ESTIMATING W O LF KILL RATES
933
Developm ent and applieation of a ratio
estim ator to estim ate w olf k ill rates
and variance in a m ultiple-prey system
Mark Hebblewhite, Paul C. Paquet, Daniel H. Pletscher, Robert B.
Lessard, and Carolyn J. Callaghan
Abstract
E stim ating n u m b e r o f p rey k ille d by ca rn ivo re s such as w o lv e s {Cam's lupus) per u n it tim e ,
o r k ill rate, is im p o rta n t fo r the c o n se rva tio n and m an a g e m e n t o f ca rn ivo re s and th e ir
prey. W e re vie w e d p u b lis h e d m ethods to estim ate w o lf k ill rates and fo u n d th e m in c o n ­
sistent and la c k in g a basis in statistical s a m p lin g theory. W e d e ve lo p e d a general statis­
tic a l e stim a to r fo r k ill rate and va ria n ce using ra tio -v a ria b le s a m p lin g theory. W e illu s ­
trate o u r ra tio e stim a to r by e stim a tin g w o lf k ill rates in a m u ltip le -p re y system in B anff
N a tio n a l Park, A lb e rta , fo r w in te rs fro m 1 986 to 2 0 0 0 . W e used s n o w tra c k in g and
ra d io te le m e try to locate 4 2 9 k ills d u rin g 195 s a m p lin g intervals c o v e rin g 1 ,2 9 4 days.
M e a n k ill rate by w o lf packs (expressed as the n u m b e r o f k ills /d a y /p a c k , k/d/p) was 0.33
k/d/p, m ost o f w h ic h , 0 .23 k/d/p, w e re e lk (Cervus elaphus), the m ost a b u n d a n t ungulate.
K ill-ra te estim ates w e re va ria b le despite intensive s a m p lin g effort. The m ean k ill rate o f
0 .33 k/d /p had a p o o le d 9 5 % c o n fid e n c e interval o f 0 .2 9 to 0 .3 7 . A n intensive s a m p lin g
e ffo rt o f > 6 - 8 in d iv id u a l s a m p lin g intervals c o v e rin g a p p ro x im a te ly 2 5 % o f the w in te r
was re q u ire d to m in im iz e s a m p lin g v a ria tio n . W e co m p a re d o u r m e th o d to 3 o th e r p u b ­
lished m ethods fo r e stim a tin g k ill rates and e xa m in e d the bias and p re cisio n o f k ill-ra te
m ethods using s im u la tio n s . O u r ratio e stim a to r a p p ro a ch was the least biased and m ost
precise w h e n c o m p a re d to o th e r approaches. Ratio estim ators p ro v id e a sta n d a rd ized
m e th o d to estim ate k ill rates in o th e r p re d a to r-p re y systems and w ill fa c ilita te c o m p a ri­
son across studies and e x a m in a tio n o f patterns o f k ill-ra te v a ria tio n .
Key w o r d s
B anff N a tio n a l F^rk, Canis lupus, Cervus elaphus, elk, k ill rate, k ill-ra te va ria n ce , m u lti­
ple prey, p re d a to r-p re y , ratio estim ator, w o lf
The recolonization of gray wolves (Canis lupus)
throngh dispersal (Boyd and Pletscher 1999) and
their reintrodnction (Fritts et al. 1997) across western N orth America are restoring the wolf to ecosystems w ith mnltiple prey species w here the dominant nngnlate often is elk (C ervus elaphus).
Knowledge of the im pact of predation by wolves
on elk and other nngnlates w onld assist wildlife
managers in ensnring snstainable nngnlate harvests
after w olf recolonization (Boyce 1992). The impact
of wolves on nngnlates also is im portant in determining w hether wolves are keystone species (Estes
1996, Terborgh et al. 1999). Wolf predation can
limit popnlations of moose (Alces alces), caribon
(R a n g ifer tarandus), and w hite-tailed deer
(Odocoileus virginanus)( M.essiet 1991,
1992,
A d d r e s s fo r M a r k H e b b l e w h i t e a n d D a n ie l H. Ple ts c her: W il d li fe B io lo gy P r o g ra m , S c h o o l o f Forestry, U n iv e rs it y of M o n t a n a , M is ­
s o u la , MT, 5 9 8 0 1 , USA; p r e s e n t a d d r e s s fo r H e b b l e w h i t e : D e p a r t m e n t of B io lo gic al S c i e n c e s , U n iv e rs it y o f A lb e rta , E d m o n to n ,
AB, T 5 C 2E9 C a n a d a ; e m a il: m a r k . h e b b l e w h l t e @ u a l b e r t a . c a . A d d r e s s fo r Paul C. Paque t: Faculty of E n v i r o n m e n ta l D e sig n , U n i ­
ve rs it y of Cal ga ry , Cal ga ry , AB, T 2 N 1 N 4, C a n a d a . A d d r e s s fo r R o b e r t B. Ees sard: D e p a r t m e n t o f R e n e w a b l e R e s o u rc e s , U n iv e rs i­
ty of A lb e rta , E d m o n to n , AB, T 5 C 2E9, C a n a d a . A d d r e s s fo r C a r o ly n C a l l a g h a n : D e p a r t m e n t o f Zo o lo g y , U n iv e rs it y o f G u e l p h ,
G uelph, O N , N I G 2W 1, C anada.
W ildlife S ociety Bulletin 2 0 0 3 , 31 (4 ):9 3 3 -9 4 6
Peer refereed
934
Wildlife Society Bulletin 2003, 3 1 ( 4 ):9 3 3 - 9 4 6
Bull elk killed b y w o l v e s — l o c a t e d t h r o u g h s n o w - b a c k t r a c k i n g .
Gasaway et al. 1992, Messier 1994, Boertje et al.
1996). However, w olf-elk systems w ith multiple
prey species have received less research attention
than single-prey systems. Despite strong selection
of elk by wolves in such systems (Huggard 1993«,
Weaver 1994), predicting the impact of wolves on
elk will be difficult w ithout quantitative analyses of
w olf-elk dynamics. The consequences of wolf pref­
erence for elk to population dynamics will depend
on how the w olf functional (the kill rate, or num ber
of prey killed p er predator p er unit time) and
num eric responses (num ber of wolves) change
w ith elk density (Holling 1959). Understanding the
dynamics of w olf kill rates in multiple-prey systems
could provide a theoretical basis for management
similar to wo If-m oose systems (Messier 1994,
Orians et al. 1997).
Kill rate has a long history as a metric (Cowan
1947, Schaller 1972). Kill rates have been variously
estimated using ground observations (Murie 1944),
snow tracking (Cowan 1947, Huggard 1993b), daily
aerial tracking (Peterson 1977), infrequent aerial
observations (Fuller and Keith 1980, Fuller 1989),
and a com bination of the various m ethods.
Methods differ among studies in defining sampling
intervals for estimating kill rate (Huggard 1993b,
Dale et al. 1995, Ballard et al. 1997), and biases have
not been addressed for all m ethods (sensu Fuller
1989). Furtherm ore, estimates of precision associ­
ated w ith kill rates have only recently begun to be
reported, yet statistical m ethods differ across stud­
ies (Hayes et al. 2000, Jedrzejewski et al. 2000).
Therefore, despite long-standing use, estimation of
kill rates has developed w ithout formal statistical
treatment.
We first reviewed m ethods used to estimate kill
rates and then developed a kill-rate estimator based
on ratio-variable sampling theory w ith variance
estim ators to accom m odate different sampling
designs. We consider the scope of inference for kill
rate-estimation to be an entire w inter because kill
rates typically are applied for an entire w inter (e.g..
Messier 1994, Hebblewhite et al. 2002). We illus­
trate our ratio estim ator by estimating w inter kill
rates and kill-rate variance of recolonizing wolves
for 23 wolf pack-years in a multiple-prey system in
Banff National Park (BNP), Alberta, from 1986 to
2000. We examine the statistical perform ance (bias,
precision) of our kill-rate estim ator com pared to 2
other published m ethods using simulations. We dis­
cuss implications of our kill rates to w oh-prey
dynamics in BNP and potential applications of esti­
mation of kill-rate variance in w o h -p rey systems.
