (AICc), the difference in
AICc values between the ith model and the model with the lowest AICc value (ΔAICc),
Akaike weights (wi), and number of parameters (K) are presented
Model
AICc
ΔAICc
wi
K
Deviance
199.2
0.00
0.30
5
188.3
φ (.) p(T2)A f(sex)
φ (.) p(T2) f(.)
200.7
1.44
0.14
4
192.1
φ (.) p(.) f(.)
200.8
1.57
0.13
3
194.5
φ (.) p(T2) f(sex + ld)B
201.5
2.27
0.09
6
188.2
φ (sex) p(T2) f(sex)
201.6
2.35
0.09
6
188.3
φ (.) p(T2) f(sex X ld)
202.1
2.82
0.07
7
186.3
φ (.) p(T2) f(.+ ld)
202.9
3.70
0.05
5
192.0
2
φ (.) p(T ) f(.+ d)
203.0
3.76
0.05
5
192.1
φ (1) p(T2) f(0)
203.1
3.85
0.04
2
198.9
φ (sex) p(t) f(sex)
203.9
4.62
0.03
8
185.5
φ (1) p(.) f(0)
208.2
8.92
0.00
1
206.1
φ (1) p(.) f(.)
210.3
11.04
0.00
2
206.1
A 2
T denotes that capture probability variation was simulated as a quadratic trend
B
ld denotes that the log of the distance of mean capture of bear from grid edge
Model averaged estimates of demographic parameters suggested that emigration rates
were equal for males and females, however, immigration rates were higher for female
bears (Table 4).
Table 4: Model averaged estimates of Pradel model parameters
Parm
Sex
Estimate
SE
LCI
Male
0.76
0.11
0.49
φ (emigration)
Female
0.76
0.11
0.49
f (immigration)
Male
0.14
0.11
0.03
Female
0.32
0.20
0.07
UCI
0.91
0.91
0.50
0.73
Tests for uniform distribution of bears in core and edge areas suggested that density was
uneven on the study area grid (χ2=17.1 df=5, p=0.004), a conclusion that was reasonably
obvious from observation of mean capture locations (Figure 3). Therefore using the coreextrapolation to correct density estimates was not valid. Instead, we used GPS collared
bear movements during 2004 and previous years to estimate the average number of bears
on the sampling grid over the course of the DNA study.
4.3.
Population estimation model selection
AICc model selection results suggested that models with time and heterogeneity variation
were most supported by the data (Table 5). These models constrained per-session capture
probabilities to be equal for all sessions except the 4th session. The most supported
model also suggested that capture probabilities of bears varied as a function of the log of
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
13
their distance from the edge of the sampling grid. A model with sex-specific capture
probability variation was also partially supported by the data (Mth p(sex X linear trend)).
Models that assumed equal capture probabilities of bears, temporally constant capture
probabilities, or associations between capture probability and precipitation were less
supported.
Table 5: MARK Huggins closed model AICc model selection. Akaike Information Criteria
(AICc), the difference in AICc values between the ith model and the model with the lowest
AICc value (ΔAICc), Akaike weights (wi), and number of parameters (K) are presented
Model
AICc
ΔAICc
wi
K
Deviance
A
B
202.0
0.00
0.20
6
189.4
Mth2 πi (.), (θ1& θ2(.) X t4 ) +ld
Mth2 πi (.), θ1& θ2 (.)X t
202.1
0.14
0.19
9
182.9
Mth2 πi (.), θ1& θ2 (.)X t4
202.7
0.73
0.14
5
192.3
204.0
2.09
0.07
4
195.8
Mth p (sex X T)
Mth2 πi (.), (θ1& θ2 (.)X t4) +ld+ld2
204.1
2.17
0.07
7
189.4
204.3
2.31
0.06
4
196.0
Mt p(t4) +ld + ld2
Mth p(sex+t4)
204.4
2.40
0.06
3
198.2
205.0
3.01
0.04
4
196.7
Mth p(sex+t4) + ld
Mt p(t)
205.3
3.37
0.04
4
197.1
205.7
3.76
0.03
5
195.3
Mth p(sex+t4) X ld
Mt p(sex)
206.3
4.33
0.02
8
189.3
206.4
4.44
0.02
5
196.0
Mth p((sex+t4) + ld +ld2
Mo (p(.))
