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Dan Prenzlow, Director, Colorado Parks and Wildlife • Parks and Wildlife Commission: Marvin McDaniel, Chair • Carrie Besnette Hauser, Vice-Chair
Marie Haskett, Secretary • Taishya Adams • Betsy Blecha • Charles Garcia • Dallas May • Duke Phillips, IV • Luke B. Schafer • James Jay Tutchton • Eden Vardy
�CHAPTER 17
Wolf Kill Rates: Predictably Variable?
Matthew S. Becker,* Robert A. Garrott,* P. J. White,† Rosemary Jaffe,*
John J. Borkowski,{ Claire N. Gower,* and Eric J. Bergman*
*Fish and Wildlife Management Program, Department of Ecology, Montana State University
†
National Park Service, Yellowstone National Park
{
Department of Mathematical Sciences, Montana State University
Contents
I. Introduction
II. Methods
A. Wolf Tracking and Kill Detection
B. Kill Rate Estimation
C. Evaluating Kill Rate Variation
D. Evaluating the Functional Response for Elk
III. Results
A. Variations in Elk Kill Rate
B. Variations in Bison and Total Kill Rates
C. Functional Response
IV. Discussion
V. Summary
VI. References
Appendix
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365
Theme
The ability of predators to successfully capture and kill prey is affected by the abundance and diversity of the
prey assemblage, and such variation is a fundamental driver of ecosystem dynamics because per capita
consumption rate strongly influences the stability and strength of community interactions. Descriptions of
predatory behavior in this context typically include the functional response, specifically the kill rate of a predator
as a function of prey density. Thus, a major objective in studying predator–prey interactions is to evaluate the
strength of the numerous factors related to the kill rate of a predator, and to subsequently determine the forms of
its functional response in natural systems because different forms have different consequences for ecosystem
dynamics. Recent controversies over the nature of predation focus on the respective roles of prey and predator
abundance in affecting the functional response. However, resolution requires more direct measures of kill rates
in natural systems. We estimated wolf (Canis lupus) kill rates in a tractable and newly established wolf–elk
(Cervus elaphus)–bison (Bison bison) system in the Madison headwaters area of Yellowstone National Park
during winters 1998–1999 to 2006–2007 to document the transition from over seven decades without wolves to
The Ecology of Large Mammals in Central Yellowstone
R. Garrott, P. J. White and F. Watson
ISSN 1936-7961, DOI: 10.1016/S1936-7961(08)00217-0
Copyright # 2009, Elsevier Inc.
All rights reserved.
�340
Becker et al.
a well-established top predator population. Wolf abundance, distribution, and prey selection varied during the
study, concurrent with variations in the demography, distribution, and behavior of elk and bison. These dynamics
enabled us to evaluate factors influencing variations in wolf kill rates and the forms of their functional response.
I. INTRODUCTION
The role of a predator in regulating or destabilizing prey populations is widely believed to depend on
the form of its functional response (Murdoch and Oaten 1975, Oaten and Murdoch 1975, Hassell
1978). There are three general forms of the functional response (Holling 1959). A predation rate
increasing linearly with prey density is know as a Type I response, which is considered unrealistic for
most predators due to lack of constraints in handling time or the time needed to capture and subdue a
prey item. Consequently, Type I responses are most likely confined to predators with few handling
constraints such as filter feeders (Jeschke et al. 2004). A more plausible description of predation
behavior is the Type II response, which exhibits a decelerating predation rate with increasing prey
density reflective of predator satiation (Holling 1959). This form of predatory behavior is thought to be
common in natural systems consisting of a single prey species or a specialist predator (Peckarsky 1984)
and is considered destabilizing because it is inversely density-dependent (Oaten and Murdoch 1975;
Hassell 1978, 2000). However, Fryxell et al. (2007) argue that, while Type II responses can be
destabilizing in the solitary prey and predator systems from which most predator–prey theory has
been developed and refined, such a response can also be stabilizing if prey and predators aggregate in
groups. The sigmoidal Type III functional response can be exhibited when more than one prey type
exists and a predator is plastic in its foraging (Holling 1959). A Type III functional response can be
generated by a variety of factors such as prey-switching (Murdoch 1969), selective foraging, and
learning by the predator (Tinbergen 1960, Real 1979), or use of refuges by prey (Taylor 1984). Because
a Type III response implies density-dependent predation, such behavior is considered to have a
stabilizing influence on ecosystems (Oaten and Murdoch 1975).
Due largely to the dramatically different ecosystem trajectories that can ensue with different
predator behaviors, an immense body of work has been performed on determining the drivers of kill
rates. Kill rates are influenced by encounter rates and prey density (Holling 1959), the presence of
alternative prey (Murdoch 1969), environmental factors (Thompson 1978, Anderson 2001), and prey
distribution (Real 1979, Cosner et al. 1999, Pitt and Ritchie 2002). However, the causal factors driving
the ultimate form of the functional response have been the subject of considerable debate. The longstanding belief that forms were driven by prey density alone (‘‘prey-dependence’’) has been challenged
by the idea that predator density can also appreciably influence per capita consumption (‘‘predator
dependence’’). Strong predator dependence, typically denoted by the ratio of predators to prey (‘‘ratiodependence’’) has been vigorously debated as an alternative to the prey dependent functional response
(Arditi and Ginzburg 1989, Berryman 1992, Abrams 1994, Akcakaya et al. 1995, Abrams and Ginzburg
2000). While this debate has not been resolved, it is likely that both prey and predator numbers
influence predation and more empirical studies are needed (Abrams and Ginzburg 2000, Schenk
et al. 2005).
Wolf–ungulate systems have received substantial attention in studies of kill rates, largely due to the
strong scientific and societal interest in assessing the effects of wolves on prey populations. Kill rates of
wolves are extremely variable and influenced by prey density, pack size, and snow pack (see reviews in
Hebblewhite et al. 2003, Mech and Peterson 2003). There is considerable disagreement regarding the
nature of wolf foraging behavior, but few studies of functional responses due to the difficulties inherent
in data collection (Mech and Peterson 2003). Thus, various scientists have advocated that wolf kill rates
were best described as constant (Eberhardt 1997), ratio-dependent (Vucetich et al. 2002, Jost et al.
2005), or prey-dependent (Messier 1994, 1995; Messier and Joly 2000; Varley and Boyce 2006). Recent
�Chapter 17
.
Wolf Kill Rates: Predictably Variable?
341
analyses from the long-term wolf–moose dataset of Isle Royale indicated that ratio-dependence best
described the nature of wolf predation in this system (Vucetich et al. 2002, Jost et al. 2005), and that the
inclusion of a ratio-dependent functional response in kill rate analyses may provide significant insights
into discrimination between types of predation.
Disagreement also stems, in part, from the inherent difficulty in accurately assessing kill rates in
wolf–ungulate systems. These metrics are so difficult to measure that some scientists question the
appropriateness and/or feasibility of estimating the functional response for describing wolf–ungulate
systems (Eberhardt 1997, Marshal and Boutin 1999, Person et al. 2001, Mech and Peterson 2003) while
others contend that distinguishing between functional response forms is not ecologically critical (Dale
et al. 1994, Van Ballenberghe and Ballard 1994). Instead, some scientists suggest monitoring causespecific mortality and recruitment rates of prey species (Kunkel et al. 2004) or changes in ungulate
carrying capacity (Bowyer et al. 2005). In addition, the methods and metrics used to calculate kill rates
vary widely. Thus, comparisons between systems and methods are often not possible (Hebblewhite
et al. 2003). Lastly, the estimation of kill rates is subject to variable observer effort, weather conditions,
and movement of wolves. Only recently have investigators attempted to account for these sources of
variability (Jaffe 2001, Hebblewhite et al. 2003, Smith et al. 2004).
We evaluated drivers of kill rates and the forms of wolf functional response using long-term
predation data collected during winters 1998–1999 through 2006–2007 from a tractable wolf–elk–
bison system in the Madison headwaters area of Yellowstone National Park that experienced substantial
seasonal and annual variation in prey abundance, predator abundance, and snow pack. Data on wolf
numbers and kills were collected daily for each winter (15 November–21 April) of the study period
(Figure 17.1). There are various ecological justifications for employing several different metrics to
evaluate wolf kill rates. Metrics employing kills as the unit of measure are likely more appropriate than
biomass for evaluating the effects of wolf predation on ungulate prey populations (Hayes and Harestad
2000, Hayes et al. 2000). Furthermore, it is useful to distinguish between kills per pack and kills per wolf
because wolves are group hunting predators with a rigid social hierarchy and the pack is typically the
hunting unit rather than individual wolves (Mech 1970). Questions concerning wolf population
dynamics and their food acquisition are better addressed using a metric of biomass (Mech and
Peterson 2003). Consequently, we calculated metrics of kills/pack/day, kills/wolf/day, and kg/wolf/
day for each pack. Our objectives were to: (1) describe temporal trends in kill rates within and among
winters and wolf packs; (2) determine the primary factors driving trends in total wolf kill rates, as well
FIGURE 17.1 Wolves from the Hayden pack feed on an elk kill in the Madison headwaters. Estimating the rate at
which a predator kills prey is fundamental to understanding predator–prey dynamics (Photo by Shana Dunkley).
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as kill rates for elk and bison for various metrics; and (3) assess the form of wolf functional response for
elk. We predicted that kill rates would be positively influenced by the abundance of elk and bison and
negatively influenced by wolf abundance. Also, we predicted that kill rates would be positively related
to pack size when calculated per pack, and negatively related to pack size when calculated per wolf.
Further, we predicted wolf functional responses would be best described by ratio dependence due to the
strong potential for predator dependence.
II. METHODS
A. Wolf Tracking and Kill Detection
We conducted intensive kill rate investigations in the primary winter ranges of bison and elk in the
Madison headwaters area (31,000 ha), with concurrent investigations of these prey species allowing
collection of wolf predation data in a tractable area with a well-described ungulate prey base. We
documented wolf kill rates during 15 November through 21 April each winter from the establishment
of a resident pack in 1998–1999 (Chapter 15 by Smith et al., this volume) through 2006–2007. Our
sampling unit was radio-collared wolf packs that incorporated the study area as part of their territory.
Wolves were aerially darted from helicopters by National Park Service biologists and fitted with VHF
telemetry collars. A total of 37 wolves from four packs were collared during the course of the study
(Chapter 15 by Smith et al., this volume).