Study area
Banff National Park (BNP), Alberta, is 6,641 km^ in
area and is located in the Canadian Rocky Mountains
on the eastern slope of the Continental Divide. The
climate was characterized by short, dry summers
and long, cold winters w ith infrequent w arm weath­
er caused by Chinook winds. Topography was
extrem e in the Canadian Rockies (elevation
1,400-3,400 m), and approximately half of BNP was
rock and ice imusable to wolves and their prey
(Holroyd and VanTighem 1983). Average maximum
snow depth varied from 50 cm at the tow n of Banff
to 75 cm in Lake Louise and was higher in side val­
leys (Holland and Coen 1983). Prey populations in
the study area were diverse, including the primary
prey species, elk (Huggard 1993«), and the second­
ary prey species, mule deer iOdocoileus hem ionus),
white-tailed deer, moose, and bighorn sheep (.Ovis
canadensis). The primary study area within BNP
Estimating w olf kill rates • H e b b le w h ite et al.
was defined by wolf-pack territories and was -3000
km^ (see Hebblewhite et al. 2002).
M ethods
Estimating kill rate
A variety of methods have been nsed to estimate
kill rates in the literature (Table 1). Aerial telem etry
is nsed to estimate kill rate as a function of num ber
of days wolves are relocated from the air on a kill
(Mech 1977, Fuller and Keith 1980). Differences in
prey handling times affects the probability of locat­
ing wolves on a kill, biasing kill rates. Fuller and
Keith (1980) and Fuller (1989) developed methods
for correcting for this bias. The more com m on
approach (Table l),th e m ethod w e nsed, combines
ground and aerial tracking and radioteiemetry to
estimate kill rate in continnons tracking periods
(Huggard 1993a, Dale et al. 1995, Murphy 1998,
Hayes et al. 2000). Fuller (1989) contended that
ground m ethods w ere more accurate; yet potential
biases in ground m ethods remain untested.
For example, m ethods of defining the start and
end of a continnons ground-monitoring or tracking
T a ble 1.
935
period, called the predation period by Hayes et al.
(2000), vary among studies (Table 1). Herein, we
call the predation period the sampling interval for
consistency w ith statistical treatm ents (Thompson
1992). Researchers assumed that because the
length of time betw een kills (kill interval) before
and after the sampling interval is unknown, includ­
ing days before and after first and last kill in a sam­
pling interval w onld bias kill rates. To minimize this
presum ed bias, Ballard et al. (1997) removed the
first day sampled in a sampling interval, Hayes et al.
(2000) ended a sampling interval if the w olf pack
had not been seen for >3 days, and Dale et al.
(1995) truncated a sampling interval to the day
after the first kill and ending the day of the last
observed kill. Murphy (1998) and Jedrzejewski et
al. (2000) adopted the Dale et al. (1995) m ethod to
reduce this assumed bias, yet earlier researchers did
not consider this potential bias and did not trun­
cate sam pling intervals (e.g., Peterson 1977,
Huggard 1993b,Table 1), making direct comparison
among studies difficult. To onr knowledge, no
quantitative assessment of this potential bias has
been conducted.
R e v i e w o f c o m m o n kill- rate e s t i m a t i o n m e t h o d s in w o i f - p r e y s y s t e m s a n d s e l e c t e d o t h e r p r e d a t o r - p r e y s y ste m s .
S tu dy
Prey s p e c i e s
M ethod
N o t e s a n d m e a s u r e of kill rate
M e c h 1 9 7 7 , Fritts a n d
M e c h 1981
W h ite-tailed d e er
Aeria l e s t i m a t e
C a l c u l a t e d as n u m b e r of r e l o c a t i o n s p e r kill.
Fulle r a n d Keith 1 9 8 0
M oose
Aeria l e s t i m a t e
C o r r e c t e d for ti m e s p e n t a t c a r c a s s a n d
m o n i t o r i n g inte rval, kil ls /re lo c a tio n flight.
P e te rs o n 1 9 7 7
M oose
Aeria l e s t i m a t e
C o r r e c t e d u s in g Fuller a n d Keith's (1 98 0)
c o r r e c t i o n factor, r e p o r te d a s days/kill.
Fulle r 1 9 8 9
W h ite-tailed d e er
Aeria l e s t i m a t e
C o r r e c t e d for n u m b e r of d a y s s p e n t o n kill,
r e p o r te d a s D a ys/kill/w olf.
C a r b y n 1 9 83
Elk
A e r i a l / g r o u n d t r a c k in g
K il ls /d ay/p ac k o r w o lf .
M e s s i e r a n d C re te 1 9 8 5
M oose
Aeria l tr a c k in g
C o n tin u o u s sessions defined as relocations
s e p a r a t e d < 5 2 h o u rs , Kiiis/day/woif.
T h u r b e r a n d Peterson 1993
M oose
Aeria l tr a c k in g
Kill rate d e t e r m i n e d fr o m kill inte rval, days/kill.
D a le e t a i . 1 9 9 4 , 1 9 9 5
C aribou, m oose
Aeria l tr a c k in g
3 0 - d a y t r a c k in g p e rio d s , kill rate e s t i m a t e d d a y
after first kill to d a y o f last kill, kiiis/woif/day.
B al lard e t al. 1 9 9 7
M oose, c aribou
Aeria l tr a c k in g
C a l c u l a t e d kii is /d a y /w o if after r e m o v i n g first
s a m p l i n g day.
H a y e s e t al. 2 0 0 0
M oose
Aeria l tr a c k in g
interval e n d e d after 3 d a y s o f lost c o n t a c t .
M o n i t o r i n g in te n s it y d e p e n d e n t o n p r e y availability.
Huggard 19 93a
Elk
S n o w tr a c k in g , scats®
D a ys s i n c e last kill, i n c l u d e d p r o b a b l e m is s e d kills.
Kunkel 1 9 9 7
Elk
S n o w tra c k in g
R e p o r te d a s w o l f kills / k il o m e t e r tr a c k in g .
J e d r z e j e w s k i e t al. 2 0 0 0
Red de er , w i l d b o a r
S n o w tra c kin g,
te le m e t ry , scats®
C o n s e c u t i v e kills d e f i n e d a s < 5 d a y s apar t,
e s t i m a t e d ti m e s i n c e last kill aft er 1®* kill
e x c l u d e d , kiiis/da y/woif, v a r i a n c e e s t i m a t e d .
A e r ia l e s ti m a t o r s
A e r ia l tr a c k in g
C r o u n d tr a c k in g
® U s e d s c a ts to id e nti fy m is s e d kills d u r i n g p e r i o d s w h e r e c o n t i n u o u s m o n i t o r i n g w a s lost d u e to w e a t h e r , w o l f m o v e m e n t s , etc.
936
Wildlife Society Bulletin 2003, 31(4):933-946
Furthermore, few studies determ ined variance in
kill rate associated w ith kill-rate estimates, and
th o se th at did used different approaches
(Jedrzejewski et al. 2000, Hayes et al. 2000). O ther
researchers have noted lack of rigor in kill-rate esti­
mates and have developed com plem entary m eth­
ods to rigorously estimate kill rates and variance in
kill rate w here continuous tracking is not possible
(Smith et al. 2003). Below, w e develop a statistical
estimator that addresses some of these deficiencies
in com m on m ethods to provide 1) a framework for
estimating any type of kill rate (e.g. num ber of kills
or kilograms [kg] of prey killed p er day p er pack or
p er woli), and 2) a basis for estimating variance in
kill rate in systems w here continuous tracking is
possible under several different sampling designs.