208.1
6.18
0.01
1
206.1
208.2
6.28
0.01
3
202.1
Mt P(precip+precip2)
Mb sex
208.7
6.74
0.01
4
200.4
208.7
6.74
0.01
4
200.4
Mb p(sex) c(sex)
Mth2 πi (.), θ1& θ2 (.)+t
209.5
7.53
0.00
7
194.7
209.5
7.59
0.00
2
205.5
Mt P(precip)
Mo (sex)
210.2
8.22
0.00
2
206.1
210.5
8.57
0.00
3
204.4
Mh2 πi (.) p1&2(.)
Mh2 πi (sex), θ1& θ2 (sex)
216.8
14.84
0.00
6
204.2
A
t4 denotes that capture probabilities were constrained to be different for session 4 but
equal for other sessions
B
ld denotes that the log of the distance of mean capture of bear from grid edge
A plot of bear capture probability as a function of mean distance of capture from the edge
of the sampling grid suggest that bears in the central area of the trapping grid had higher
capture probabilities than bears on the edge of the sampling grid (Figure 4). This was
most likely due to closure violation reducing the capture probabilities of bears near the
grid edge. As a result population estimates from closed models will represent bears that
were present on the grid and the surrounding area (termed the superpopulation of bears).
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
14
0.7
0.65
Capture probabilities
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
5
10
15
20
25
30
Mean distance (km) of capture from grid edge
Figure 4: Estimated capture probabilities of bears as a function of distance of mean
capture from the grid edge. Estimates are from model Mth2 πi (.), (θ1& θ2(.) X t4) +ld in
Table 4. Standard errors of estimates are shown as error bars.
A model with sex-specific capture probability variation (Mth p(sex X linear trend) was
marginally supported by the data with a ΔAICc score of 2.08. Average capture
probability estimates from this model for males and females were 0.41 (SE=0.08,
CI=0.26 to 0.58) and 0.39 (SE=0.08 CI=0.25 to 0.55). This suggests that male capture
probabilities were marginally higher than females. Point estimates of capture probability
for the mixture distributions in the most supported model in Table 4 (Mth2 πi (.), (θ1&
θ2(.) X t4) +ld) suggested that a segment of the population displayed lower capture
probabilities. Estimates from this model suggested that 70.5% of the population had a
mean capture probability of 0.62 (CI=0.45 to 0.77) and 29.5% of the population had a
mean capture probability of 0.00007 (CI=0 to 1). This result suggests that a proportion of
the population displayed lower capture probabilities, however, this segment was still
estimated by the Mth2 model.
Program CAPTURE model selection also suggested that models with time and
heterogeneity variation were most supported by the data. For the pooled sex data set,
time variation was detected (χ2=10.6, df=3, p=0.014) and model Mth was chosen as the
most appropriate estimation model.
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
4.4.
15
Superpopulation and population estimates
Superpopulation (the population of bears on the sampling grid and surrounding area)
were estimated using program CAPTURE models and the Huggins MARK models
(Table 6). There was general agreement between estimators that incorporated
heterogeneity. Precision was influenced by the complexity of models, namely, estimators
that allowed temporal and heterogeneity variation (Mth models) in capture probabilities
exhibited slightly lower precision. The time variation only models (Mt-Chao) showed
lower estimates and higher precision, however, it is likely that point and variance
estimates from these models will be negatively biased due to heterogeneity variation.
Mark model averaged estimates were similar to other models but precision was lower,
presumably due to the added variance caused by model selection uncertainty.
Table 6: Superpopulation estimates of grizzly bears for the Alberta area 3B and 4B
Inventory project. Estimates correspond to bears in the sampling grid and surrounding
area.
Estimator
Males and females
Mh (Chao)
Mh (jackknife)
Mt (Chao)
Mth
Mth2 πi (.), (θ1& θ2(.) X t4) +ld
MARK model ave.
females
Mh (Chao)
Mh (jackknife)
Mt (Chao)
Mth
Mth2 πi (.), (θ1& θ2(.) X t4) +ld
MARK model ave.
males
Mh (Chao)
Mh (jackknife)
Mt (Chao)
Mth
Mth2 πi (.), (θ1& θ2(.) X t4) +ld
MARK model ave.
4.5.