The number and sizes of wolf packs using the study area were dynamic within and among winters
(Chapter 15 by Smith et al., this volume). Thus, we used ground observations, snow-tracking, and
counts during aerial tracking flights by park biologists to estimate the wolf population. We estimated
the wolf population in wolf days, defined as one wolf in the study area for one day. We used roads
traversing each river drainage in the study area (Chapter 2 by Newman and Watson, this volume) to
rapidly sample for wolf presence daily through the winter. Sampling began at dawn with ground crews
of three to four people covering all roads by snowmobile or vehicle and using strategic high points in
the landscape to facilitate telemetry triangulations (White and Garrott 1990) and observations of
wolves. When possible, multiple locations were obtained in early morning and evening each day. We
also recorded any uncollared wolves detected opportunistically via tracks or observations to aid in the
estimation of the wolf population using the study area. In addition, biologists studying elk and bison
routinely covered backcountry areas to assist with wolf detection.
When wolves were located, we used visual scans and monitoring of avian scavengers in the vicinity to
detect kills. Ravens preferentially associate with wolves in winter, and an average of 28 ravens (Corvus
corax) were present at fresh wolf kills on the northern elk winter range in Yellowstone National Park
(Stahler et al. 2002), with slightly lower averages in the Madison headwaters area (D. Stahler, National
Park Service, personal communication). This association facilitated the detection of kills. We also
conducted extensive snow-tracking after wolves departed the area to further facilitate kill detection
(Huggard 1993, Dale et al. 1995, Je˛drzejewski et al. 2000, Jaffe 2001, Hebblewhite et al. 2003, Bergman
et al. 2006; Figure 17.2). We necropsied ungulate carcasses to determine cause of death, species, sex, age,
condition, and percent consumed. Wolf kills were inferred from collective evidence of subcutaneous
hemorrhaging indicative of injuries sustained before death, signs of struggle or chase at the kill site,
blood trails, signs of predator presence, and our knowledge of wolf movements and activities. We
documented frequent spring grizzly bear (Ursus arctos) predation on bison during the latter years of the
study. Thus, when both bears and wolves were present on a kill, we classified it based on the patterns of
injury and subcutaneous hemorrhaging. Bears typically attacked the head and spine, while wolves
attacked the flanks, hindquarters, and underside of the neck. Similarly, mountain lion (Puma concolor)
kills of elk were determined based on characteristics of the kill site and patterns of injury. Kills were
�Chapter 17
.
Wolf Kill Rates: Predictably Variable?
343
FIGURE 17.2 Snow tracking the Gibbon pack along Nez Perce Creek in the Firehole river drainage. A combination of
telemetry, scanning, and tracking was employed on a daily basis to detect wolf kills in the study area (Photo by Shana
Dunkley).
sexed using the presence of genitalia, horns, antlers, or pedicels, and aged based on size and patterns of
tooth eruption and replacement (Fuller 1959, Hudson et al. 2002). When available, an incisor or canine
was removed from adult ungulates and aged using cementum annuli (Moffitt 1998, Hamlin et al. 2000).
B. Kill Rate Estimation
Daily estimates of wolf numbers and kills detected for the wolf population at large and for each pack
were used to estimate minimum kill rates each winter and for three winter periods of approximately eight weeks each that corresponded to early (15 November–6 January), middle (7 January–
27 February), and late winter (28 February–21 April), ending near the mean pack denning date, after
which packs were considerably less cohesive (Jaffe 2001; Chapter 15 by Smith et al., this volume). The
kills/pack/day metric was calculated by dividing the number of kills by the number of days in the
sampling period in which the respective pack was detected, while the kills/wolf/day metric was
calculated by dividing the number of kills by the estimated wolf days for a given pack for each period.
Winter and winter period estimates of kills/wolf/day for the entire population were calculated from
pooled estimates of wolf days and kills for a given period or winter. Estimates for kg/wolf/day were
derived by summing the biomass of all kills for a given pack and dividing by the estimated wolf days for
that period. We classified all kills into species, sex, and age, and used biomass estimates for elk and
bison obtained from Murie (1951) and Meagher (1973), respectively. Elk bulls, cows, and calves were
estimated at 287, 236, and 116 kg, respectively, while bison adults and calves were estimated at 500 and
136 kg, respectively. Bison adult age classes of both sexes can vary dramatically in weight depending on
age (Berger and Peacock 1988). Thus, we did not use separate categories for males and females in
biomass estimation. We assumed 75% edible biomass for each prey item (Peterson 1977), but did not
account for scavenger loss or incomplete consumption of carcasses. Therefore, the kg/wolf/day metric
was considered an index rather than an absolute measure of consumption per wolf. For each pack,
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we estimated total kill rates and separate kill rates for elk and bison using all three kill rate metrics for
each winter period. We also estimated these kill rates using the same metrics for the entire wolf
population during each winter and winter period.
C. Evaluating Kill Rate Variation
We used multiple linear regression techniques (Neter et al. 1996) to evaluate factors affecting variation
in wolf kill rates within and among winters and packs. Because the Madison headwaters was a multipleprey system with ungulate species differing substantially in abundance and defenses (Garrott et al.
2007), we evaluated variations in kill rates for each wolf pack using three separate response variables:
total kill rates, kill rates for elk, and kill rates for bison. For each response variable, we calculated three
kill rate metrics (kills/wolf/day, kills/pack/day, and kg/wolf/day) during a given winter period, comprising nine analyses in total. We developed eight covariates to evaluate the influences of prey
abundance, wolf pack size, and snow pack. These covariates were judiciously selected from factors
reported to be influential in the kill rate literature, as well as from our knowledge of the study system.
We used six covariates to describe wolf prey abundance, including elk abundance (ELKall), bison
abundance (BISONall), and the respective abundances of elk adults and calves (ELKadt, ELKcalf ) and
bison adults (BISONadt) and calves (BISONcalf ). We estimated prey abundance for the entire study area
rather than just the drainages encompassed by a particular pack’s territory because multiple packs
overlapped spatially and temporally, larger packs were more dominant (Chapter 15 by Smith et al. and
Chapter 16 by Becker et al., this volume), and kill rates were estimated for each pack over nearly an
8-week period during which bison movement between wolf pack territories was considerable (Chapter
12 by Bruggeman et al., this volume). We estimated the abundances of adult elk and calves during early,
middle, and late winter using replicate mark–resight techniques and age composition data (Chapters
11 and 23 by Garrott et al. and Chapter 16 by Becker et al., this volume). Estimates of bison abundance
and age class were obtained via ground survey counts of the bison winter range in the study area.
Surveys were conducted every 10–16 days during winter, with observers recording the number,
location, sex, and age class of all bison sighted (Chapter 12 by Bruggeman et al. and Chapter 16 by
Becker et al., this volume). The effect of snow pack on prey vulnerability was estimated using a metric
of accumulated snow-water equivalents (SWEacc, Garrott et al. 2003). We used a validated model
describing snow pack dynamics (Chapter 6 by Watson et al., this volume) to estimate a mean daily
SWE for the study area, and accumulated mean daily SWE values from the typical start of the first
snowfall (1 October) until the end of a given winter period. By estimating the duration and severity of
snow pack and its weakening effect on prey, we considered SWEacc an indicator of prey physiological
condition and SWEacc explained substantially more variation in wolf prey selection in the Madison
headwaters area than the mean SWE on the ground at the time of a kill (Chapter 16 by Becker et al., this
volume). Also, we calculated a mean wolf pack size (WOLFpack) for each winter period from daily
estimates of size for a given pack.
We developed and evaluated a priori hypotheses in the form of 12 candidate models fitted to each of
the kill rate metrics (Appendix A). Thus, we performed three separate analyses for total kill rates and
kill rates of elk and bison. To facilitate comparison of coefficient estimates, all covariates were centered
and scaled prior to analysis by subtracting the midpoint and dividing by half of the range, resulting in
values between �1 and 1. We assessed potential colinearity between covariates using variance inflation
factors and did not use covariates with values >6 in the same model (Neter et al. 1996). Covariates that
were not used in the same model due to strong colinearity were BISONadt and BISONcalf, and all bison
covariates with SWEacc because increasing snow pack resulted in increased bison migration into the
study area (Chapter 12 by Bruggeman et al., this volume). We fitted all models in R version 2.4.1
(R Development Core Team 2006). Models were compared using Akaike’s Information Criterion
corrected for small samples (AICc; Burnham and Anderson 2002). We calculated Akaike weights and
�Chapter 17
.
Wolf Kill Rates: Predictably Variable?
345
evaluated the importance of each covariate by its predictor weight (wp), which we calculated by
summing the Akaike weights for all models containing the covariate (Burnham and Anderson 2002).
Goodness of fit was evaluated using adjusted R-squared values for each model, and covariate coefficients were evaluated for direction (i.e., positive or negative) and stability among different models.
Next, we fit moderated and pseudo-threshold forms to all prey covariates to determine if the fit was
improved. Lastly, we performed exploratory analyses fitting kill rates to a ‘‘per-pack’’ scale, wherein
prey abundance was estimated for each pack’s territory, to determine if covariate relationships and
model selection results were affected.
Elk were considerably more vulnerable to wolf predation than bison in the Madison headwaters,
though wolves increasingly selected bison in late winter at high bison:elk ratios (Chapter 16 by Becker
et al., this volume). Thus, we predicted elk abundance would be positively related to elk kill rates and
total kill rates, but negatively related to bison kill rates, for all three metrics. Similarly, we predicted
bison abundance covariates would be negatively correlated with elk kill rates of all three metrics
because the migration of bison into the system would provide a large alternative food source that
could decrease kill rates of elk. We predicted that increasing abundance of either or both prey species
would be positively related to total kill rates. In social predators such as wolves, pack size has an
important influence on kill rates because larger packs can make more kills, but acquire less food
per capita (Thurber and Peterson 1993, Schmidt and Mech 1997). Thus, we predicted pack size would
be positively related to kills/pack/day, but negatively related to kills/wolf/day and kg/wolf/day for total
kill rates and kill rates of elk and bison. Lastly, snow pack has a considerable debilitating influence on
prey, both in their ability to escape from predation and on their physiological condition (Chapter 16 by
Becker et al., this volume). Thus, we predicted that SWEacc would be positively related to kill rates of all
three metrics (Appendix A).