Application o f the ratio estimator to kill
rates
Consider a typical sequence of events for a pred­
ator. Predators search for suitable prey and, given an
encounter, have a probability of making a kill. Once
a kill has been made, predators spend time consum­
ing or handling the prey. Predators’time consists of
bouts of time spent alternately searching for and
handling prey. We sample this repeating sequence
of events for a w olf pack and estimate kill rate as the
num ber of kills (Pj) located p er unit time (x^) per
wolf pack (yfX p kills/day/pack, or k/d/p). Kill rate
is, by definition, a ratio variable consisting of 2 sepa­
rate variables, num ber of kills made by wolves and
num ber of days it took wolves to make kills. Thus,
kill rate must be treated differently than if it w ere a
single parameter, especially w ith respect to vari­
ance, because both the num erator and denominator
may vary independently betw een samples (Cochran
1977, Thom pson 1992). This fimdamental property
of kill rates has not been appreciated in the litera­
ture (Table 1), and can lead to biased estimates of
kill rate and its variance (Thompson 1992).
Consider a sampling design in w hich wolves are
continuously m onitored during sampling intervals
and all kills are located within each sampling inter­
val, w hich w e consider the sampling unit. Sampled
intervals are interspersed w ith unsam pled inter­
vals, and by assuming that these intervals are
approximately randomly distributed, kill rate can be
estimated using a ratio variable (Thom pson 1992).
N um ber of days in each sampling interval in most
study designs (unless strictly controlled) will itself
be a random variable, corresponding to a modelbased design (Thom pson 1992). Assuming that
wolves make zero kills in zero days, the relationship
betw een num ber of kills and num ber of days can
be described by the zero-intercept linear m odel y^
= ^Xp w here j^—the num ber of kills in sampling
interval i (or kg of prey killed in sampling period i
for kg/d/w ), x^ = the num ber of days in sampling
interval / and P=kill rate (Thompson 1992). Given
this linear model, kill rate, p, is then estimated by
P =
i=^ Vi
Z ni=V
(equation 1,Thom pson 1992:73), w here i= th e sam­
pling period, 1 to n, and n = total num ber of sam­
pling periods. To estimate total num ber of kills (T)
made during the total population of days in a win­
ter period (X), use F = P x. This estim ator has been
used correctly in the past by others (e.g.. Messier
and Crete 1985,Thurber and Peterson 1993); how ­
ever, variance has been ignored or calculated incor­
rectly. For example, Jedrzejewski et al. (2000) and
Hayes et al. (2000) calculated variance assuming all
sampling intervals w ere of equal length, equivalent
to an unw eighted variance estimate, w hen sam­
pling intervals w ere not equal.
Ratio-variable theory provides means to estimate
kill-rate variance correctly. To calculate variance for
kill rate, w e could use standard formulas from
Cochran (1977) and Thom pson (1992);
( 1)
var(p) = | 1 - —
X
n {n -V )
w here x = to tal num ber of days sampled, X is total
num ber of days in the population, and
n
X =
i~\
or m ean num ber of days p er sampled interval.
However, this variance estimator applies only to
sampled intervals, and assumes the num ber of kills
p er sampling interval are not random (i.e., are
fixed) and only random sampling of intervals
selects different intervals. However, if w e instead
consider that num ber of kills (Pj) p er sampling
interval is itself a random variable, more likely in
field settings, a different approach to estimating
variance is required (Thom pson 1992).
Estimating w olf kill rates • H e b b le w h ite et al.
Researchers often are interested in total nnm ber
of kills made in a season (or year) for ecological
(Thnrber and Peterson 1993) or predator-prey
stndies (Messier 1994). Typically researchers nse
sampled intervals to predict kill rate, or the nnm ber
of kills, across the w hole w inter based on sampled
intervals. These applications can be accom m odated
nsing Thom pson’s (1992) model-based design for
ratio-estimators if w e consider the entire w inter as
the popnlation for estimation of variance. To esti­
mate variance in this setting, w e nse information in
sampled and nnsam pled intervals to estimate killrate variance across an entire w inter in a predictive
sense (Thom pson 1992). From Thom pson (1992),
an nnbiased estimate of variance in kill rate across
the entire winter, w hich incorporates a finite popn­
lation size adjnstm ent based on the proportion of
total periods sampled, is
v ar(P )=
N iN -n }
X ^n
"i=l
(2 )
w here A^= total nnm ber of sampling intervals (sam­
pled and nnsam pled) in the popnlation (typically
dnring the time period of interest, e.g., winter), X=
total nnm ber of days in the stndy x= X / N or mean
nnm ber of days in the popnlation of sampling
intervals (sampled and imsampled), Xg> = mean
length in days of imsampled periods, and ;Cj=mean
length in days of sampled periods. To estimate vari­
ance in Y, mnltiply Eqnation 2 by nnm ber of days in
a w inter period (X). Dividing kill rate by pack size
and variance by the delta m ethod (i.e., by pack
size^) derives per-capita kill rates (i.e.,
kill/day/woft).
In applications w here the main objective is to
estimate kill-rate variance from sampled intervals or
w here interval length was fixed a priori by the
investigator, w e recom m end nsing Eqnation 1 to
estimate variance, eqnivalent to a design-based
approach (Thom pson 1992). However, w e feel that
the typical objectives of kill-rate stndies are to make
inferences to an entire w inter or season (e.g.
M essier 1994, T hnrber and P eterson 1993).
Therefore, w e recom m end Eqnation 2, the modelbased approach for estimating kill rate and vari­
ance.
Assnmptions of the ratio m ethod are 1) all kills
are located within each sampling interval, 2) sam­
pling intervals are sampled at random, 3) all from
seqnential sampling intervals are nncorrelated, and
937
for the model-based approach of Eqnation 2,4) that
the variance in y^ is proportional to x^ (Cochran
1977,Thom pson 1992). Smith et al. (2003) discnss
statistical m ethods to accom m odate violations of
the first assnmption. We made efforts to ensnre that
intervals w ere sampled randomly in onr stndy.
While onr ability to m onitor wolves was partially
dependent on w eather conditions for aerial telem e­
try and for gronnd tracking, wolves seemed to hnnt
in all w eather conditions. Therefore we felt that the
assnmption of random sampling was satisfied in
onr stndy. Fnrther research, perhaps w ith GPS col­
lars and detailed climatic data, is needed to test the
assnmption that w olf movem ent is random w ith
respect to weather, w hich wonld inflnence sam­
pling. We tested the third assnmption, nncorrelated
seqnential y^, nsing Pearson’s correlations betw een
seqnential y^ w ithin each pack-year for all intervals
collected dnring the stndy. Finally, for the modelbased approach, as length of sampling interval
increases, variance in y^ may asymptote. However,
this will co-vary w ith pack size, as variance in nnm­
ber of kills made by small packs did not appear to
asymptote w ith increasing sample interval length
(nnpnblished data).
Estimating kill rates o f wolves in BNP
We illnstrate the application of the ratio estima­
tor nsing data collected on wolves in a mnltipleprey system in BNP. Wolves w ere captnred nsing
modified foot-hold traps (Livestock Protection Co.,
Alpine, Tex.) w ith trap transm itters (Advanced
Telemetry Systems, Isanti, Minn.) in the snmmer, or
by aerial darting or net-gnnning from rotary-wing
aircraft dnring winter. Wolves w ere chemically
immobilized nsing Ketamine-Xylazine, Telazol, or a
Telazol-Xylazine m ixtnre nnder veterinary direction
and ontfitted w ith a VHP radiocollar (LOTEK wire­
less, Newmarket, Ont.). Parks Canada approved
captnre and handling protocol. We m onitored
wolves in 5 different packs dnring the w inter (01
Novem ber-30 April, 181 days, 182 in leap years)
betw een 1986-1987 and 1999-2000; the Spray,
Castle, Bow Valley, Cascade, and Fairholme w olf
packs (see Hebblewhite et al. 2002). Wolf-pack ter­
ritories ranged from approximately 500-2,000 km^
(95% m inim nm convex polygon, H ebblew hite
2000). We m onitored w olf packs nsing gronnd and
aerial radioteiemetry and snow tracking to locate
kills and maintain continnons records of w olf activ­
ity for as long as possible. We nsed mean traveling
pack size observed on aerial telem etry flights
938
Wildlife Society Bulletin 2003, 3 1 ( 4 ):9 3 3 - 9 4 6
(average num ber of wolves traveling and feeding
together in a winter, Messier 1985) to calculate percapita kill rates.