N̂
SE
95% CI
53
54
48
57
60
53
8.3
6.6
5.6
10.9
10.9
12.1
44
46
42
46
47
29
80
73
66
92
94
77
15.7%
12.2%
11.7%
19.1%
18.2%
24.3%
29
28
26
33
32
29
6.9
4.4
4.4
8.8
6.5
6.4
24
24
23
25
25
16
55
42
43
64
53
41
23.8%
15.9%
17.0%
26.7%
20.3%
22.4%
23
22
21
25
27
24
4.9
3.5
2.7
6.9
5.6
5.7
20
20
19
20
21
13
42
35
32
53
46
36
21.4%
15.9%
13.0%
27.8%
20.8%
23.2%
CV
Simulation tests of mark-recapture model estimators
A key question was whether heterogeneity (Mh) model estimators were robust to the
levels of time variation observed in the data set. Simulations were run in which time
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
16
variation as estimated from the mark-recapture data was simulated with heterogeneity
variation based upon observed levels in Mth2 MARK models and expert-based models.
Simulations results suggested that heterogeneity estimators showed reasonable robustness
to time variation, with both estimators exhibiting mean squared error scores that were
lower than the CAPTURE Mth model (Table 7). For this reason, these estimators, that
display higher precision, are most appropriate for this data set compared to the more
complex and less precise Mth models. Of the Mh estimators, the Chao estimator displayed
better performance than the jackknife estimator with confidence interval coverage that
was closer to 95%. Therefore, we conclude the Mh (Chao) estimator is most appropriate
for this data set.
Table 7: Simulation results from data and expert based simulations.
Estimation model
PRB
CV
MSE
C.I. Coverage
Data-based simulations
Mh (jackknife)
0.74
13.0
13.8
89.20
Mh (Chao)
-2.72
14.0
16.7
94.00
-10.36
11.1
21.4
85.70
Mt (Chao)
Mth
6.40
16.2
22.6
89.90
Expert based simulations
Mt (Chao)
-2.72
11.4
14.1
94.40
7.21
14.5
21.7
88.40
Mh (Chao)
Mh (jackknife)
12.55
13.0
25.5
69.10
8.79
17.8
26.6
87.50
Mth
4.6.
Estimation of average number of bears on the sampling grid and density
Seventy nine percent of GPS bear locations were on the sampling grid during the time
period of DNA sampling in 2003 and 2004. Using this figure, superpopulation estimates
were scaled to average population size on the sampling grid (Table 8). This resulted in
average values of 42 and 48 for the Mh Chao and Mth2 πi (.), (θ1& θ2(.) X t4) +ld models.
Coefficients of variation for average number of bears on the sampling grid were below
20% for the Mh (Chao) model suggesting adequate precision of average N̂ estimates.
Table 8: Superpopulation estimates, proportion of GPS collar locations on sampling grid
and the estimated average population size of bears on the 3B4B sampling grid.
Estimation model
Superpopulation
estimate
Estimator
Mh (Chao)
Mth2 πi (.), (θ1& θ2(.) X t4)
+ld
Integrated Ecological Research
N̂
53
S.E.
8.3
60
10.9
Proportion GPS bears Average N̂ on sampling
grid
on grid
Ave N̂
Estimate
SE
SE CIL CIR
CV
79.70%
6.18%
42
7.39
36
55
17.51
79.70%
6.18%
48
9.44
40
64
19.77
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
17
The average population size estimates were then divided by grid area (8820 km2) to
obtain density estimates for the 3B and 4B area (Table 9).
Table 9: Estimates of density ( D̂ )for the Alberta 3B and 4B Inventory area. Density is
expressed as bears per 1000km2
D̂
Estimator
Mh (Chao)
Mth2 πi (.), (θ1& θ2(.) X t4) +ld
4.79
5.33
SE
0.84
1.07
CIL
4.10
4.38
CIR
6.28
7.33
5. Discussion
The levels of precision of population estimates are comparable to the most precise
population estimates from grizzly bear projects in British Columbia and Alberta
(Table 10). Of the projects listed in Table 10, only 1 project has obtained a higher level
of precision. This was due to higher capture probability of bears in the 3B and 4B study
area. The capture probabilities of bears were the highest ever observed in a DNA markrecapture project suggesting that the sampling design was very efficient in capturing
bears. We speculate that a-priori remote sensing based site selection and the relatively
small grid cell size helped in obtaining this higher level of precision.