D. Evaluating the Functional Response for Elk
Multiple regression analyses evaluated the influence of various prey, predator, and environmental
influences on the kill rates of wolf packs within the study area. To describe the average rate of elk
consumption per wolf (i.e., the functional response), however, we fit traditional functional response
models (Holling 1959) to wolf kill rate data. Functional response curves were fit for the metric of elk
kills/wolf/day estimated at a winter range scale, whereby a single elk kill rate for the study area during a
given winter period was estimated by pooling all elk kills and dividing by the estimated wolf days in the
period. Bison and elk abundance covariates were then estimated for the entire study area for each
period as described for the kill rate variation analyses, and wolf abundance was estimated by dividing
the total wolf days for a given winter period by the number of days in the period. While wolves killed
bison to varying degrees within and among winters, this change in diet appeared to be heavily
moderated by circumstances that increased bison vulnerability, such as severe winters and high wolf:
ungulate ratios (Chapter 16 by Becker et al., this volume) We also did not have sufficient kill rates
on bison across a wide range of bison densities and therefore did not fit a wolf functional response
for bison.
We fit elk kill rate data to seven a priori models to evaluate the form of wolf functional responses.
The models were categorized into four groups, namely a null model of constant kill rate, preydependent Type II and Type III responses, ratio-dependent Type II and Type III responses, and preyand ratio-dependent Type III responses with two prey. The generalized prey-dependent Type II and
Type III equations (17.1a) and (17.1b), respectively from Holling’s (1959) disk equation were:
aN
1 þ ahN
ð17:1aÞ
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Becker et al.
aN 2
1 þ ahN 2
ð17:1bÞ
where a is the elk attack rate, h is the handling time of a single prey item, and N is elk abundance. While
prey abundance is certainly of essential importance in kill rates, there are a growing number of findings
demonstrating the importance of predator dependence (Reeve 1997, Vucetich et al. 2002, Jost et al.
2005, Schenk et al. 2005, Tschanz et al. 2007). Thus, we used the Type II and Type III ratio-dependent
models (17.2a) and (17.2b), respectively, from Arditi and Ginzburg (1989) denoted as:
aN
P þ ahN
ð17:2aÞ
aN 2
P þ ahN 2
ð17:2bÞ
where P is wolf abundance. Different functional responses can be exhibited for different species in
multiple prey systems (Messier 1995), and indirect effects can exist between prey species sharing a
predator (Holt 1977). Thus, we fit Type III prey-dependent and ratio-dependent functional responses
that incorporated the abundances of both elk and bison. The structures of these models were adapted
from Garrott et al. (2007), with Type III prey-dependent and ratio-dependent equations (17.3a) and
(17.3b) as:
g1 ¼
g1 ¼
aN1
1 þ a N2
hc b m
ðN2 =N1 Þb�1 þ a N1 h
aN1
P þ a N2 hc b m ðN2 =N1 Þb�1 þ a N1 h
ð17:3aÞ
ð17:3bÞ
Where subscripts 1 and 2 denote elk and bison respectively, the proportionality constant, c, measures
‘‘the bias in the predator’s diet to one prey species’’ and relates the ratio of prey eaten to their relative
abundance (Murdoch 1969, p. 337), and b is the extent of prey switching (Chapter 16 by Becker et al.,
this volume). A value of c less than one indicates a preference for that prey, a value of b greater than one
indicates prey switching (Elliott 2004, Greenwood and Elton 1979), and m is the biomass ratio between
bison and elk. We estimated fixed quantities of c and b from prey selection data in the Madison
headwaters to be 0.229 and 2.091, respectively (Chapter 16 by Becker et al., this volume), with m ¼ 2. A
complete explanation of these equations is available in Appendix B.
Determining a priori the appropriate scale at which to evaluate functional responses can be difficult
(Jost et al. 2005). However, we considered the effects of predator abundance and interference reflected
in ratio dependence to be most pronounced on a population scale, where predator abundance was not
pack size but rather wolf numbers. Thus, we fit models on a study area-wide scale. Elk populations
typically declined modestly from fall to spring (Chapter 16 by Becker et al. and Chapter 23 by Garrott
et al., this volume) and therefore division into winter periods of approximately eight weeks each
minimized the possibility of prey depletion bias (Jost et al. 2005). We fitted models and estimated
parameter coefficients using the nls function from the nlme package in R version 2.4.1 (R Development
Core Team 2006). We determined model and predictor weights and employed diagnostics similar to
the multiple linear regression analysis. Based on previous empirical evaluations of wolf functional
responses from Isle Royale (Vucetich et al. 2002, Jost et al. 2005), we predicted that ratio-dependent
models would be more supported than prey-dependent models, and that the elk:wolf ratios would be
positively related to wolf per capita kill rates on elk.
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Wolf Kill Rates: Predictably Variable?
347
III. RESULTS
We detected and followed each radio-collared pack for an average of 35.1 days (95% CI ¼ 31.6, 38.6)
during each winter period, or �66% of the time. Wolves were not in the Madison headwaters study
area during the remainder of the time. We detected 688 ungulates (i.e., 274 elk calves, 276 elk adults,
79 bison calves, 59 bison adults) killed by wolf packs during the winters of 1998–1999 through 2006–
2007. The mean number of kills per period for each pack was 13.9 (range ¼ 2–37, SD ¼ 7.1) and mean
pack size for which we estimated kill rates was 10.4 (range ¼ 3.6–21.1, SD ¼ 4.4). We censored kill rate
estimates from four wolf packs for five predation periods that had inadequate tracking efficiencies due
to wolves spending little time in the study area or poor tracking conditions.
Elk population estimates for the study area ranged from 290–664 in autumn to 174–577 in spring,
with the population decreasing 5–42% during winter. Dramatic changes in elk distribution and
abundance occurred over the course of the study (Chapter 21 by White et al. and Chapter 23 by
Garrott et al., this volume), with a progressive decrease beginning in 2003–2004 and continuing
through 2006–2007. The distribution of elk also changed from approximately equal proportions in
each drainage prior to 1997–1998 to 84% of the population residing in the Madison drainage by the
end of 2006–2007 (Chapter 21 by White et al., this volume). We detected no discernible trend in bison
abundance over the 9-year study. However, pronounced seasonal trends were evident with bison
numbers generally increasing as winter progressed as animals migrated into the study area from the
Hayden and Pelican valleys (Chapter 12 by Bruggeman et al., this volume). The mean numbers of bison
recorded in the study area for the three periods used to estimate kill rates each year ranged from 234 to
1,356. The wolf population increased steadily from seven wolves during the first winter to a peak of
�45 animals in multiple packs during winter 2004–2005, followed by an abrupt decrease in wolf
abundance in the ensuing winters (Chapter 15 by Smith et al., this volume).
We collected kill rate data from four main radio-collared wolf packs during 26 winter predation
periods, for a total of 36 measures of kill rate for each metric during 1998–1999 through 2006–2007.
Total kills/wolf/day ranged from 0.020 to 0.083 (mean ¼ 0.042; 95% CI ¼ 0.037, 0.048), with elk kills/
wolf/day ranging from 0.007 to 0.074 (mean ¼ 0.033; 95% CI ¼ 0.028, 0.039) and bison kills/wolf/day
ranging from 0.000 to 0.045 (mean ¼ 0.009; 95% CI ¼ 0.013, 0.005). Total kills/pack/day ranged from
0.125 to 0.810 (mean ¼ 0.400; 95% CI ¼ 0.346, 0.450), while elk kills/pack/day ranged from 0.070 to
0.810 (mean ¼ 0.320; 95% CI ¼ 0.260, 0.381), and bison kills/pack/day ranged from 0.000 to 0.265
(mean ¼ 0.076; 95% CI ¼ 0.049, 0.103). Total kg/wolf/day ranged from 1.7 to 19.1 (mean ¼ 6.6; 95%
CI ¼ 5.6, 7.7), while elk kg/wolf/day ranged from 1.1 to 9.4 (mean ¼ 4.6; 95% CI ¼ 3.9, 5.4), and bison
kg/wolf/day ranged from 0.0 to 16.3 (mean ¼ 2.0; 95% CI ¼ 1.0, 3.0). Mean total kill rates did not
significantly differ from mean elk kill rates across winter periods for all metrics except the late winter
kg/wolf/day estimates, while mean bison kill rates increased from early to late winter (Figure 17.3).
Mean elk kill rates were similar across all winter periods and were higher than mean bison kill rates for
early and middle winter periods. However, the two kill rates had overlapping confidence intervals for
the late winter period (Figure 17.3). We pooled kill rate data across all wolf packs to estimate total
kills/wolf/day and elk kills/wolf/day for each winter period and winter, providing 26 and nine measures
of kill rates, respectively. Winter range kill rate estimates for each winter exhibited steady decreases,
with the lowest kill rate corresponding to peak wolf abundance in 2004–2005, before sharply increasing
in 2005–2006 and decreasing again in 2006–2007 (Figure 17.4). There was also an inverse relationship
between kill rates and consumption among winters, with the highest carcass consumption and lowest
variance occurring in 2004–2005 when wolf numbers peaked and kill rates were at their lowest, and the
lowest carcass consumption occurring when wolves first established in the system in 1998–1999
(Chapter 15 by Smith et al., this volume; Figure 17.5). Aside from these two extremes, carcass
consumption did not vary substantially among winters.
�348
Becker et al.
A
0.06
Kills/wolf/day
0.05
Total kill rate
Elk kill rate
Bison kill rate
0.04
0.03
0.02
0.01
0.00
Early winter
Mid winter
Late winter
Early winter
Mid winter
Late winter
Early winter
Mid winter
Late winter
B
0.60
Kills/pack/day
0.50
0.40
0.30
0.20
0.10
0.00
C
12.0
Kg/wolf/day
10.0
8.0
6.0
4.0
2.0
0.0
FIGURE 17.3 Kill rate summaries and 95% confidence intervals by winter period (early ¼ 15 November–6 January;
middle ¼ 7 January–27 February; late ¼ 28 February–21 April) for wolves in the Madison headwaters area of
Yellowstone National Park during 1998–1999 through 2006–2007 using the following metrics: (A) kills/wolf/day,
(B) kills/pack/day, and (C) kg/wolf/day.
�Chapter 17
.
349
Wolf Kill Rates: Predictably Variable?
A
0.09
4000
Total kills/wolf/day
Wolf days
0.08
3500
3000
0.06
2500
0.05
2000
0.04
1500
0.03
Estimated wolf days
Total kills/wolf/day
0.07
1000
0.02
500
0.01
0.00
0
1998–99 1999–00 2000–01 2001–02 2002–03 2003–04 2004–05 2005–06 2006–07
B
0.09
1998–99
0.08
Total kills/wolf/day
0.07
2
R = 0.70
0.06
2005–06
2001–02
2006–07
2002–03
1999–00
2000–01
0.05
0.04
2003–04
0.03
2004-05
0.02
0.01
0
0
500
1000
1500
2000
2500
Estimated wolf days
3000
3500
4000
FIGURE 17.4 (A) Observed trends in estimated wolf days and winter kill rates (kills/wolf/day) in the Madison
headwaters area of Yellowstone National Park during winters 1998–1999 through 2006–2007 and (B) the correlation
between the two metrics.