We partitioned total kill rates into specific kill
rates for elk, mule deer, white-tailed deer, moose,
and bighorn sheep nsing the nnm ber (or kg) of a
particular prey species killed by wolves p er sam­
pling interval. Converting nnm ber of prey to kilo­
grams of prey killed p er day p er w olf allows com­
parison among stndies w ith different prey species
and woh-pack sizes. We did not include prey scav­
enged by wolves in kill rates, although this m ethod
can be applied to estimate scavenging rates (e.g.,
Huggard 1993b). To estimate mass in kilograms
(kg) of prey consumed, w e adjusted p ercent of the
carcass edible (assum ed 75% for all prey species
following Peterson 1977) by wolves in each sam­
pling interval nsing m ean mass for each species,
age, and sex class dnring w in ter in BNP
(Hebblewhite 2000). Kill rates w ere corrected for
p ercent of carcass consum ed (Huggard 1993b) by
adjusting by the average percent of carcasses con­
sum ed in BNP (75%, Hebblewhite 2000) to more
accurately reflect tru e consum ption rate.
However, consum ption rates w ere not adjusted for
losses to scavengers and thus only approxim ate
actual consum ption rates (Creel 1997). To exam­
ine factors affecting kill rates, w e nsed kg of prey
available to calculate kg prey consnm ed/day/w olf
to control for variance in pack size and prey
species com position. We tested w hether kill rates
differed by prey species across all years nsing
ANOVA (Sokal and R ohlfl995).
Evaluation o f kill-rate estimators
We com pared onr ratio estimator to that of Dale
et al. (1995) and Ballard et al. (1997) (herein called
the Dale and Ballard methods, respectively) by com­
paring m ethods w ith onr BNP data set and nsing
statistical simulations. We applied these 3 estima­
tors to onr BNP data set and com pared the Dale and
Ballard m ethods to onr ratio estimator nsing paired
f-tests. We also tested for biases w ith varying sam­
pling interval lengths. The length of time spent
tracking wolves in a sampling interval should not
inflnence interval kill rate, only sampling variance
(Thom pson 1992). We com pared m ethods for this
bias w ith onr BNP data by testing w hether tracking
interval length (Xj) affected interval kill rate ( y / x ^
nsing linear regression. We also com pared methods
for calculating variance in kill rate nsing design and
model-based approaches of Eqnation 1 and 2 to
that of treating sampling intervals as equal in
weight. Finally, w e investigated how nnm ber of
individual sampling intervals and total percent of
w inter tracked affected kill-rate variance (SE in
k /d /p ) nsing nonlinear regression to provide guid­
ance for future stndies.
Next, w e evaluated the statistical perform ance
(bias, precision) of these 3 estimators (ratio. Dale,
Ballard) nsing randomization simulation modeling
(Manly 1997). W ithin individual sampling intervals,
kill rates w ere normally distributed (one sample
Kolmogorov-Smirnoff test, LilUefors P=0.09, w = 195
intervals). Therefore, w e generated 100 sequences
of kills 181 days in length (i.e., one w inter) at ran­
dom from a normal distribution described by mean
kill rate and standard deviation (SD) estimated from
this stndy (see results. Table 2). We then sampled
these 100 random sequences of kills at random
nsing sampled (A^) and nnsampled (A^-) intervals of
m ean length and SD equal to period lengths in onr
stndy (Table 2). We also investigated the effect of
sampling intensity (both % w inter tracked and
tracking interval length) in simulations. We set min­
imnm sampling intensity to 25% of the w inter
because estimates w ere unstable below this sam­
pling intensity (see results). This generated a series
of kill sequences based on a know n kill rate and a
set of sampled intervals based on sampling intensi­
ty estimates in onr stndy w ith w hich to compare all
3 estimators. We nsed the 3 m ethods to estimate
kill rates from these random sampling sequences.
We com pared precision and bias of simulated kill
rates to know n kill rate nsed to generate the
sequences. Finally, w e evaluated these 3 methods
over a range of kill rates (from 0.05-0.4 kills/d/w)
to ensnre that onr simulations w ere robust across a
range of kill rates possible in other stndies. We also
estimated kill-rate variance nsing onr model-based
ratio-variance estimator (Eq. 2) for all 3 methods to
explore how variance in kill rate is affected by the
3 methods.
We conducted statistical analyses nsing SYSTAT
8.0 (Wilkinson 1998). For ANOVA we assessed dif­
ferences betw een classes nsing post-hoc Bonferroni
mnltiple comparisons procedures that controlled
for experiment-wise error rate. We assessed nor­
mality w ith norm al p-p plots, and variance
homoskedasticity w ith Levene’s T-test in ANOVA
and residual plots in regression analyses. We pro­
grammed statistical simulations in R-t (Departm ent
of Statistics, University of Auckland, software avail­
able at http://w w w .r-project.org).
Estimating w olf kill rates • H e b b le w h ite et al.
939
T a ble 2. S n o w - t r a c k i n g d a t a u s e d to e s ti m a te w i n t e r kill ra te s o f w o l v e s in Banff N a t i o n a l Park, C a n a d a , fr o m 1 9 8 6 - 2 0 0 0 . For e a c h
w o l f p a c k - y e a r, t h e n u m b e r o f s a m p l i n g in te rv a ls (N), m e a n tr a c k i n g p e r i o d in d a y s (Xj), n u m b e r of d a y s t r a c k e d (n), tota l n u m b e r
o f d a y s (X), % o f t h e w i n t e r p e r i o d t r a c k e d , n u m b e r o f kills f o u n d (y,), a n d m e a n tr a v e li n g w o l f p a c k s iz e a r e r e p o r t e d . Total w o l f
kill rates in k ills /d a y /p a c k (k/d/p) a n d kg p r e y k i i i e d /d a y /w o if (kg /d/w) w e r e e s t i m a t e d w i t h o u r m o d e l - b a s e d , ratio e s t i m a t o r o f w i t h ­
i n - w i n t e r v a r i a n c e in kill rate. Kill ra te s c a l c u l a t e d aft er D a le e t ai. (1 9 9 5 ) a n d Ballard e t al. (1 9 9 7 ) m e t h o d s a n d t h e u n w e i g h t e d
s t a n d a r d e rro r s (SE) a r e p r e s e n t e d fo r c o m p a r i s o n .
# of
sam pling
W olf
in te rv als
pack -ye ar®
(N)
SP 8 6 / 8 7
1
M ean
len gth
in d ays
# D a ys
tracked
bs)
in)
8.0
8
% w inter
# of
t r a c k e d kills (y,)
4.4
4
Ratioestim ator
W olf
pack
siz e
kill
k/d/p
rates
kg /d /w b
Ballard
e t ai.
(1 9 9 7 )
k/d/p
4.0
0.40
7 .9 6
0.40
D a le
e t ai.