Table 10: DNA mark-recapture projects conducted in British Columbia
Project
Model
Jumbo
UCR 97
Mh (Chao)
Mh (Chao)
45
55
7.1
9.5
UCR 98
Mt (Chao)
92
29.8 32.4%
Kingcome
Mh (Chao)
102 20.7 20.3%
Alberta 3B4B
UCR 96
Granby Kettle
Parsnip
Flathead
Prophet
Mh (Chao)
Mh (Chao)
Mt (Chao)
Mth
Mh (Chao)
Mth
53
108
46
386
156
166
1
N̂
SE
8.3
23.8
16.4
48.4
47.5
26.2
CV
15.8%
17.3%
15.7%
22.0%
35.7%
12.5%
30.5%
15.8%
p̂
Cell Sites
Sessions
Reference1
2)
Size(km moved?
0.26
25
no
4
(Strom et al. 1999)
0.20
25
no
5
(Boulanger et al.
2004a)
0.12
25
no
5
(Boulanger et al.
2004a)
0.19
49
yes
5
(Boulanger and
Himmer 2000)
yes
4
0.33
49
0.16
64
yes
4
(Boulanger et al 2004)
0.13
64
yes
5
(Boulanger 2000)
0.18
64
yes
4
(Mowat et al. 2005)
0.10
64
yes
4
(Boulanger 2001)
0.17
81
yes
5
(Poole et al. 2001)
Further details on British Columbia projects can be found in Boulanger et al. (2002)
In terms of management of bears, the estimate of average number of bears on the grid at
any one time is most applicable since it directly pertains to the 3B and 4B management
areas. There was a substantial difference between estimates of the superpopulation of
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
18
bears on the sampling grid and surrounding area and the average number of bears on the
sampling grid (Table 8). This was most likely due to the concentration of bears on the
western side of the study area (Figure 3). Basically, many of the bears caught on this
edge of the grid partially live in Jasper Park as estimated from GPS bear movements
across grid boundaries. Superpopulation estimates will be influenced by the duration of
sampling and the degree of permeability of grid border areas. For this reason, these
estimates are less useful for management purposes.
Estimates of population size for the 3B and 4B area assume that the sampling grid
sampled all of the bears in the 3B and 4B area. Transect sites to the east only captured 1
bear so it is reasonable to assume that the population size of bears in this area is
negligible. One smaller area in the southwest corner of the grid was also not sampled.
We assumed this area also contains a negligible number of bears since much of it is rock
and ice habitat.
A GPS collar-based grizzly bear study was conducted on the northern part of the study
area from 1999-2004. One potential issue as a result of this study is aversion of bears to
bait sites given that they are similar to snare sites used for live capture. There are various
reasons to believe this issue did not significantly bias estimates. First, (Boulanger et al.
2004b) used data from a 1999 DNA inventory effort in the Foothills Model Forest (near
Hinton) to show that GPS collared bears exhibited capture probabilities that were similar
to bears without collars. Second, the main effect of collaring would potentially be
reduced capture probabilities of bears that were collared. In this case the heterogeneity
models used for population estimates are robust to unequal capture probabilities and
therefore a large degree of bias would not be expected. Third, four bears in the DNA
study area were monitored using GPS collars in 2004. Of these 3 were resident (all GPS
locations on the sampling grid) and 1 was a part time resident (31% of GPS locations on
the grid). Of these, 1 was captured at a DNA bait site. The superpopulation estimate of
bears for the sampling grid was 53 whereas only 39 were actually captured at DNA sites.
Therefore, it is plausible that the 3 GPS bears that were not captured could be included in
the estimated non-captured segment of the population. The only way bias would occur
would be if a significant proportion of bears exhibited zero probability of capture. While
capture for collaring may reduce capture probabilities there is no current evidence that it
makes it impossible to capture bears that have been previously GPS collared.
The density estimate from this project (4.9 bears/1000km2) is lower than most bear
projects conducted in British Columbia and Alberta (Boulanger et al. 2004c; Mowat et al.
2005). The density estimate from the 1999 DNA inventory in the Foothills Model Forest
study area was 14.9 bears/1000 km2 (SE=6.5, CI=9.9-27.1). This study area overlapped
the northwest corner of the 3B4B study area but also included Jasper National Park
(Figure 5). Plots of mean capture locations of bear identified in each inventory effort
show that the 1999 study sampled areas of relatively high bear density. For example, the
majority of bears captured in the 2004 inventory were captured in the area that
overlapped the 1999 FMF grid. In contrast, the 3B4B had large areas (e.g. the entire
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
19
eastern section) with few or no bears. This large difference in bear distribution relative to
grid placement, potential changes in density since 1999, and low precision of the 1999
estimate contributed to the difference in estimates from these two studies.