A. Variations in Elk Kill Rate
We fitted 12 a priori models to 36 elk kill rate estimates of kills/wolf/day, kills/pack/day and kg/wolf/
day, respectively, from resident wolf packs across nine winters. Model selection results supported one
top model for both the kills/wolf/day and kills/pack/day metrics, with Akaike model weights (wk) of
0.46 and 0.40, respectively (Table 17.1). The most-supported model structure was identical for both
metrics, consisting of covariates for total elk abundance (ELKall) and pack size (WOLFpack). For both
metrics, all other models had DAICc > 2 and primarily differed in the substitution of elk age class
covariates (ELKad and ELKcalf ) for total elk abundance, though confidence intervals for elk calves
overlapped zero (Table 17.3). Several models also included covariates for bison abundance, wolf
population, and snow pack, but coefficient estimates overlapped zero. Predictor weight for total elk
abundance (ELKall) for kills/wolf/day and kills/pack/day was 0.70 and 0.68, respectively, while the
predictor weight for wolf pack size (WOLFpack) was 0.99 and 0.86, respectively. Elk abundance was
�350
Becker et al.
100
Percent carcass consumption
2004-05
90
2005-06
2006-07
2002-03
2000-01
R2 = 0.58
1999-00
80
2001-02
2003-04
70
1998-99
60
50
0.02
0.03
0.04
0.05
0.06
Total kills/wolf/day
0.07
0.08
0.09
FIGURE 17.5 Relationship between total wolf kill rates and percent of carcass consumption with 95% confidence
intervals for the Madison headwaters area of Yellowstone National Park during winters 1998–1999 through 2006–2007.
Peak carcass consumption occurred in winter 2004–2005 and corresponded to peak wolf numbers, decreasing elk
abundance, and low kill rates. The lowest carcass consumption occurred in winter 1998–1999 when wolves first
became established in the Madison headwaters, elk were most abundant, and kill rates peaked.
significant in all models because models estimating elk abundance by age class accounted for the
remaining predictor weights (0.29 and 0.30 for kills/wolf/day and kills/pack/day, respectively;
Table 17.3).
Model results for kg/wolf/day were less clear, with three most-supported models differing primarily
in whether total elk abundance or age class abundance was used (Table 17.1) and one model including
snow pack (SWEacc). Additional predictors aside from elk abundance and wolf pack size (predictor
weights of 0.66 and 0.97, respectively) contributed little explanatory power (Tables17.1 and 17.3).
Consistent with our predictions, the elk abundance covariates ELKall and Elkad were positively related
to kill rates of all three metrics. Wolf pack size was negatively related to kills/wolf/day and kg/wolf/day,
but positively related to kills/pack/day (Table 17.3). Coefficients for significant top model covariates
were stable and fitting modified and pseudo-threshold forms to prey abundance covariates did not
improve model fits. Substantially more variation was explained by the most-supported model for elk
kills/pack/day than the top models of kills/wolf/day and kg/wolf/day (r2adj of 0.51 vs. 0.37 and 0.36,
respectively).
B. Variations in Bison and Total Kill Rates
We fitted 12 a priori models to 36 bison kill rate estimates of kills/wolf/day, kills/pack/day, and kg/wolf/
day, respectively, from resident wolf packs across nine winters. Model selection results supported four
top models for kills/wolf/day, one top model for kills/pack/day, and two top models for kg/wolf/day
(Table 17.2). Akaike model weights (wk) for the most-supported kills/wolf/day models were SWEacc
(0.29), BISONcalf (0.20), BISONcalf and WOLFpack (0.20), and Elkall and SWEacc (0.18; Table 17.2).
Model weight for the most-supported kills/pack/day model with a single covariate structure, BISONcalf,
was 0.58. Model weights for the most-supported models for kg/wolf/day were BISONcalf (0.50) and
BISONcalf and WOLFpack (0.22; Table 17.2). For kills/wolf/day, confidence intervals for only the
WOLFpack and BISONcalf coefficients did not overlap zero. Similarly, coefficient estimates for kills/
pack/day and kg/pack/day indicated that BISONcalf was the only significant predictor (Table 17.4).
�Chapter 17
.
351
Wolf Kill Rates: Predictably Variable?
TABLE 17.1 A priori model structure and results from top models for multiple linear regression analyses of
elk kill rates by resident wolf packs in the Madison headwaters area of Yellowstone National
Park during 1998–1999 through 2006–2007
Model structure and metric
Elk kills/wolf/day
ELKall þ WOLFpack
ELKad þ ELKcalf þ WOLFpack
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKall þ WOLFpack þ SWEacc
ELKall þ BISONall þ WOLFpack
ELKad þ ELKcalf þ WOLFpack þ SWEacc
ELKall
Elk kills/pack/day
ELKall þ WOLFpack
ELKad þ ELKcalf þ WOLFpack
ELKall þ WOLFpack þ SWEacc
ELKall þ BISONall þ WOLFpack
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKad þ ELKcalf þ WOLFpack þ SWEacc
ELKall
ELKad þ ELKcalf þ SWEacc
ELKad þ ELKcalf þ BISONcalf
ELKall þ SWEacc
ELKad þ ELKcalf
ELKall þ BISONall
Elk kg/wolf/day
ELKall þ WOLFpack
ELKad þ ELKcalf þ WOLFpack
ELKall þ WOLFpack þ SWEacc
ELKall þ BISONall þ WOLFpack
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKad þ ELKcalf þ WOLFpack þ SWEacc
ELKall þ SWEacc
DAICc
wk
r2adj
0.00
2.44
2.53
2.67
2.71
5.31
10.22
0.46
0.13
0.13
0.12
0.12
0.03
0.00
0.37
0.38
0.35
0.35
0.36
0.33
0.12
0.00
2.33
2.64
2.71
2.87
4.47
4.73
5.58
5.64
5.87
6.81
7.20
0.40
0.12
0.11
0.10
0.09
0.04
0.04
0.02
0.02
0.02
0.01
0.01
0.51
0.50
0.49
0.49
0.51
0.49
0.41
0.45
0.45
0.42
0.40
0.40
0.00
1.63
1.74
2.53
2.68
4.36
11.53
0.39
0.17
0.16
0.11
0.10
0.04
0.00
0.36
0.36
0.36
0.34
0.37
0.34
0.11
Covariate codes are total numbers of elk and bison (ELKall,, BISONall), numbers of adult elk and bison (ELKadt, BISONadt),
numbers of calf elk and bison (ELKcalf, BISONcalf ), wolf pack size (WOLFpack), and accumulated snow pack (SWEacc).
Predictor weights for SWEacc and BISONcalf in kills/wolf/day analyses were 0.55 and 0.45, respectively,
while the predictor weight for BISONcalf was 0.88 and 0.92 for kills/pack/day and kg/wolf/day,
respectively. Consistent with our predictions, bison calf abundance and increasing snow pack were
positively related to kill rates of all three metrics (Table 17.4). Coefficients for significant top model
covariates were stable and fitting modified and pseudo-threshold forms to prey abundance covariates
did not improve model fits. The top models for kills/wolf/day and kills/pack/day explained similar
amounts of variation relative to kg/wolf/day (r2adj of 0.40 and 0.39 vs. 0.34, respectively).
Top models for total kill rates primarily reflected the top elk kill rate models with the addition of
SWEacc (Appendix A). There was one most-supported model for kills/wolf/day with a weight of 0.73
and the structure of ELKall, WOLFpack, and SWEacc. There were three top-ranking models for kills/
pack/day that differed in their inclusion of SWEacc and the covariates for elk abundance, with model
weights of 0.32, 0.19, and 0.17, respectively. The kg/wolf/day analysis supported an identical top model
to the analysis for kills/wolf/day, with a weight of 0.61. All predictors for these two analyses had
coefficient estimates with confidence intervals that did not overlap zero, while confidence intervals for
ELKcalf and SWEacc coefficient estimates in the kills/pack/day analysis included zero (Appendix A). Elk
abundance and SWEacc were positively related to kill rates of all metrics. Pack size was positively related
to kills/pack/day and negatively related to kills/wolf/day and kg/wolf/day (Appendix A).
�352
Becker et al.
TABLE 17.2 A priori model structure and results from top models for multiple linear regression analyses of
bison kill rates by resident wolf packs in the Madison headwaters area of Yellowstone National
Park during 1998–1999 through 2006–2007
Model structure and metric
Bison kills/wolf/day
SWEacc
BISONcalf
BISONcalf þ WOLFpack
ELKall þ SWEacc
ELKall þ SWEacc þ WOLFpack
ELKad þ ELKcalf þ BISONcalf
ELKad þ ELKcalf þ SWEacc þ WOLFpack
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
BISONall þ WOLFpack
Bison kills/pack/day
BISONcalf
BISONcalf þ WOLFpack
ELKad þ ELKcalf þ BISONcalf
SWEacc
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKall þ SWEacc þ WOLFpack
ELKall þ SWEacc
ELKad þ ELKcalf þ SWEacc þ WOLFpack
BISONall
Bison kg/wolf/day
BISONcalf
BISONcalf þ WOLFpack
ELKad þ ELKcalf þ BISONcalf
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKall þ SWEacc
SWEacc
ELKall þ SWEacc þ WOLFpack
BISONall
DAICc
wk
r2adj
0.00
0.72
0.73
0.89
3.52
4.29
4.87
5.89
12.01
0.29
0.20
0.20
0.18
0.05
0.03
0.03
0.02
0.00
0.39
0.38
0.40
0.40
0.38
0.37
0.39
0.37
0.18
0.00
2.31
3.88
5.42
5.53
5.70
6.26
8.12
10.09
0.58
0.18
0.08
0.04
0.04
0.03
0.03
0.01
0.00
0.39
0.37
0.37
0.29
0.37
0.34
0.30
0.33
0.19
0.00
1.59
2.21
5.06
5.67
5.92
8.35
9.81
0.50
0.22
0.16
0.04
0.03
0.03
0.01
0.00
0.34
0.34
0.35
0.33
0.26
0.22
0.23
0.13
Covariate codes are defined in Table 17.1.