(1 995)
(k/d/p)
0.413
SE o f (k/d/p)
D e sig n - M o d e l Un­
based
based
w e ig hte d ® ratiob
ratio®
____ f
____ f
____ f
SP 8 7 / 8 8
7
7.3
51
28.0
24
5.8
0.47
9.15
0.4 3
0.455
0.052
0.017
0.029
SP 8 8 / 8 9
10
5.2
52
28.7
14
5.0
0.28
4.15
0.32
0.333
0.0 41
0.007
0.014
SP 8 9 / 9 0
7
1 3 .6
95
52.5
38
4.6
0.40
7.52
0.40
0.423
0.013
0.009
0.007
SP 90/91
6
7 .7
46
25.4
22
6.2
0.48
7 .1 4
0 .5 5
0.577
0.080
0.019
0.036
SP 9 1 /9 2
12
6.3
75
41.2
22
6.0
0.29
5.08
0.2 7
0.282
0.078
0.008
0 .0 1 2
CT 90/91
7
6 .6
46
25.4
22
5.6
0.48
3.73
0.48
0.500
0.0 51
0.014
0 .0 2 3
BVP 9 3 / 9 4
5
6.0
30
16.6
11
5.3
0.37
6.09
0.19
0.200
0.0 31
0.007
0.014
BVP 9 4 / 9 5
11
7.4
81
44.8
19
8.4
0 .2 3
2.10
0.32
0.333
0.035
0.008
0.010
BVP 9 5 / 9 6
14
6 .6
93
51.1
24
5.3
0.26
4.68
0.29
0.308
0.026
0 .0 1 2
0.010
BVP 9 6 / 9 7
15
5.5
83
45.9
20
5.9
0.24
3.21
0.47
0.438
0.022
0.006
0.007
BVP 9 7 / 9 8
7
9 .4
66
36.5
5
2.8
0.08
2.59
0.10
0.100
0.0 31
0.008
0.011
BVP 9 8 / 9 9
12
5
60
33.1
16
2.3
0.2 7
8.11
0.44
0.462
0.035
0.008
0.010
BVP 9 9 / 0 0
8
13.0
104
57.1
12
2.1
0.11
4.89
0.16
0.166
0.013
0.005
0.005
CA 9 1 /9 2
4
5.2
26
14.4
9
4.0
0.35
9.85
0.36
0.385
0.083
0.009
0 .0 3 2
CA 9 3 / 9 4
7
3.7
26
14.4
8
4.0
0.31
8.93
0.22
0.235
0.052
0.005
0.018
CA 9 4 / 9 5
8
5.3
42
23.2
13
6.0
0.31
6.08
0 .2 5
0.2 73
0.050
0.008
0.019
CA 9 5 / 9 6
14
7.4
103
56.6
42
8.9
0.41
4.34
0.48
0.508
0.019
0 .0 1 3
0.007
CA 9 6 / 9 7
9
8.2
74
13.3
35
13.1
0.47
3.58
0.56
0.596
0.053
0 .0 2 2
0 .0 2 3
CA 9 7 / 9 8
10
6 .8
68
37.6
24
15.2
0.35
2.38
0.39
0.406
0.025
0.009
0.007
CA 9 8 / 9 9
9
4.9
44
24.3
14
12.3
0.30
2.30
0.32
0.333
0.037
0.008
0.015
CA 9 9 / 0 0
4
6.3
25
13.7
7
6.3
0.28
2 .2 2
0.31
0.333
0.033
0.008
0.020
PR 9 9 / 0 0
8
5 .8
46
25.3
12
2 .1 g
0.26
11.35
0.40
0.417
0.039
0.007
0 .0 1 3
X
8. 5
7 .0
68.8
31.0
23.7
6.1
0.328
0.346
0.363
0.0 41
0.010
0.016
—
41 7®
—
0 . 119 b
0 . 07 b
—
5.43 1
Total® or
p o o l e d SE^
195®
—
1,294®
0.08b
0.36b
—
® A b b r e v i a t i o n s a r e CA - C a s c a d e p a c k , SP- S pray p a c k , CT - C as tl e p a c k , PR- F a ir h o lm e, BVP- B ow V a ll e y pa ck,
^ K g/d a y/w o if c o r r e c t e d fo r 7 5 % e d i b l e fr o m P e te rs on (1 9 7 7 ) a n d a n a v e r a g e of 7 5 % a c t u a l l y c o n s u m e d fr o m th is study.
U n w e i g h t e d s t a n d a r d e r r o r c a l c u l a t e d fr o m u n w e i g h t e d m e a n kill rate fo l l o w i n g J e d z r e je w s k i et ai. (2 0 0 0 ) w i t h a finite p o p u ­
latio n s iz e a d j u s t m e n t a c c o u n t i n g for % w i n t e r tr a c k e d ( T h o m p s o n 1 992) .
^ D e s i g n - b a s e d r a tio - e s tim a to r s t a n d a r d e rr o r fo r k/d /p o v e r t h e e n tire w i n t e r u s in g e q u a t i o n 1.
® M o d e l - b a s e d r a tio - e s tim a to r s t a n d a r d e r r o r fo r k /d /p o v e r t h e e n ti re w i n t e r u s in g e q u a t i o n 2.
f
O n l y 1 interval c o l l e c t e d fo r S p r a y 1 9 8 6 / 8 7 p a c k , h e n c e n o v a r i a n c e e s ti m a te ,
g Pack s iz e e s t i m a t e d fr om s n o w tr a c k in g , n o w o l v e s w e r e ra d i o - c o i i a r e d in th is p a c k .
Results
We m onitored 18 radiocollared wolves during
w inter in 5 packs for 23 w olf pack-years betw een
1986 and 2000. We m onitored wolves during 195
sampling intervals, and located 429 kills made by
wolves over 1,294 days (Table 2). We tracked packs
an average of 8.5 sampling intervals p er year aver­
aging 7.0 days in length (Table 2), for a m ean of
59.5 days/winter sampled, or 31% (5-57%) of the
w inter study period (Table 2). Mean pack size was
6.1 wolves, ranging from 2-18 (Table 2).
Kill rates
Using the ratio estimator, average wolf-pack kill
rate was 0.33 k /d /p (Table 2), com posed of a mean
of 0.23 elk/d/p, 0.04 mule deer/d/p, 0.022 w hite­
tailed d eer/d /p , 0.015 m oose/d/p, and 0.017
940
Wildlife Society Bulletin 2003, 3 1 ( 4 ):9 3 3 - 9 4 6
Ta ble 3. S u m m a r y o f m e a n w i n t e r w o l f kill ra te s fr o m all w o l f p a c k s c o m b i n e d in kills/
d a y / p a c k a n d kg o f p r e y k ille d /d a y /w o lf fo r t h e 5 p r e y s p e c i e s in Banff N a t i o n a l Park, fr om
1986-2000.
<^22. a = 0.05/2 ■
2 .0 1 , P
0.06). We fonnd that the
slope of th e m odel
(k/d/Wj)= Po + P
, w here
K il ls /d ay/p ac k
B io m a s s in k g /d a y /w o if ^
is period length in days,
Species
X
R ange
P o o le d SE
X
R ange
P o o le d SE
was not dhferent than 0
Elk a
0.230
0 .0 4 -0 .4 0
0.025
4.23
0.7 0 -8 .4 8
0.36
for both th e ratio and
M ule de er
0.039
0. 0 1 - 0 . 1 2
0.010
0.1 7
0.0 6 -0 .6 6
0.11
Ballard m ethods (P=0.31,
W hite-tailed d e er
0.022
0. 0 1 - 0 . 1 2
0.007
0.19
0.0 2 -0 .4 8
0.14
0.11, respectively), sngM oose
0.015
0 .0 1 -0 .0 8
0.004
0.64
0.1 3 -2 .5 4
0.31
ge sting that bias in kill
B ig h orn s h e e p
0.017
0 .0 1 -0 .0 9
0.004
0.16
0.0 2 -0 .2 9
0.11
rate w ith sampling inter­
Total
0.330
0 .1 1 -0 .4 8
0.080
5 .4 2
2.2 2 -1 1 .3 5
0.36
val length was negligible.
However,
th e
slope
® Z o n e s p e c i f i c kill rate s ( H e b b l e w h i t e 2 0 0 0 ) o f el k d o n o t s u m to tota l e lk kill rate s b e c a u s e
z o n e s p e c i f i c kill rate s o n l y i n c l u d e y e a r s w h e r e w o l v e s u s e d a s p e c i f i c z o n e , w h i l e th e total
approached statistical sig­
e lk kill rate is a n a v e r a g e o f all w o l f - p a c k y e a r s in all z o n e s .
nificance for th e Dale
^ K g/d a y/w olf c o r r e c t e d for 7 5 % e d i b l e fr o m P e te rs on (1 9 7 7 ) a n d a n a v e r a g e o f 7 5 % a c t u ­
m ethod
(Pj = -0.002,
ally c o n s u m e d fr om th is study.