V
&
V
&
V
&
&
V
S
#
S
#
S
#
#&
S
V&
V#S
V&
&
V#S
S
#
V
S&
#
S
#
S
#
V
&
&
V
&&
VV
&
V &
V
V
&
S #
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V #S
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V
VS&
&
V
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V
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S#
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S
S &
#
V
S
S #
##
S
#
S
S#
#
S
S
S#
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FMF 1999 study area
S FMF 1999 bear mean capture location
#
V
&
S
#
V
&
S
#
S
#
&#S&
V
V#S #S
V
&
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3B4B 2004 mean capture locations
30
0
30 Kilometers
Figure 5: The Foothills 1999 study area compared to the 2004 Alberta 3B4B study area.
Population estimates from this study correspond to the entire grizzly bear population
including dependent offspring. Age cannot be identified using DNA methods and
therefore dependent offspring (cubs and yearlings) cannot be separated from adults in the
data set. Boulanger et al. (2004b) found that a small proportion of cubs from radio
collared female bears are also sampled using single barbed wire sites in British Columbia.
It is also highly likely that yearlings would also be sampled using DNA bait sites.
Analysis of reproductive data from GPS collared bears in the area suggest that the
proportion of cubs was lower in the population in 2004 so it is likely that they do not
constitute a large proportion of the estimate.
Other sampling procedures were tested in this project such as use of a 2nd bottom wire
and fixed sites. The results of the analysis of these procedures will be documented in
future reports.
The success of this grizzly bear population inventory project was in large part due to the
data sets that have been collected for this area through the Foothills Model Forest Grizzly
Bear Research Project over the past 6 years. Having remote sensing based habitat maps,
resource selection function models, and GPS bear movement data has proved invaluable
Integrated Ecological Research
July 29, 2005
Alberta 3B and 4B Grizzly Bear Inventory Project
20
in the design and analysis of this DNA population inventory. This highlights the need to
have these types of data sets in place before embarking on inventory efforts in other bear
management units in Alberta.
6. Literature cited
Boulanger, J. 2000. Granby-Kettle/Boundary Forest District 1997grizzly bear DNA
mark-recapture inventory project: Statistical analysis and population estimates.
Kamloops: Ministry of Environment, Lands, and Parks.
Boulanger, J. 2001. Analysis of the 1997 Elk Valley and Flathead valley DNA markrecapture grizzly bear inventory projects. Cranbrook, BC: Ministry of
Environment, Lands, and Parks.
Boulanger, J., and S. Himmer. 2000. Kingcome (1997) DNA mark-recapture grizzly bear
inventory project final report. Naniamo: Ministry of Environment, Lands, and
Parks.
Boulanger, J., and B. McLellan. 2001. Closure violation in DNA-based mark-recapture
estimation of grizzly bear populations. Canadian Journal of Zoology 79:642-651.
Boulanger, J., B. N. McLellan, J. G. Woods, M. F. Proctor, and C. Strobeck. 2004a.
Sampling design and bias in DNA-based capture-mark-recapture population and
density estimates of grizzly bears. Journal of Wildlife Management 68(3):457469.
Boulanger, J., G. Stenhouse, and R. Munro. 2004b. Sources of heterogeneity bias when
DNA mark-recapture sampling methods are applied to grizzly bear (Ursus arctos)
populations. Journal of Mammalogy 85:618-624.
Boulanger, J., G. C. White, B. N. McLellan, J. G. Woods, M. F. Proctor, and S. Himmer.
2002. A meta-analysis of grizzly bear DNA mark-recapture projects in British
Columbia. Ursus 13:137-152.
Boulanger, J., J. Woods, B. N. McLellan, M. F. Proctor, and C. Strobeck. 2004c.
Sampling design and capture probability bias in DNA based mark-recapture
estimates of grizzly bear populations. Journal of Wildlife Management 68:457469.
Burnham, K. P., and D. R. Anderson. 1998. Model selection and inference: A practical
information theoretic approach. New York: Springer. 353 p.