C. Functional Response
Fitting seven functional response models to 26 pooled wolf kill rate estimates for each winter period
yielded overwhelming support for the Type II ratio-dependent model (wk ¼ 0.70), and ratiodependent models comprised 0.92 of the model weights (Table 17.5). Coefficient values for estimated
attack rate (a) and handling time (h) in the top model were 0.002 (95% CI ¼ 0.001, 0.004) and 13.9
days (95% CI ¼ 7.9, 19.9), respectively. The predicted functional response from the most-supported
model increased rapidly at low elk:wolf ratios before gradually approaching an asymptote of approximately 0.058 kills/wolf/day (Figure 17.6A). One value in the data appeared to be an extreme outlier that
could potentially influence the asymptotic value and model results (Figure 17.6A). Thus, we removed
this data point and refit all models during an exploratory analysis. The asymptotic value decreased to
0.048 kills/wolf/day, coefficient values for attack rate and handling time changed to 0.004 (95% CI ¼
0.001, 0.007) and 18.79 (95% CI ¼ 13.8, 23.8), respectively, and model selection results remained
unchanged. The Type II functional response was also the most-supported prey-dependent model, but
overall had little support and no clear asymptote as elk abundance increased (Figure 17.6B). The twoprey functional response models for both prey-dependent and ratio-dependent models were not
supported by the data.
�TABLE 17.3 Coefficient values (Bi), lower and upper 95% confidence intervals (in parentheses), and predictor weights (wp) for the best approximating models
for each kill rate metric identified through AIC model comparison techniques for elk kill rates by resident wolf packs in the Madison headwaters
area of Yellowstone National Park during 1998–1999 through 2006–2007
Metric and model
Elk kills/wolf/day
Predictor weight
ELKall þ WOLFpack
Elk kills/pack/day
Predictor weight
ELKall þ WOLFpack
Elk kg/wolf/day
Predictor weight
ELKall þ WOLFpack
ELKadt þ ELKcalf þ
WOLFpack
ELKall þ WOLFpack
þ
SWEacc
ELKall
ELKadt
ELKcalf
BISONall
BISONcalf
WOLFpack
SWEacc
0.70
0.020
(0.011, 0.029)
0.29
0.29
0.12
0.13
0.99
�0.019
(�0.028, �0.009)
0.15
0.68
0.161
(0.115, 0.207)
0.30
0.30
0.11
0.11
0.86
0.131 (0.037, 0.225)
0.19
0.66
2.19
(0.97, 3.41)
0.31
0.31
0.11
0.10
0.20
2.19
(0.82, 3.56)
0.06
(�1.57, 1.68)
0.97
�2.87
(�4.18, �1.56)
�2.91
(�4.22, �1.60)
�2.62
(�4.03, �1.21)
2.20
(0.98, 3.40)
Boldface type indicates confidence intervals do not span zero. Covariate codes are defined in Table 17.1.
0.67
(�0.73, 2.06)
�TABLE 17.4 Coefficient values (Bi), lower and upper 95% confidence intervals (in parentheses), and predictor weights (wp) for the best approximating models
for each kill rate metric identified through AIC model comparison techniques for bison kill rates by resident wolf packs in the Madison headwaters
area of Yellowstone National Park during 1998–1999 through 2006–2007
Covariate
Metric and model
Bison kills/wolf/day
Predictor weight
SWEacc
BISONcalf
BISONcalf þ
WOLFpack
ELKall þ SWEacc
ELKall
0.23
ELKadt
ELKcalf
0.08
0.08
BISONall
0.00
�0.003
(�0.009, 0.003)
BISONcalf
WOLFpack
0.45
0.30
0.013
(0.007, 0.019)
0.011
(0.005, 0.017)
�0.005
(�0.011, 0.001)
SWEacc
0.55
0.015 (0.009, 0.021)
0.015 (0.009, 0.021)
Bison kills/pack/day
Predictor weight
BISONcalf
0.07
0.13
0.13
0.00
0.88
0.094
(0.055, 0.133)
0.26
0.11
Bison kg/wolf/day
Predictor weight
BISONcalf
0.04
0.20
0.20
0.00
0.92
3.44
(1.89, 4.99)
3.23
(1.62, 4.84)
0.27
0.07
BISONcalf þ
WOLFpack
Boldface type indicates confidence intervals do not span zero. Covariate codes are defined in Appendix A, Table 1.
�0.85
(�2.61, 0.92)
�Chapter 17
A
.
355
Wolf Kill Rates: Predictably Variable?
0.14
Observed
0.12
Elk kills/wolf/day
Predicted
0.1
0.08
0.06
0.04
0.02
0
0
B
40
20
60
80
Elk:Wolf ratio
100
300
400
Elk abundance
500
120
140
0.14
Observed
0.12
Predicted
Elk kills/wolf/day
0.1
0.08
0.06
0.04
0.02
0
0
100
200
600
700
FIGURE 17.6 Predicted and observed functional response curves for elk from wolf predation data in the Madison
headwaters area of Yellowstone National Park during winters 1998–1999 to 2006–2007, including (A) a Type II ratiodependent curve and (B) a Type II prey-dependent curve.
TABLE 17.5 Results from functional response analyses of wolf kill rates on elk during 1998–1999 through
2006–2007 in the Madison headwaters area of Yellowstone National Park
Model
Structure
DAICc
wk
Type II ratio-dependent
Type III ratio-dependent
aN
p þ ahN
aN 2
p þ ahN 2
aN
1 þ ahN
aN 2
1 þ ahN 2
0.00
2.49
0.70
0.20
6.47
6.94
7.00
0.03
0.02
0.01
7.88
0.01
8.27
0.01
Type II prey-dependent
Type III prey-dependent
Two-prey prey dependent Type III
Two-prey ratio dependent Type III
Constant
1
g1 ¼ 1 þ aN h c b mðNaN=N
Þb�1 þ aN h
2
2
1
1
2
2
1
1
1
g1 ¼ P þ aN hc b mðNaN=N
Þb�1 þ aN h
a
�356
Becker et al.
IV. DISCUSSION
Adding a top predator to an ecosystem can result in profound demographic, spatial, and behavioral
changes in prey and predator populations (Taylor 1984, Berger et al. 2001). Evaluating these dynamics
requires descriptions of a predator’s per capita consumption for a particular prey (Abrams and
Ginzburg 2000). Thus, the functional response is a critical component embedded in virtually every
predator–prey model. While prey abundance is essential for these descriptions, the influence of
predator abundance on kill rates and functional responses is controversial because prey-dependent
and predator-dependent models often make considerably different predictions about ecosystem
dynamics (Abrams and Ginzburg 2000). We demonstrated that factors driving variation in wolf kill
rates on elk and bison in the Madison headwaters area of Yellowstone differed between prey species. Kill
rates on elk were primarily influenced by elk abundance and wolf pack size, while kill rates on bison were
primarily influenced by the abundance of bison calves and snow pack severity. The form of the wolf
functional response for elk was strongly Type II ratio-dependent, further supporting the importance of
satiation and predator dependence in wolf–ungulate systems (Vucetich et al. 2002, Jost et al. 2005).
Elk were the preferred and primary prey for wolves in the Madison headwaters area, even though
bison were more abundant during winter (Chapter 16 by Becker et al., this volume). Thus, elk
abundance significantly influenced variations in both total kill rates and kill rates on elk, similar to
findings from other multiple-prey systems where elk were the primary prey (Hebblewhite et al. 2003).
Discriminating between adult and calf elk abundance did not improve model fit relative to overall elk
abundance, even though wolves preferred elk calves and calves were consumed faster due to their
smaller size. The significance of total elk abundance and adult elk abundance in explaining kill rate
variation on elk was likely due to an overall decrease in the elk numbers during the latter years of the
study, which included a substantial increase in adult mortality (Chapter 23 by Garrott et al., this
volume). Calf survival was strongly affected by wolves (Chapter 23 by Garrott et al., this volume) and
typically decreased through winter due to starvation mortality prior to wolf recolonization (Chapter 11
by Garrott et al., this volume), perhaps explaining why calf abundance was a poor predictor of kill rate
variation on elk.
The Madison headwaters supported a two-prey system, making complex, indirect effects possible
between prey that share a predator (Chapter 24 by Garrott et al., this volume). However, the abundance
of alternative bison prey explained little variation in wolf kill rates on elk. Rather than decreasing elk
kill rates, increasing bison abundance, particularly calves, was correlated with increased bison kill rates
and total kill rates (Table 17.1; Figure 17.1). Similarly, accumulated snow pack did not appear to
explain variation in wolf kill rates on elk in our study system, contrary to findings from other studies
(Mech et al. 2001, Hebblewhite et al. 2002).
We detected a significant relationship between pack size and kill rates across all metrics for elk kill
rates, consistent with findings from other studies (Thurber and Peterson 1993, Schmidt and Mech
1997, Hayes et al. 2000). Larger packs killed more frequently than smaller packs and per capita kills and
gross food availability decreased with increasing numbers of wolves in a pack. However, the importance
of pack size on kill rates differs across systems. For example, in linear regression analyses, Hayes et al.
(2000) found that variability in kill rates was best explained by pack size, while in multiple regression
analyses Je˛drzejewski et al. (2002) found it explained little variance relative to snow cover, though the
size and range of pack sizes in their study was limited. In terms of total biomass acquired per wolf per
day, our estimates were well above the minimum estimated intake of 1.6 kg/wolf/day (Mech 1970),
particularly in late winter when kg/wolf/day was significantly higher due to the increased predation on
bison. However, these estimates certainly reflect maximum intake because we did not account for
scavenger loss or incomplete consumption (Figure 17.5). While smaller packs had a higher kg/wolf/day
kill rate, the net disparity between large and small packs may not have been great given that larger packs
�Chapter 17
.
Wolf Kill Rates: Predictably Variable?
357
are less prone to scavenger loss (Vucetich et al. 2004) and grizzly bears were common at wolf kills in
spring and frequently usurped wolf kills (Ballard et al. 2003, Garrott unpublished data). Thus, while
larger packs had lower gross per capita biomass at kills, they also were likely more effective at avoiding
scavenger loss and increasing food acquisition (Vucetich et al. 2004) such that the difference in food
intake was not as pronounced. Alternatively, smaller packs may have consumed less food at higher
kill rates.
In contrast to kill rates on elk, wolf pack size poorly explained variation in kill rates on bison because
larger packs did not kill bison more frequently. Kill rate variation on bison was best explained by the
abundance of bison calves or by snow pack severity, but was not significantly affected by elk abundance.