SE(Pi) = 0.001, P = 0.06),
sngge sting a negative bias
in kill rate as length of
bighorn sh eep/d/p (Table 3). Kill rates in kg/d/w sampling interval increased (e.g. -0.04 k /d /p for a
differed among the 5 prey species (ANOVA,
8g= 20-day m onitoring interval). On average, the
36.70, P < 0 .0005,Table 3). The kill rate of elk was design-based approach provided estimates of vari­
highest (P < 0.0005), whereas kill rates for the 4 ance that w ere on average 30% smaller than modelalternate prey species w ere similar (Table 3, all based variance estimates (design-based average SE=
comparisons P>0.50). Using the model-based SEs 0.01, model-based average SE = 0.016,Table 2). The
presented in Table 2, the 95% confidence interval nnw eighted kill rate SE averaged 0.041, higher than
(Cl) for total woh-pack kill rate ranged from the model-based ratio estimator by approximately
+0.01-0.12 k /d /p w ithin years. Pooled across pack- 70% (Table 2). The SE of k /d /p declined exponen­
years, the model-based 95% Cl was 0.33 ±0.04 tially in BNP, stabilizing w hen nnm ber of sampled
(0.29-0.37 k /d /p ) or 52-67 kills dnring a 181-day intervals was greater than ~6 (Fignre la ) and per­
w inter period (Table 2). However, pooling across cent w inter tracked was greater than -20-25%
years artificially rednces w ithin-year variation (Fignre lb).
Simnlations w ith kill rate set at 0.33 k /d /p
becanse this only acconnts for sample variance and
does not allow inference to nnsam pled winters. show ed that the Dale m ethod overestimated kill
W oh packs killed an average of 41.8 elk (95% C.l. rate by an average of 0.10 k /d /w (0.42 k/d/w, Fignre
34.9-48.7), 7.1 mnle deer (3.3-10.9), 3.9 w hite­ 2a), com pared to the Ballard (0.34 k/d/w ) and ratio
tailed deer (0.6-7.3), 2.7 moose (0.5-4.9), and 3.1 (0.33 k/d/w ) m ethods (Fignre 2b, c). In addition to
bighorn sheep (0.5-5.6) p er 181-day winter. Total being biased, the Dale m ethod was least precise
per-capita consnm ption rates in kg prey (Fignre 2a, d), followed by the Ballard (Fignre 2b, d)
killed/day/w oh (adjnsted for percent edible and and ratio m ethod (Fignre 2c), w hich was m ost pre­
consnm ed) averaged 5.42 k g /d /w (Table 3). cise (Fignre 2d). Of the 3 methods, variance
Hebblewhite (2000) reported on species- and zone- declined the m ost for the Dale m ethod as sampling
specific kill rates for each w o h pack-year. Finally, intensity increased (show n here by % of the w inter
kill rates of seqnential sampling intervals were sampled), w hereas both the Ballard and ratio
nncorrelated (r= 0.08, P = 0.23, n = 170), satisfying m ethod had more stable variance estimates (Fignre
the third assnm ption of Thom pson (1992).
2d). Resnlts w ere similar w hen we varied the nnm­
ber and length of sampling intervals (nnpnblished
data), so herein w e report resnlts only for percent
Comparison o f kill-rate estimators
Both Dale and Ballard m ethods estimated higher w inter sampled. Similarly, we fonnd that varying
kill rates (0.36 k/d/p, and 0.35 k/d/p,Table 2) than kill rate betw een 0.05-0.4 did not affect onr resnlts,
the ratio m ethod (0.33 k/d/p, paired /-test. Dale vs. namely that the ratio m ethod was m ost precise and
Ratio / 22. a = 0 .05/2' =2.33, p= 0.03; Ballard vs. Ratio accnrate across this range (nnpnblished data).
Estimating w olf kill rates • H e b b le w h ite et al.
P 0.02
w 0.01
0.00
2
4
6
8
10
12
14
16
Number of intervals tracked
0.04
0.03
CD
UJ
0.02
0.01
0.00
0.1
0.2
0.3
0.4
0.5
0.6
Proportion winter tracked
Figure 1. R e l a tio n s h ip b e t w e e n th e m o d e l - b a s e d ratio e sti m a te
of th e SE of kill rate in k il ls /d a y /p a ck a n d th e a) n u m b e r of in te r­
vals tr a c k e d a n d b) p e r c e n t of th e w i n t e r t ra c k e d fr om s n o w
tra c k in g d a t a c o l l e c t e d in Banff N a tio n a l Park fr om 1 9 8 6 - 2 0 0 0 .
The b e s t fitting n o n l i n e a r c u r v e s a r e a) Y = O.OOSx^-®® + 2 . 4 1 ,
= 0 . 5 9 , P < 0 . 0 5 , a n d b) Y = 0 . 0 8 6 x ^ - 8 8 , p2 ^ q . 3 6 , P < 0 . 0 5 .
Discussion
Per-capita wolf-kill rates in BNP varied substan­
tially across years (overall X = 5 .4 8 kg/d/w, 0.33
k /d /p ,+0.01-0.12,Table 3), and w ere comparable to
published kill-rate estimates (corrected for % edi­
ble) from other stndies. Wolves in Minnesota were
reported to kill 1.5-5.8 kg of primarily white-tailed
d eer/d /w (Mech 1977). Thnrber and Peterson
(1993) reported wolves killed approximately 6.2 kg
of m oose/d/w on Isle Royale, Michigan. Wolves in a
mnltiple-prey system in Gates of the Arctic Wildlife
Refuge, Alaska, w ere reported to kill a m ean of 6.9
kg of p rey/d/w (range 4.1-12.0 kg/d/w. Dale et al.
941
1995), >90% of w hich was caribon. Wolves preyed
equally on migratory caribon and moose in Alaska,
killing a reported average of 5.3 kg of prey/d/w
(Ballard et al. 1997). Carbyn (1983) reported that
wolves in a m nltiple-prey system in Riding
Moimtain National Park, Manitoba, killed a mean of
6.9 kg/d/w, 78% of w hich was elk. In another mnl­
tiple-prey system in Bialowieza primeval forest in
Poland, wolves w ere re p o rted to kill 7.7
kg/day/wolf, 68% of w hich was European red deer
(.Cervus elaphus) (Jedrzejewski et al. 2000).
Unfortunately, differences w e reveal in kill-rate
m ethods may render direct comparisons among
stndies unreliable (Table 1). For example, the Dale
and Ballard m ethod estimated higher kill rates than
onr m ethod (Table 2). The Ballard and ratio
m ethods w ere similar, likely becanse the Ballard
m ethod excluded only one day. The Dale m ethod
likely was higher becanse the probability of ending
a sampling interval dnring w olf tracking (i.e., of
ending sampling) increased w ith nnm ber of days
sampled due to w eather or wolf movements. By
excluding periods before and after the first and last
kill, the Dale m ethod is biased against including
longer intervals w ithout kills, thereby overestimat­
ing kill rate. Kill rates reported in Dale et al. (1995)
may therefore overestimate w o h kill rates, w ith
implications for w oh-caribon fimctional response
m odels p re sen ted by Dale et al. (1994).
Furtherm ore, by truncating days before and after
th e first and last kill, the Dale m ethod also
increased sampling variation (Eq. 2, Table 2).
Finally, we fonnd evidence for a slight negative bias
in kill rate w ith increasing sampling interval length
nsing the Dale m ethod, possibly becanse this
m ethod excluded longer intervals w ithout kills.
Considering predation as a function of handling
and search times makes the potential bias assumed
by Dale et al. (1995) and others seem unlikely For
example, the aerial m ethods of Fuller and Keith
(1980) w ere biased becanse they only sampled han­
dling time to estimate kill rate, and handling time
varied w ith prey size, thereby affecting probability
of detection on aerial telem etry flights. However,
continnons groimd or aerial tracking samples both
search and handling time and w onld not have the
same bias as Fuller and Keith’s (1980) aerial m eth­
ods. Therefore, truncation likely was unwarranted.
However, both the Ballard and Dale m ethods were
designed for nse in one long (30-day) sampling
interval, w here such truncation might not bias kill
rate. Unfortunately, these m ethods have been
942
Wildlife Society Bulletin 2003, 3 1 ( 4 ):9 3 3 - 9 4 6
ITT
d
-O
Q)
H
O
: I. -.