Huggins, R. M. 1991. Some practical aspects of a conditional likelihood approach to
capture experiments. Biometrics 47:725-732.
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Kendall, W. L. 1999. Robustness of closed capture-recapture methods to violations of the
closure assumption. Ecology 80(8):2517-2525.
Mowat, G., D. C. Heard, D. R. Seip, K. G. Poole, G. Stenhouse, and D. Paetkau. 2005.
Grizzly Ursus Arctos and black bear U. americanus densities in the interior
mountains of North America. Wildife Biology 11:31-48.
Nielsen, S. E. 2004. Habitat and grizzly bear density estimates for the 2004 DNA census
of grizzly bear management areas 3B & 4B, west-central Alberta, Canada.
Edmonton: University of Alberta.
Nielsen, S. E., M. S. Boyce, G. B. Stenhouse, and R.H.M.Munro. 2002. Modeling
grizzly bear habitats in the Yellowhead ecosystem of Alberta: Taking
autocorrelation seriously. Ursus 13:45-56.
Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson. 1978. Statistical inference
from capture data on closed animal populations. Wildl. Monographs 62:1-135.
Paetkau, D. 2003. Genetical error in DNA-based inventories: insight from reference data
and recent projects. Molecular Ecology 12:1375-1387.
Pledger, S. 2000. Unified maximum likelihood estimates for closed models using
mixtures. Biometrics 56:434-442.
Poole, K. G., G. Mowat, and D. A. Fear. 2001. DNA-based population estimate for
grizzly bears Ursus Arctos in northeastern British Columbia, Canada. Wildlife
Biology 7:105-115.
Seber, G. A. F. 1982. The Estimation of Animal Abundance. London: Charles Griffin and
Company. 654 p.
Strom, K., M. Proctor, and J. Boulanger. 1999. Grizzly bear population survey in the
Central Purcell Mountains, British Columbia. Calgary, AB; Nelson, BC: Axys
Consulting, University of Calgary, and Integrated Ecological Research.
White, G. C. 1996. NOREMARK: Population estimation from mark-resighting surveys.
Wildlife Society Bulletin 24:50-52.
White, G. C., D. R. Anderson, K. P. Burnham, and D. L. Otis. 1982. Capture-recapture
and removal methods for sampling closed populations. Los Alamos, NM: Los
Alamos National Labratory.
White, G. C., and K. P. Burnham. 1999. Program MARK: Survival estimation from
populations of marked animals. Bird Study Supplement 46:120-138.
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White, G. C., K. P. Burnham, and D. R. Anderson. 2002. Advanced features of program
MARK. In: Fields R, Warren RJ, Okarma H, Seivert PR, editors. Integrating
People and Wildlife for a Sustainable Future: Proceedings of the Second
International Wildlife Management Congress. Gödölló, Hungary. p 368-377.
White, G. C., and T. M. Shenk. 2001. Population estimation with radio marked animals.
In: Millspaugh JJ, Marzluff JM, editors. Design and Analysis of Radio Telemetry
Studies. San Diego, California: Academic Press.
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Zar, J. H. 1996. Biostatistical analysis. London: Prentice-Hall. 662 p.
7. Appendix 1 Details about simulation methods
7.1.1.
Expert opinion simulations
One of the most likely forms of capture probability variation in grizzly bear data sets is
heterogeneity variation due to females and cubs displaying reduced capture probabilities.
Unfortunately, age cannot be determined from DNA therefore making this a nonidentifiable form of heterogeneity. This form of heterogeneity is difficult to detect given
the lower power of model selection routines with sample sizes typical of bear populations
(Boulanger et al. 2002). The problem of low cub capture probabilities is especially
relevant to the current design of most DNA mark-recapture projects that use a single
strand of barbed wire that cubs may be able to go under without getting snagged. The
Alberta 3B and 4B project was unique in that it used a 2nd strand of barbed wire to
capture cubs. For this reason, cub capture probabilities were increased in simulation
compared to those conducted by Boulanger et al. (2004)
Boulanger et al. (2004) used data from known female bear with cubs monitored during
DNA sampling to estimate the probability of capture of cubs.
In addition, they
hypothesized capture probabilities of other age classes of bears based on their height
relative to hair snag wire. Parameters used for simulations are presented in Table 2.