Due to the strong correlation between increasing snow pack and bison migration into the study system
(Chapter 12 by Bruggeman et al., this volume), we were unable to distinguish between the respective
influences of snow pack and bison calf abundance. Wolves strongly selected for bison calves and
predation coincided with increases in bison abundance and bison abundance relative to elk. Bison
predation typically occurred in late winter when ungulates were likely in their worst physical condition
due to prolonged nutritional deficits (Chapter 12 by Bruggeman et al. and Chapter 16 by Becker et al.,
this volume). Bison are considerably more formidable prey than elk in both defenses and anti-predator
behaviors (MacNulty et al. 2007; Chapter 16 by Becker et al., this volume). Thus, wolf predation was
largely opportunistic and primarily occurred when bison vulnerability increased late winter. Social
carnivores typically take larger prey with larger foraging groups (Rosenzweig 1966, Gittleman 1989,
Creel and Creel 1995). However, bison selection by wolves in the Madison headwaters was negatively
related to pack size because the largest packs (22 wolves) selected primarily elk, while smaller packs
(six wolves) killed the most bison (Chapter 16 by Becker et al., this volume). This likely does not reflect
an opposite trend from that observed in other systems so much as favorable conditions for killing bison
occurred for these packs. In the more severe winters many bison were in very poor nutritional
condition and therefore very vulnerable to wolf predation (Chapter 16 by Becker et al., this volume;
Figure 16.12), and this vulnerability was likely not substantially increased with more wolves in a pack.
In addition several of the largest packs occurred in winters when elk were abundant and widely
distributed throughout the system and wolf competition for them was low. An alternative explanation
for this relationship is that particular packs learned how to kill bison more efficiently, which would also
affect kill rates but would not be reflected in pack size. While we acknowledge that learning can assume
an important role in predation, virtually all wolf packs had experience in killing bison and specific wolf
packs did not exhibit a constant increase in bison kills as would be expected if they were simply
improving their efficiency. For example, the Gibbon pack killed 63% bison in winter 2005–2006, but
only 15% in 2006–2007 (Chapter 16 by Becker et al., this volume).
Approximately 0.45–0.57 of the variation in total wolf kill rate was explained by the abundance of
preferred prey, wolf pack size, and snow pack severity. While all of these factors have been identified as
having important influences on wolf kill rates, we are not aware of other studies that conducted
separate evaluations of factors affecting kill rates for multiple prey species. Nevertheless the mostsupported models across all kill rate metrics for elk and bison explained 0.36–0.51 of the respective kill
rate variation for each species (Tables 17.1 and 17.2), leaving a substantial amount of kill rate variation
unexplained. This finding likely reflects the complexities of a multiple-prey system wherein species
differed substantially in their relative vulnerability to wolves and vulnerability differed among age
classes (Chapter 16 by Becker et al., this volume). Also, there were complex interactions between
heterogeneous landscapes, climate, and multiple, overlapping packs that undoubtedly contributed to
variations in kill rate. In addition to these complexities, the appropriate scale at which to analyze kill
rates and functional response is often unclear. Strong arguments can be made for using a per-pack
scale, whereby kill rates and prey abundance are estimated for each pack’s respective territory. However,
kill rates could also be evaluated at a ‘‘mixed’’ scale, where kill rates are estimated for each pack but prey
abundance is estimated for the entire system, or at a study area scale, where kill rates and prey
abundance are estimated for the entire wolf and ungulate populations (Jost et al. 2005). In exploratory
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Becker et al.
analyses, we evaluated kill rates on the per-pack scale and results were quite similar to the study-area
scale analyses presented in this chapter. However, the per-pack scale models explained substantially less
variation in kill rates with less distinction between models, possibly because much of the variation was
related to decreases in elk abundance in the system during the latter years. Thus, estimating prey
abundance at the study-area scale served as an umbrella for describing these influences. In addition, the
study-area scale was appropriate for evaluating the shape of the functional response given that it is a
description of the average per capita consumption rate of wolves. Nevertheless, the potential for
differing results at differing scales should be heeded.
It is essential to avoid biases in the detection efficiency of kills when evaluating variation in kill rates
and the subsequent shape of the functional response. For example, one potential reason for the effect of
pack size on per capita kill rates is purely methodological in that larger packs can consume prey faster
and potentially make kills more difficult to detect (Mech and Peterson 2003). Kill detection probability
can also be subject to substantial variation in observer effort, weather conditions, and wolf movement,
which can translate into substantial differences in kill detection both within and among studies
(Jaffe 2001, Hebblewhite et al. 2003, Smith et al. 2004). We were able to obtain accurate estimates of
wolf kill rates by evaluating wolf kill rates in a relatively small, tractable area defined by a high density
ungulate winter range and employing intensive ground-based monitoring and tracking on a daily basis
through each winter. Jaffe (2001) evaluated kill detection efficiency in the Madison headwaters and
determined that the methods were effective in detecting at least 75% of the kills made by wolf packs in
the study area, and subsequent estimates have improved efficiency to �85% (Garrott unpublished
data). No systematic biases were detected across prey types or pack sizes that would indicate inaccurate
estimates.
Similar to other studies of functional responses, wolves exhibited a Type II curve for elk (Dale et al.
1994, Hayes and Harestad 2000, Vucetich et al. 2002). One of the primary ways an asymptotic, Type II
functional response can arise is through predator satiation. However, this response may also be more
likely to occur if a prey item is preferred (Holling 1959, Messier 1995). If a prey item is preferred
relative to an alternative prey, then the functional response can be destabilizing (Eubanks and Denno
2000). Thus, the incorporation of a Type III functional response for elk in modeling wolf predation for
Greater Yellowstone systems (Boyce 1993, 1995; Varley and Boyce 2006) may underestimate the effects
of wolf predation on a preferred prey if alternative prey are considerably less vulnerable (Chapter 24 by
Garrott et al., this volume). Alternatively, recent investigations advocate that a Type II response can be
stabilizing with social predators and prey (Fryxell et al. 2007). However, the dynamic nature of elk
grouping strategies on fine and coarse temporal scales in response to variability in predation risk and
habitat in this system (Chapter 19 by Gower et al., this volume) make application of this idea difficult.
Using data from Dale et al. (1994) and Messier (1991, 1994), Eberhardt (1997) demonstrated that
wolf Type II functional response curves for moose and caribou increased rapidly before reaching
asymptotic values at approximately 0.021 and 0.089 kills/wolf/day, respectively, across a wide range of
prey densities.
The asymptotic value for elk functional response curves in the Madison headwaters was �0.058
kills/wolf/day, similar to the mean total kill rates (90% of which were elk) reported by Smith et al.
(2004) of 0.061 and 0.068 kills/wolf/day for wolf packs elsewhere in Yellowstone during winters 1995–
1996 through 1999–2000. When the asymptotic value for wolf functional responses on elk in the
Madison headwaters and caribou from Dale et al. (1994) are converted to moose equivalents (one
moose equivalent to two elk or three caribou; Keith 1983), the resultant values are both 0.029 moose/
wolf/day respectively, remarkably similar to the value calculated by Eberhardt (1997) for moose. Such
consistency suggests a relatively uniform asymptotic wolf kill rate across a wide variety of wolf–
ungulate systems and ungulate densities (Eberhardt 1997, Eberhardt et al. 2003). In the Madison
headwaters, the functional response curve appears relatively constant across a wide range of values
aside from an outlier (where wolves had a very high kill rate in the first winter of their establishment in
the study area) and the three lowest values. High initial kill rates occurring on a naı̈ve prey base and the
�Chapter 17
.
Wolf Kill Rates: Predictably Variable?
359
extremely low elk:wolf ratios we recorded at peak wolf abundance (approximately one wolf per 10 elk)
possibly represented transitory extremes in the system. Given the potential for a continuing decrease in
elk abundance (Chapter 24 by Garrott et al., this volume), additional estimates at very low elk:wolf
ratios may be possible.
The multiple-species dependent models incorporating elk and bison abundance in describing elk
functional response were not supported likely due to a variety of reasons. While it is possible that the
model structure is inappropriate for these multiple-species interactions, bison did not appreciably
influence variation in elk kill rates due to differences in vulnerability between the two prey species.
Therefore, application of these models may be more appropriate in multiple prey systems where prey
species do not differ so substantially (Chapter 24 by Garrott et al., this volume). Alternatively, given
that the Madison headwaters is still a developing two-prey system, the lack of fit may simply be due to a
strong wolf preference for elk, with increased bison predation only under circumstances such as severe
winters and high bison:elk ratios (Chapter 16 by Becker et al., this volume); therefore additional
estimates with continued decreases in elk abundance and increases in bison:elk ratios may provide
better fits as wolves may increasingly kill bison (Chapter 24 by Garrott et al., this volume). Regardless,
we strongly advocate the continued development of multiple-species dependent functional response
models to describe multiple-prey systems (Garrott et al. 2007, Tschanz et al. 2007).
Ratio-dependent models describe the functional response well for numerous predators and parasitoids (Arditi and Akcakaya 1990), and are supported by controlled experiments in natural settings
(Reeve 1997) and field studies (Vucetich et al. 2002, Jost et al. 2005). The extensive long-term kill rate
data from the Isle Royale wolf–moose system strongly indicated that the functional response for moose
was best described by ratio dependence, and our investigations in a multiple prey system yielded similar
results. Ratio dependence can arise through a variety of factors, including direct and hostile interference among predators, non-random foraging, the presence of prey refugia, changes in prey behavior
resulting in less vulnerable prey with increasing predators, and differential vulnerability among the
prey population (Charnov et al. 1976, Hassell 1978, Arditi and Ginzburg 1989, Abrams 1994). Elk were
considerably more vulnerable than bison and preferred by wolves during our study. There was also
differential vulnerability of prey across age classes (Chapter 16 by Becker et al., this volume).
In addition, wolves were territorial and inter-pack strife was the major cause of mortality (Chapter
15 by Smith et al., this volume). Wolves had specific patterns of foraging and aggregated disproportionately in the Madison headwaters relative to the rest of their territories (Chapter 15 by Smith et al.,
this volume) to target areas of high elk vulnerability (Bergman et al. 2006). Furthermore, landscape
characteristics apparently created refugia in certain areas that contributed to a large-scale change in elk
distribution within our study system (Chapter 21 by White et al., this volume). Consequently, increases
in wolf numbers and decreases in the elk:wolf ratios negatively affected elk kill rates by increasing
competition and intra-specific strife between packs and increasing anti-predator behaviors by elk
(Chapters 18–20 by Gower et al., this volume).