O
d
it
I -
I
II
I
^
^T"n"T-f|-ttt
°° o ° o o < ^
0.30
i .. j i . .. 4 i- -4 ik
0.35
0.40
0.45
0.50
-i4 l
0.55
.
P P -0
CM
d
—^
0.60
0.30
I
I
I
0.35
0.40
0.45
^
0.50
I
r
0.55
0.60
0.55
0.60
Percent winter sam pled
CO
T TT t
II
. .(5 .0
ooo
Percen t winter sam pled
d
JtT tT tT iItt
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0.35
0.40
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0.45
0.50
0.55
0.60
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0.30
0.35
0.40
0.45
0.50
Percent winter sam pled
Figur e 2. Results o f 1 0 0 kill- rate e s t i m a t i o n s i m u l a t i o n s fo r a) th e D a le e t al (1 9 9 5 ), b) Bal lard e t al. (1 997) , a n d t h e c) ratio-varia n c e e s t i m a t o r a p p r o a c h e s , a n d d) t h e e s t i m a t e d SE in kill rate fo r all 3 a re s h o w n . A c c u r a c y a n d p r e c i s i o n of all kill-rate e s t i m a ­
to rs w e r e e v a l u a t e d at a k n o w n kill rate of 0 . 3 2 8 1 kil ls /d a y /p a ck , s a m p l e d w i t h s a m p l i n g c h a r a c t e r i s t i c s m a t c h i n g t h o s e in BNP
(Table 1) o v e r a r a n g e of s a m p l i n g intensity.
applied in settings w ith shorter sampling intervals
w ithont consideration of these biases (Mnrphy
1998Jedrzejew ski et al. 2000). Fnrtherm ore, given
the difficnlty of monitoring wolves for long peri­
ods, w e feel that stndies will more commonly have
several shorter sampling intervals (e.g., Hnggard
1993a, Mnrphy 1998, Jedrzejewski et al. 2000,Table
1). In addition, estimating kill rate in one long inter­
val cannot provide variance estimates, yet snch kill
rates freqnently are extrapolated to entire winters.
Based on onr simnlations, in designs w ith >1 short­
er (-5 -1 3 days,Table 3) sampling intervals, the ratio
approach was the least biased and m ost precise
m ethod w e examined (Fignre 2). While the Ballard
and Dale m ethods may be appropriate for a single
long interval (>30 days), onr estimator provides a
framework for estimating kill rate across several
periods in more com mon designs of shorter sam­
pling intervals. Therefore, becanse of the problems
w ith previons estim ators, statistical basis, and
enhanced perform ance of the ratio estimator, we
recom m end the ratio m ethod for estimating kill
rate in fntnre stndies of w olf-prey systems.
Deciding w hich formnla to nse for estimating
variance in kill rate (eq.l or eq.2) will depend on
stndy design objectives and will be more of a philo­
sophical choice (see Thom pson 1992; p 95). The
standard design-based approach provided more
precise standard errors than th e model-based
approach (Table 2). However, we feel that in most
Estimating w olf kill rates • H e b b le w h ite et al.
w olf-prey studies, a more conservative (i.e., w ider
confidence intervals) approach to estimating vari­
ance may be w arranted because of inferences typi­
cally made by managers and the public (i.e., extrap­
olating w olf kill rates to entire seasons/years).
Furtherm ore, the 2 estimators provided nearly
identical estimates in cases w here length of sam­
pled and unsam pled intervals w ere equal (i.e. the
second term in Equation 2 equals 1,Table 2, unpub­
lished data), but the model-based variance estimate
was larger w hen sampled interval length varied.
This further suggested that treating interval length
as a random variable and including this in variance
estimation (Eq. 2) is warranted.
The double-sampling kill-rate m ethod developed
by Smith et al. (2003) can address violation of the
assumption that all kills are located within a sam­
pling interval, and is applicable w hen continuous
m onitoring is not possible. Their approach adjust­
ed kill rates within a sampling interval to estimate
an imbiased kill rate for that sampling interval, and
our ratio-variable m ethod used m onitored sampling
intervals to estimate kill rate (or the num ber of prey
killed) in periods that w ere not m onitored (i.e.,
over a w hole winter). These 2 m ethods could easi­
ly be com bined to estimate kill rates and variance
in kill rate by substituting Smith et al.’s (2003) esti­
mate of num ber of kills from the double-sampling
m ethod in sampling interval (i) for iuEquation 1.
Kill rates of elk in BNP w ere higher than kill rates
of alternate prey, w hich w ere approximately equal
(Table 3). Kill rates w ere ranked elk > mule deer=
white-tailed deer > moose = bighorn sheep, similar
to Weaver’s (1994) review of N orth American
w olf-elk studies. Elk w ere the most abundant
ungulate in the study area, w ith population esti­
m ates varying from -5 0 0 -2 ,0 0 0 elk from
1986-2000 (Hebblewhite et al. 2002). Hebblewhite
et al. (2002) reported the w o h kill rates reported
here interacted w ith snow depth to limit elk popu­
lation growth in the Bow Valley of BNP. Huggard
(1993a) indicated that wolves strongly selected elk
at the individual and herd levels during 1989-1991
in BNP. W oh selection for elk may be a function of
increased encounter rates (Huggard 1993a) and
attack success (Hebblewhite and Pletscher 2002)
by wolves preying on group-living elk. Further
research is required to test w hether w o h predation
on elk is density-dependent (e.g.. Messier 1994).
Parks Canada (unpublished data) estim ated
approximately 1,000 bighorn sheep in the study
area; thus the im pact of w inter w o h predation on
943
S n o w t r a c k in g w o l v e s o n t h e g r o u n d to lo c a te w o lf- k ille d prey.
bighorn sheep in our study area (6-10 killed by
wolves p er w inter) should be minimal. In compar­
ison, Hurd (1999) showed that low-density moose
populations (-7 5 in the study area) w ere declining
due to low adult survival following w o h recolo­
nization, and predation by wolves was a leading
cause of mortality. The relatively higher impact of
w inter w o h predation (5-8 m oose/w inter) is con­
sistent w ith these declines. Uihortunately assessing
the impact of w o h predation on mule deer (14-16
mule deer/w inter) and white-tailed deer (8-12
white-tailed deer/w inter) is dhficult because popu­
lation estimates are unavailable for our study area.
We believe that exploration of variation in kill
rates will provide valuable insights into preda­
to r-p rey dynamics, such as in functional response
estimation. Typical regression approaches to esti­
mating functional responses fail to account for vari­
ation in kill rate (Dale et al. 1994, Messier 1994, but
see Messier and Joly 2000). Incorporating kill-rate
variance will only exacerbate statistical dhficulties
described by Marshal and Boutin (1999) that
plague estim ation of the w oh-prey functional
response. Furtherm ore, incorporation of kill-rate
944
Wildlife Society Bulletin 2003, 3 1 ( 4 ):9 3 3 - 9 4 6
variance into m anagement applications of kill rates
to nngnlate harvest m anagement (e.g., Keith 1983)
will increase the rigor of snch applications. Onr
kill-rate estimates, w hen nsed w ith associated vari­
ance estimates, will provide an nnderstanding of
the potential im pact of wolves on elk popnlations
thronghont w estern N orth America.
Conclusions
W inter w olf kill-rate estimates varied considerably
despite intensive m onitoring (Table 2, 3). A sam­
pling effort of >25% of the entire w inter in >6-8
sampling intervals appeared to stabilize sample vari­
ance in kill-rate estimates in BNP to reasonable lev­
els (Fignre 1,2). Fnrthermore, throngh simnlations,
w e evalnated kill-rate estimator perform ance at a
sampling intensity of np to 55% of the winter. Even
at these high sampling intensities, w e fonnd snbstantial variation in kill rates (Fignre 1,2). Therefore
w e recom m end that researchers sample at least
>25% o f the w in ter to estim ate kill rates.