Proportions of each age and sex class (Page-sex) are from McLellan (1989). Psnag , the
proportion of each cohort snagged relative to adult males, was hypothesized upon bear
height relative to barb wire and trap encounter rates for yearlings and females. Pcohort was
the capture probability of each age and sex cohort. Pcohort was the product of Psnag and
Pcohort for adult males. For cubs, Pcohort was also estimated from 7 cubs and their 4 radio
collared mothers that were tracked during the Upper Columbia River 1996 DNA markrecapture project (Boulanger et al. submitted). The mean capture probability for
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simulations ( p ) was estimated as
4
∑P
age − sex
23
pcohort where j is a given age and sex class.
j =1
Male capture probability (pcohort) was varied to meet the desired mean capture probability
level for a simulation.
Table 11: An example of heterogeneity simulation parameters. Proportions of age class
Page-sex), expected number of bears (E(N)), proportion of age class snagged (psnag), capture
probability (pcohort).
psnag
pcohort
Age
Sex
Page-sex
Cub
males and females
0.215
0.5
0.25
Yearlings males and females
0.175
0.75
0.32
Adult
females
0.305
0.75
0.32
Adult
males
0.305
1.00
0.43
One set of simulations was conducted with heterogeneity variation parameterized in
Table 11. Another set of simulations was conducted with time and heterogeneity
variation. The level of time variation was directly estimated from the Alberta 3B and 4B
study (Table 12) Time variation was standardized so that it would vary so that the mean
capture probability level from any simulation was constant. This was done by first
estimating capture probability for each sample session pˆ j as:
pˆ j =
nj
Nˆ
where nj is the number of bears captured for session j and N̂ is the population size
estimate. The average capture probability p was then estimated by the sum of individual
capture probabilities divided by the number of sessions. A capture probability
“multiplier” was then estimated by
mult j =
pˆ j
p
The capture probabilities for any cohort in Table 12 for a given session were therefore
equal to its individual capture probability times the multiplier for that session (Table 3).
Using this approach ensured that the mean population capture probability level did not
change even when time variation was simulated. This is the same approach used in the
simulation module in program CAPTURE (Otis et al. 1978).
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Table 12: Time parameterization for simulations based upon animals caught in Alberta
3B4B DNA mark-recapture project.
Statistic
j=1
2
3
4
22
36
18
42
Animals caught nj
pˆ j for period j
0.450
0.518
0.405
0.225
multiplier for p
7.2.
1.127
1.296
1.014
0.563
Data-based simulations
A set of simulations was conducted based on levels of heterogeneity and time variation
estimated directly from mark-recapture data. For this analysis, levels of heterogeneity
were estimated by mixture models. As described in Boulanger et al. (2002), Mh mixture
models use a mixture of ≥2 capture probabilities to model heterogeneity of a single
capture probability. This allows bimodal or multimodal distributions that may arise from
heterogeneity of capture probabilities to be modeled. For example, the overall capture
probability for an encounter history where a mixture of A distributions is used is
A
∑π θ
i =1
i
v
i
(1 − θ i )
t −v
, where v equals the number of captures of the animal for t occasions,
π i is the probability the animal has capture probability θ i , with the sum of the π i forced
to equal 1. Thus, for A = 2, π 2 = 1 − π 1 . Different values of capture probabilities were
simulated for each mixture distribution according to Table 5.
8. Appendix 2 : Background information on mark-recapture issues
Several fundamental mark-recapture concepts must be defined to ensure adequate
understanding of the concepts discussed in this report.
8.1.
Definition of a model and estimator
Mark-recapture estimation represents an improvement from traditional count-based
census methods. With traditional methods bears would be counted or trapped and the
number trapped would be the estimate of population size. Inherent in this is the
assumption that all animals have been trapped or counted, otherwise the estimate of
population size would be lower than the actual population size. In mark-recapture
estimation the percentage of animals captured is estimated. This percentage is called a
capture probability. This concept can be expressed by the following formula:
M
Nˆ =
pˆ
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In the above formula, M is the census of animals, p̂ is the estimate of capture probability,
and N̂ is the estimate of population size. With traditional census methods p̂ is assumed
to equal one.