While there is a vast literature on anti-predator behaviors in prey (Caro 2005), less is known about
the effectiveness of this decision-making in actually reducing predation (Lima 2002), and elk demonstrate a variety of anti-predator responses to wolves (Hebblewhite and Pletscher 2002, Creel et al. 2005,
Creel and Winnie 2005, Gude et al. 2006, Hebblewhite and Merrill 2007). We detected a strong
correlation between mean winter group size for elk and wolf abundance, as well as an increase in elk
group size variance with increasing wolf abundance, which we interpret as an elk behavioral response to
increasing predation risk (Chapter 19 by Gower et al., this volume). In addition, substantial changes in
elk abundance, recruitment, and distribution occurred during the study period (Chapter 21 by White
et al. and Chapter 23 by Garrott et al., this volume), such that in the latter winters elk became more
concentrated and predictable in areas that apparently provided refuge and escape habitat (Chapter 21
by White et al., this volume). If an elk group is the unit of encounter rather than an individual
(Huggard 1993), and herd sizes are variable on fine temporal scales as elk respond to immediate
wolf threats by grouping and using escape terrain (Chapter 19 by Gower et al., Chapter 21 by
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Becker et al.
White et al., and Chapter 24 by Garrott et al., this volume), then we would expect kill rates to decrease
with increased wolf numbers due to prey depression, as fewer vulnerable individuals would be available
despite being relatively predictable in their locations. There was a negative correlation between
mean winter group size for elk and winter wolf kill rates on elk (R2 ¼ 0.66; Figure 17.7), indicating
that these behavioral responses may have been effective at reducing predation risk. Whether these
adjustments were part of the transitory dynamics of a newly-established system with prey adapting
to the novel presence of a top predator or whether such plasticity in prey responses can be expected
as the system continues to develop is unknown. Regardless, wolf kill rates on large herbivores in
the Madison headwaters area were likely strongly dependent on the physical, behavioral, and environmental vulnerability of their prey (Chapter 24 by Garrott et al., this volume), in addition to encounter
rates.
A
16.0
14.0
0.070
12.0
0.060
10.0
0.050
8.0
0.040
6.0
0.030
4.0
0.020
2.0
0.010
Elk kills/wolf/day
Mean elk group size
0.080
Mean elk group size
Elk kills/wolf/day
0.000
0.0
1998-99 1999-00 2000-01 2001-02 2002-03 2003-04 2004-05 2005-06 2006-07
B
0.080
1998–99
Elk kills/wolf/day
0.060
1999–00
2001–02
2002–03
2005–06
0.040
2006–07
2000–01
R2 = 0.66
2003–04
0.020
2004–05
0.000
5.0
7.0
9.0
11.0
Mean elk group size
13.0
15.0
FIGURE 17.7 (A) Observed trends in mean winter elk group size (Chapter 19 by Gower et al., this volume) and winter
wolf kill rates (kills/wolf/day) on elk in the Madison headwaters area of Yellowstone National Park during 1998–1999
through 2006–2007 and (B) the correlation between the two metrics.
�Chapter 17
.
Wolf Kill Rates: Predictably Variable?
361
FIGURE 17.8 Members of the Hayden pack scavenging on an adult, female bison carcass that was repeatedly
revisited by wolves. Increased wolf use of the Madison headwaters area was strongly correlated with decreased kill rates
and increased carcass consumption. Scavenging and revisitation of old kills also appeared to increase with decreased
elk abundance and increased wolf use of the system (Photo by Shana Dunkley).
Though it is extremely difficult, if not impossible, to disentangle the respective influences of all these
different factors into models of predator–prey interactions, we concur with other investigators that
ratio-dependence is a parsimonious means of describing the effects of predator density on per capita
consumption rates (Jost et al. 2005; Figure 17.8). Precise ratio-dependence or prey dependence is likely
rare in nature, and can change within systems (Abrams and Ginzburg 2000, Schenk et al. 2005, Tschanz
et al. 2007). Thus, further studies are necessary before generalizations can be made. However, our
evaluation of a newly-established large mammal predator–prey system further corroborates the
importance of considering predator population density in understanding the nature of predation.
V. SUMMARY
1.
2.
3.
We estimated wolf kill rates in a tractable and newly established wolf–elk–bison system in the
Madison headwaters area of Yellowstone National Park during winters 1998–1999 to 2006–2007 to
document the transition from more than seven decades without wolves to a well-established, top
predator population.
Multiple regression analyses of 501 elk and bison kills made by the four primary packs in the study
system indicated that variations in kill rates (kills/wolf/day, kills/pack/day, and kg/wolf/day) of elk,
the preferred prey, were positively related to elk abundance, negatively related to pack size for
per capita kill rates, and positively related to pack size for kills/pack/day.
Variations in kill rates of bison were positively related to snow pack and bison calf abundance.
Increases in bison kill rates were not related to reductions in elk kill rates, but simply served to
increase total kill rates.
�362
4.
5.
6.
7.
Becker et al.
Elk abundance was a poor single predictor of variations in wolf kill rates and describing the shape
of the functional response. Also, the duration and accumulation of snow pack did not significantly
influence wolf kill rates on elk due to a concurrent, progressive increase in the numbers of
nutritionally-deprived bison migrating into the area from higher elevations.
The functional response of wolves for elk was best described by a Type II ratio-dependent model
that rapidly increased at low elk:wolf ratios before approaching an asymptote at approximately
0.058 kills/wolf/day across a wide range of elk:wolf ratios.
Potential mechanisms generating ratio-dependence in the Madison headwaters were likely
substantial increases in wolf abundance concurrent with elk responses resulting in distribution,
abundance, and behavioral changes.
Our findings further suggest the influence of predator abundance as described by ratiodependence is important for understanding the nature of predation.
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Wolf Kill Rates: Predictably Variable?
365
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APPENDIX
Appendix A. A priori Model Lists for Evaluating Kill Rate Variation
Total and Elk Kill Rates (Kill Rate is Kills/Wolf/Day, Kills/Pack/Day, or kg/Wolf/Day)
S1. Kill Rates � ELKall
S2. Kill Rate � ELKall þ SWEacc
S3. Kill Rate � ELKall þ BISONall
S4. Kill Rate � ELKall þ WOLFpack
S5. Kill Rate � ELKall þ WOLFpack þ SWEacc
S6. Kill Rate � ELKall þ BISONall þ WOLFpack
S7. Kill Rate � ELKadt þ ELKcalf
S8. Kill Rate � ELKadt þ ELKcalf þ SWEacc
S9. Kill Rate � ELKadt þ ELKcalf þ BISONcalf
S10. Kill Rate � ELKadt þ ELKcalf þ WOLFpack
S11. Kill Rate � ELKadt þ ELKcalf þ WOLFpack þ SWEacc
S12. Kill Rate � ELKadt þ ELKcalf þ BISONcalf þ WOLFpack
Bison Kill Rates (Kill Rate is Kills/Wolf/Day, Kills/Pack/Day, or kg/Wolf/Day)
S1. Kill Rate � BISONall
S2. Kill Rate � SWEacc
S3. Kill Rate � ELKall þ BISONall
S4. Kill Rate � BISONall þ WOLFpack
S5. Kill Rate � ELKall þ BISONall þ WOLFpack
S6. Kill Rate � ELKall þ SWEacc þ WOLFpack
S7. Kill Rate � ELKall þ SWEacc
S8. Kill Rate � BISONcalf
S9. Kill Rate � ELKad þ ELKcalf þ BISONcalf
S10. Kill Rate � BISONcalf þ WOLFpack
S11. Kill Rate � ELKadt þ ELKcalf þ BISONcalf þ WOLFpack
S12. Kill Rate � ELKadt þ ELKcalf þ SWEacc þ WOLFpack
�366
Becker et al.
APPENDIX 17A.1 Predictive table for multiple regression analyses of wolf kill rate variation
Covariate
Kill rate and
metric
Elk
Kills/wolf/day
Kills/pack/day
kg/wolf/day
Bison
Kills/wolf/day
Kills/pack/day
kg/wolf/day
Total
Kills/wolf/day
Kills/pack/day
kg/wolf/day
ELKall
ELKadt
ELKcalf
BISONall
BISONadt
BISONcalf
WOLFpack
SWEacc
þ
þ
þ
þ
þ
þ
þ
þ
þ
�
�
�
�
�
�
�
�
�
�
þ
�
þ
þ
þ
�
�
�
�
�
�
�
�
�
þ
þ
þ
þ
þ
þ
þ
þ
þ
�
þ
�
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
�
þ
�
þ
þ
þ
Covariate codes are total elk and bison abundance (ELKall,, BISONall), abundance of elk and bison adults (ELKadt, BISONadt) and
calves (ELKcalf, BISONcalf ), wolf pack size (WOLFpack), and accumulated snowpack (SWEacc).
APPENDIX 17A.2 A priori model structure and results from top models for multiple linear regression
analyses of total wolf kill rates by resident wolf packs in the Madison headwaters area
of Yellowstone National Park during 1998–1999 through 2006–2007
Model Structure and Metric
Total kills/wolf/day
ELKall þ WOLFpack þ SWEacc
ELKad þ ELKcalf þ WOLFpack þ SWEacc
ELKad þ ELKcalf þ WOLFpack
ELKall þ WOLFpack
ELKall þ BISONall þ WOLFpack
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKall þ SWEacc
Total kills/pack/day
ELKad þ ELKcalf þ WOLFpack
ELKall þ WOLFpack
ELKall þ WOLFpack þ SWEacc
ELKall þ BISONall þ WOLFpack
ELKad þ ELKcalf þ WOLFpack þ SWEacc
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKad þ ELKcalf
ELKall
ELKad þ ELKcalf þ SWEacc
ELKad þ ELKcalf þ BISONcalf
Total kg/wolf/day
ELKall þ WOLFpack þ SWEacc
ELKad þ ELKcalf þ WOLFpack þ SWEacc
ELKad þ ELKcalf þ BISONcalf þ WOLFpack
ELKall þ SWEacc
ELKad þ ELKcalf þ SWEacc
ELKall þ BISONall þ WOLFpack
ELKad þ ELKcalf þ WOLFpack
ELKall þ WOLFpack
ELKad þ ELKcalf þ BISONcalf
Covariate codes are defined in Table17.1.