Unfortnnately of the stndies reviewed in Table 1,
many sampled less intensively than this level to esti­
mate kill rates. Therefore, it may be difficult to reli­
ably estimate kill rates in many systems, prom pting
a search for new approaches for the stndy of
w olf-prey relationships (see Bontin 1992, Marshal
and Bontin 1999, Joly and Patterson 2003) that
focns on estimating survival of prey (e.g., Knnkel
and Pletscher 1999) or predator selectivity Goly and
Patternson 2003) over a range of predator densities.
Global Positioning System (GPS) collar-based
approaches may provide a technological solution to
increase sampling intensity to estimate kill rate (see
Anderson and Lindzey 2003); however, GPS-collar
approaches will introduce new sources of error
throngh kill-site misspecification in logistic regres­
sion approaches. Fnrthermore, GPS-collar approach­
es freqnently will have missed intervals due to poor
GPS coverage or GPS-collar mahimction. Managers
in areas w ith recolonizing wolves should consider
these statistical sampling issues w hen designing
stndies w here the goals are to estimate kill rates.
While kill rate may be an intuitive metric collected
for decades, we encourage researchers to adopt a
standardized approach that will increase compara­
bility and rigor across stndies. We believe the ratio
estimator we present will provide snch an approach
across predator-prey stndies.
Acknowledgm ents. This research was funded by
Parks Canada, the Central Rockies Wolf Project,
John/Panl and Associates, A lberta Human
Resources and Employment, Human Resources
Development Canada, Canadian-Pacific Foimdation,
Paquet Wildlife Fund, and World Wildlife Fund. The
Banff Warden Service, Central Rockies Wolf Project,
and dozens of research assistants collected the field
data over the 15-year period. We thank T. Drummer,
J. Graham, M. Lindberg, and D. Patterson for statisti­
cal advice on ratio estimators. L. S. Mills, K.
Foresman, M. Mnsiani, and M. Schwartz provided
useful com ments on earlier versions of this manu­
script. We thank reviewers G. C. W hite and B.
Patterson for thorough reviews that substantially
improved the m anuscript. M. Hebblewhite was
supported by the L. Pengelly and the G. Bright
memorial scholarships. School of Forestry, and the J.
Paquet Wildlife Fund while at the University of
Montana, and the Canon National Parks Service
Science Scholarship for the Americas while prepar­
ing the m anuscript for publication.
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M ark H e b b le w h ite ( p h o t o )
is a d o c t o r a l s t u d e n t in t h e
D e p a r t m e n t of Bio lo gic al S c i e n c e s a t t h e U n iv e rs it y o f A lb e rta .
H e r e c e i v e d his B.Sc. in p u r e a n d a p p l i e d e c o l o g y at th e
U n iv e rs it y of G u e l p h (1 9 9 5 ) , a n d his M.S. in w ild lif e b i o l o g y
fr o m t h e U n iv e rs it y o f M o n t a n a ( 2 0 0 0 ) . H e h a s c o n d u c t e d
r e s e a r c h o n s o n g b ir d s , f r e s h w a t e r fish, u n g u la te s , a n d c a r n i ­
v o r e s t h r o u g h o u t C a n a d a , a n d h a s w o r k e d s i n c e 1 9 9 5 in Banff
N a t i o n a l Park, w h e r e h e c o m p l e t e d his M.S. o n w o l f - e l k
d y n a m i c s . His c u r r e n t re s e a r c h in te re sts i n c l u d e li nkin g u n g u ­
late p o p u l a t i o n d y n a m i c s to p r e d a t i o n risk a n d fo r a g e d y n a m i c s
w i t h a p p l i c a t i o n s t o n a t i o n a l - p a r k m a n a g e m e n t . Paul C. P aquet
is a d j u n c t p r o f e s s o r at t h e U n iv e rs it y o f Cal gary, a n d t h e W W F C a n a d a ' s c a r n i v o r e sp e c ia lis t. H e r e c e i v e d his B.A. in p h i l o s o ­
p h y of s c i e n c e fr o m S a n t a C la ra (1 970), his B.Sc. in w il d li f e
z o o l o g y fr o m A r i z o n a St a te (1 972), his M .S c . in b e h a v i o r a l
e c o l o g y fr om P o r tl a n d Sta te (1 9 8 2 ), a n d his Ph .D . in z o o l o g y o n
w o l f - c o y o t e e c o l o g y fr o m t h e U n iv e rs it y o f A lb e rta (198 8).
Paul h a s w r i t t e n m a n y a rti c le s o n large m a m m a l i a n c a r n iv o r e s .
H e h a s b e e n c o n d u c t i n g field r e s e a r c h o n w o l v e s a n d c o y o t e s
since 1972.
H e is c u r r e n t l y i n v o lv e d in c a r n i v o r e re s e a r c h
t h r o u g h o u t N o r th A m e r i c a a n d Eu rope, i n c l u d i n g n e w re s e a r c h
o n t h e e c o l o g y a n d c o n s e r v a t i o n of c o a s ta l British C o l u m b i a
w o l v e s . D a n iel H. P letsch er is d i r e c t o r o f t h e w ild lif e b i o l o g y
p r o g r a m a t t h e U n iv e rs it y of M o n t a n a .
H e h o l d s a B.S. in
w ild lif e m a n a g e m e n t fr o m t h e U n iv e rs it y of M i n n e s o t a (1 9 74 ),
a n M.S. in w ild lif e b i o l o g y fr o m K a nsa s St a te (1977), a n d a
Ph .D . in fo res tr y fr o m Yale U n iv e rs it y (1 9 8 2 ). H e h a s s e r v e d as
P r e s id e n t o f t h e M o n t a n a C h a p t e r of The W il d li fe S o c i e t y a n d
c h a i r e d t h e s t e e r i n g c o m m i t t e e fo r th e 1 9 9 9 In te rn a ti o n a l
W il d life M a n a g e m e n t C o n g r e s s in H u n g a ry . His r e s e a r c h in te r­
ests f o c u s o n p r e d a t o r - p r e y d y n a m i c s a n d e n d a n g e r e d s p e c ie s .
R o b e rt B. L essard is a d o c t o r a l s t u d e n t in t h e D e p a r t m e n t of
R e n e w a b l e R e s o u r c e s at t h e U n iv e rs it y of A lb e rta . H e r e c e i v e d
his B.Sc. in a c t u a r ia l m a t h e m a t i c s a t C o n c o r d i a U niv e rs it y
(1 9 9 0 ) a n d his M .S c . in fo r es t r e s o u r c e m a n a g e m e n t fr o m th e
U n iv e rs it y o f British C o l u m b i a (1 9 9 8 ). H e h a s w o r k e d in forest
ha r v e s t p la n n i n g , in s e c t o u t b r e a k a n d fo r e s t fire d y n a m i c s , a n d
t h e m a n a g e m e n t of p r e d a t o r - u n g u l a t e s y s te m s .
H is c u r r e n t
in te re s t is in t h e a n a l y s is a n d m o d e l i n g o f h u m a n - a l t e r e d
w ild lif e d y n a m i c s . Carolyn j. Callaghan is a r e s e a r c h a s s o c i a t e
w i t h t h e C en tra l R o ck ies W o l f P r o je c t a n d a n a d j u n c t p ro f e s s o r
a t t h e U n iv e rs it y of Calg ary. S h e r e c e i v e d a B.Sc. a n d a Ph .D .
(2 0 0 1 ) a t t h e U niv e rs it y o f G u e l p h .
H e r r e s e a r c h in te re sts
in c l u d e m a n y a s p e c t s o f c a r n i v o r e e c o l o g y a n d m a n a g e m e n t ,
i n c l u d i n g w o l f h a b i t a t u s e a n d p o p u l a t i o n p e r s i s t e n c e ; re c e n t
w o r k i n c l u d e s u s e of w ild lif e c o r r i d o r s b y c a r n i v o r e s a n d u n g u ­
lates in m o u n t a i n o u s h a b ita ts a n d in v e st ig a ti n g t h e w o l f - l i v e s t o c k re l a t i o n s h i p in th e A lb e rta fo othills.
A sso cia te ed ito r: W h i t e