An important term can be introduced here. A model is a set of
assumptions that correspond to an estimation method. In the case of a census, our model
is based on the assumption that all animals are caught. Capture probability p̂ is rarely
equal to one, and, as a result, many models have been formulated that make differing
assumptions on how p̂ varies. For any model there is a corresponding estimator. An
estimator is a set of mathematical formulae that allow an estimate using the assumptions
of the model. In the case of a count model, the estimate is simply the count of animals
caught. The subject of estimation using mark-recapture methods has seen much
theoretical attention, and, therefore, many estimators exist which are much more complex
than simple counts.
8.2.
Bias, precision and robustness
Estimates of density and population size are evaluated using two principle measures:
precision and bias. The best way to conceptualize precision and bias is to consider what a
range of estimates might look like if a project was repeated many times (Figure 10).
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Alberta 3B and 4B Grizzly Bear Inventory Project
Unbiased and precise
26
Unbiased but not precise
Goal of most markrecapture inventory
projects
Biased but precise
Biased and not precise
Figure 6: A conceptual diagram of bias and precision. Each target represents a possible set of
estimates (“shots”) from the mark-recapture experiment, if the study were repeated many times.
Lack of precision is mainly caused by low sample sizes, and bias is caused by improper model
selection. Unlike this target analogy, most mark-recapture experiments are only conducted once
(i.e. one “shot”) and the true bulls eye (true population size) is not known. Therefore, markrecapture data should be interpreted cautiously and statistically to avoid erroneous conclusions
(target figure from White et al., 1982).
Precision is the repeatability of estimates and is usually estimated by the coefficient of
variation and the width of confidence intervals. Bias is the deviation of estimates from
the true population value and is determined by how well the statistical model and
estimator fit the mark-recapture data. The goal of most mark-recapture experiments is to
minimize both bias and maximize precision therefore minimizing potential error in
estimates.
An ideal estimator of population size or density should be unbiased, precise, and robust.
Robustness is a measure of how well an estimator will perform even when its associated
assumptions about capture probability are violated. An example of a robust estimator
would be one that assumes equal capture probabilities but still gives unbiased estimates
when moderate capture probability variation exists in the data.
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8.3.
27
Key issues in optimal inventory design
Proper sampling design is critical to obtaining reliable population estimates. The
following is a list of the three main issues in the design of projects to estimate population
size (White et al. 1982).
1.
2.
3.
Meeting the assumption of geographic and demographic closure: If closed
population estimation models are used, then it is assumed that the population is
closed or “no animals leave, enter, die or are born during the sampling process.”
Violation of closure can cause substantial biases in estimates from most markrecapture models. If closure violation is occurring, mark-recapture population
estimates will pertain to the “superpopulation” of bears in the sampling grid and
surrounding area during the time that sampling was conducted. (Kendall 1999;
White 1996). For estimation of density, and comparison of different areas the
average number of bears on the sampling grid is most applicable. Radio collared
bears or the methods of Boulanger and McLellan (1999) can be used to account
for closure violation.
Sample size: Sample size is determined by the number of animals in the trapping
area, the capture probability of the population, and the number of times the
population is sampled. In general, higher population capture probabilities are
needed for smaller population sizes to obtain adequate estimates. The primary
effect of low sample size is reduced estimate precision. In addition, if sample size
is low, then not enough data will be available to determine dominant capture
probability variation in the data set leading to erroneous model selection.
Capture probability variation: Bears probably show unequal probabilities of
capture which can lead to biased population estimates. It is possible to test data to
determine the dominant type of capture probability variation, if the above issues
are met. Capture probability variation can be divided into three categories.
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a) Heterogeneity: Each animal has a unique probability of capture that is
constant throughout the study.
b) Behavior: All animals have an equal initial capture probability but this
changes after initial capture.
c) Time: The capture probability of bears changes evenly each time sampling
occurs.
Program CAPTURE and MARK have estimation models that are formulated to
accommodate each form of capture probability variation. The models (and
corresponding capture probability assumptions) are Mo (null or equal capture
(heterogeneity),
Mt(time),
Mb(behaviour),
Mth
probabilities),
Mh
(time/heterogeneity), Mbh(behaviour/heterogeneity), Mtb(time/behaviour) and
Mtbh(time/behaviour/heterogeneity).
If the goals of study design are met, the most appropriate estimation model can be used
for population estimates, which should yield the most precise and unbiased results.
Conversely, if an inventory is designed poorly, a complex, imprecise model may have to
be used (to minimize bias), or, if assumption violations are severe, no estimator will give
a reliable estimate (Otis et al. 1978).
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