DAICc
wk
r2adj
0.00
2.95
5.88
6.08
7.18
7.93
15.14
0.73
0.17
0.04
0.03
0.02
0.01
0.00
0.57
0.55
0.49
0.47
0.47
0.49
0.31
0.00
1.05
1.22
2.09
2.64
2.71
6.01
6.95
7.76
8.64
0.32
0.19
0.17
0.11
0.08
0.08
0.02
0.01
0.01
0.00
0.53
0.50
0.52
0.50
0.52
0.52
0.42
0.38
0.42
0.40
0.00
2.66
3.99
4.35
5.34
7.52
8.91
9.08
12.89
0.61
0.16
0.08
0.07
0.04
0.01
0.01
0.01
0.00
0.45
0.44
0.42
0.35
0.37
0.33
0.30
0.26
0.22
�Chapter 17
.
Metric and model
Total kills/wolf/day
Predictor weight
ELKall þ WOLFpack þ SWEacc
Total kills/pack/day
Predictor weight
ELKadt þ ELKcalf þ WOLFpack
ELKall þ WOLFpack
ELKall þ WOLFpack þ SWEacc
Total kg/wolf/day
Predictor weight
ELKall þ WOLFpack þ SWEacc
ELKall
ELKadt
ELKcalf
BISONall
BISONcalf
WOLFpack
SWEacc
0.78
0.018
(0.012, 0.023)
0.22
0.22
0.02
0.01
1.00
�0.020
(�0.029,
�0.012)
0.90
0.013
(0.005, 0.022)
0.48
0.51
0.160
(0.074,
0.246)
0.51
�0.043
(�0.145,
0.059)
0.11
0.08
0.95
0.124
(0.042, 0.205)
0.26
0.130
(0.051, 0.209)
0.131
(0.054, 0.209)
0.70
0.85
(�0.86, 2.56)
0.128
(0.043, 0.213)
0.154
(0.064, 0.243)
0.29
0.29
0.01
0.08
0.88
�2.65
(�4.62,
�0.68)
Wolf Kill Rates: Predictably Variable?
APPENDIX 17A.3 Coefficient values (Bi), lower and upper 95% confidence intervals (in parentheses), and predictor weights (wp) for the best approximating
models for each kill rate metric identified through AIC model comparison techniques for total kill rates by resident wolf packs in the Madison
headwaters area of Yellowstone National Park during 1998–1999 through 2006–2007
0.069
(�0.019,
0.158)
0.88
3.51 (1.56, 5.47)
Boldface type indicates confidence intervals do not span zero. Covariate codes are defined in Table17.1.
367
�368
Becker et al.
Appendix B. Description of Multiple Species Functional Response Models
We evaluate the relative availability of the bison and elk in the study system with the ratio of the two
prey species in the wolves’ diet. Murdoch (1969) provides the classic equation that relates the ratio of
two prey types eaten by a predator (g1/g2) to the ratio of the prey types available to the predator
(N1/N2):
�
�b
g1
N1
¼ c
ðA:1Þ
g2
N2
where subscripts 1 and 2 correspond to prey types 1 and 2, respectively; g is the functional response
(prey killed per predator per day); N is the number of prey available, and c is a selection coefficient
(Murdoch’s ‘‘proportionality constant’’) that measures the ‘‘bias in the predator’s diet to one prey
species.’’ If c ¼ 1, then there is no bias and the predator kills the two prey types in proportion to their
availability. If c > 1, the predator kills prey type 1 disproportionately, and if c < 1 prey type 2 is killed
disproportionately. The bias of a predator’s diet could be quite malleable and, thus, c may not remain
constant but change depending on the relatively availability of the two prey types and perhaps other
factors (Elton 1927). The coefficient b is a measure of the extent of prey switching with values of b > 1
denoting prey switching (Greenwood and Elton 1979). Various relationships between the ratio of
available prey and the ratio of prey types in the predator’s diet can occur using this equation. If both c
and b equal 1 (i.e., no bias in the wolves’ diet and no switching), then wolves simply consume prey in
proportion to their availability and the relationship is linear with a slope of 1. If there is a bias in the
wolves’ diet (c 6¼ and there is no prey switching (b ¼ 1), the relationship will still be linear but the slope
of the line will be <1. If switching occurs (b > 1), the relationship will be curvilinear.
The selection coefficient for two-prey systems can be influenced by a variety of factors, including
differences in ungulate abundance, body size, anti-predator behaviors and defenses, and vulnerability,
as well as variability in wolf preference for the two prey types. Thus, to capture inherent differences
between bison and elk, we decompose c into three 3 components such that c ¼ svm, where s is the
differential preference for a predator to attack prey type 1 compared to type 2, v is the differential
vulnerability of prey type 1 compared to type 2, and m is the relative nourishment of prey type 1 to
type 2. Estimates of c and b were derived from switching analyses in Chapter 16 by Becker et al., this
volume, while m estimates were based upon body mass, with bison being much larger than elk and
providing approximately twice as much nourishment than elk to wolves when killed (m ¼ 2; Murie
1951, Meagher 1973).
The first step toward this goal is development of functional response equations that incorporate two
prey types and the potential for switching. For prey-dependence, following the structure proposed by
Murdoch (1973), the functional response model for two prey types is:
g1 ¼
a1 N1
a2 N 2
; g2 ¼
1 þ a1 N 1 h 1 þ a2 N 2 h 2
1 þ a1 N 1 h 1 þ a 2 N 2 h 2
ðA:2Þ
For ratio-dependence, the functional response model for two prey types is:
g1 ¼
a1 N 1
a2 N2
; g2 ¼
P þ a 1 N 1 h1 þ a2 N 2 h2
P þ a1 N1 h1 þ a2 N2 h2
ðA:3Þ
where subscripts 1 and 2 correspond to prey types 1 and 2, respectively; g and N are defined as above; P is
the number of predators; a is the ‘‘attack rate’’ (i.e., instantaneous rate of discovering prey by one
predator) in days�1, and h is the ‘‘handling time’’ (days perpredator perprey killed) taken by one predator
for each prey killed. Switching can be incorporated into Eqs. (A.3) and (A.4) by defining m ¼ h1/h2 and
using Eqs. (A.1)–(A.3) to derive an expression for a1. It can be shown that Eq. (A.2) becomes:
�Chapter 17
g1 ¼
.
369
Wolf Kill Rates: Predictably Variable?
a2 N1 ðsvmÞb ðN1 =N2 Þb�1
1 þ a2 N1 h2 ðsvÞ
b
mbþ1 ðN1 =N2 Þb�1
þ a2 N 2 h2
; g2 ¼
a2 N2
1 þ a2 N1 h2 ðsvÞ
b
mbþ1 ðN1 =N2 Þb�1
þ a2 N2 h2
ðA:4Þ
and Eq. (A.3), for a ratio-dependent functional response, becomes:
g1 ¼
a2 N1 ðsvmÞb ðN1 =N2 Þb�1
P þ a2 N1 h2 ðsvÞ
b
mbþ1 ðN1 =N2 Þb�1
þ a2 N 2 h2
; g2 ¼
a2 N2
P þ a2 N1 h2 ðsvÞ
b
mbþ1 ðN
1 =N2 Þ
b�1
þ a 2 N 2 h2
ðA:5Þ
From Chapter 16 we estimated variables b and c, and estimated m above. Thus rearranging the
equations and substituting c ¼ svm as well as values of b and m the new equations for two-prey
functional for prey-dependent and ratio-dependent functional response models respectively are:
g1 ¼
g1 ¼
aN1
1 þ aN2 hc b mðN2 =N1 Þb�1 þ aN1 h
aN1
P þ aN2 hc b mðN2 =N1 Þb�1 þ aN1 h
�
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Wolf kill rates: predictably variable?
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<span>The ability of predators to successfully capture and kill prey is affected by the abundance and diversity of the prey assemblage, and such variation is a fundamental driver of ecosystem dynamics because </span><em>per capita</em><span> consumption rate strongly influences the stability and strength of community interactions. Descriptions of predatory behavior in this context typically include the functional response, specifically the kill rate of a predator as a function of prey density. Thus, a major objective in studying <a href="https://www.sciencedirect.com/topics/earth-and-planetary-sciences/predator-prey-interaction" title="Learn more about Predator-Prey Interaction from ScienceDirect's AI-generated Topic Pages" class="topic-link" target="_blank" rel="noreferrer noopener">predator–prey interactions</a> is to evaluate the strength of the numerous factors related to the kill rate of a predator, and to subsequently determine the forms of its functional response in natural systems because different forms have different consequences for ecosystem dynamics. Recent controversies over the nature of predation focus on the respective roles of prey and predator abundance in affecting the functional response. However, resolution requires more direct measures of kill rates in natural systems. We estimated wolf (</span><em>Canis lupus</em><span>) kill rates in a tractable and newly established wolf–elk (</span><em>Cervus elaphus</em><span>)–bison (</span><em>Bison bison</em><span>) system in the Madison <a href="https://www.sciencedirect.com/topics/earth-and-planetary-sciences/headwater" title="Learn more about Headwater from ScienceDirect's AI-generated Topic Pages" class="topic-link" target="_blank" rel="noreferrer noopener">headwaters</a> area of Yellowstone National Park during winters 1998–1999 to 2006–2007 to document the transition from over seven decades without wolves to a well-established top predator population. Wolf abundance, distribution, and <a href="https://www.sciencedirect.com/topics/earth-and-planetary-sciences/prey-selection" title="Learn more about Prey Selection from ScienceDirect's AI-generated Topic Pages" class="topic-link" target="_blank" rel="noreferrer noopener">prey selection</a> varied during the study, concurrent with variations in the demography, distribution, and behavior of elk and bison. These dynamics enabled us to evaluate factors influencing variations in wolf kill rates and the forms of their functional response.</span>
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<p>Becker, M. S., R. A. Garrott, P. J. White, R. Jaffe, J. J. Borkowski, C. N. Gower, and E. J. Bergman. 2008. Wolf kill rates: predictably variable? Pages 305-337 <em>in</em> Garrott, R.A., P.J. White and F.G.R. Watson, editors. The ecology of large mammals in central Yellowstone: sixteen years of integrated field studies. Academic Press, New York, New York, USA. <a href="https://doi.org/10.1016/S1936-7961(08)00217-0" target="_blank" rel="noreferrer noopener">https://doi.org/10.1016/S1936-7961(08)00217-0</a></p>
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Becker, Matthew S.
Garrott, Robert A.
White, P.J.
Jaffe, Rosemary
Borkowski, John J.
Gower, Claire N.
Bergman, Eric J.
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Predator–prey interactions
Prey selection
Wolf
Elk
Bison
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31 pages
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2008-11-14
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<a href="http://rightsstatements.org/vocab/InC-NC/1.0/" target="_blank" rel="noreferrer noopener">In Copyright - Non-Commercial Use Permitted</a>
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English
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Terrestrial Ecology
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Article