OIKOS 111: 101 Á/111, 2005




Spatial decomposition of predation risk using resource selection
                            Á              Á
functions: an example in a wolf   predator    system /elk            /prey




M. Hebblewhite, E. H. Merrill and T. L. McDonald


                    Hebblewhite, M., Merrill, E. H. and McDonald, T. L. 2005. Spatial decomposition
                    of predation risk using resource selection functions: an example in a wolf Á/elk
                    predator Á/prey system. Á/ Oikos 111:101 Á/111.


                    Predation is a fundamental ecological and evolutionary process that varies in space,
                    and the avoidance of predation risk is of central importance in foraging theory. While
                    there has been a recent growth of approaches to spatially model predation risk, these
                    approaches lack an adequate mechanistic framework that can be applied to real
                    landscapes. In this paper we show how predation risk can be decomposed into
                    encounter and attack stages, and modeled spatially using resource selection functions
                    (RSF) and resource selection probability functions (RSPF). We use this approach to
                    compare the effects of landscape attributes on the relative probability of encounter, the
                    conditional probability of death given encounter, and overall wolf and elk resource
                    selection to test whether predation risk is simply equivalent to location of the predator.
                    We then combine the probability of encounter and conditional probability of death into
                    a spatially explicit function of predation risk following Lima and Dill’s reformulation
                    of Holling’s functional response. We illustrate this approach in a wolf Á/elk system in
                    and adjacent to Banff National Park, Alberta, Canada. In this system we found that
                    the odds of elk being encountered by wolves was 1.3 times higher in pine forest and 4.1
                    times less in grasslands than other habitats. The relative odds of being killed in pine
                    forests, given an encounter, increased by 1.2. Other habitats, such as grasslands,
                    afforded elk reduced odds (4.1 times less) of being encountered and subsequently killed
                    (1.4 times less) by wolves. Our approach illustrates that predation risk is not necessarily
                    equivalent to just where predators are found. We show that landscape attributes can
                    render prey more or less susceptible to predation and effects of landscape features can
                    differ between the encounter and attack stages of predation. We conclude by suggesting
                    applications of our approach to model predator Á/prey dynamics using spatial predation
                    risk functions in theoretical and applied settings.


                    M. Hebblewhite and E. H. Merrill, Dept of Biological Sciences, Univ. of Alberta,
                    Edmonton, AB., Canada, T6G 2E9 (mark.hebblewhite@ualberta.ca). Á/ T. L.
                    McDonald, West Inc, 2003 Central Ave, Cheyenne, WY 82001, USA.


Ecologists are increasingly recognizing that predation     of predators to a predator-free system (Abrahams and
risk can have as important an effect as the direct effects   Dill 1989). Lima and Dill (1990) suggest, however, that
of predation on structuring communities (Abrams et al.     predation risk is more than just predator presence, and
1996, Schmitz 1998). Experimental studies have demon-      Lima (2002) criticizes experimental approaches because
strated that risk avoidance can increase energetic costs    they often ignore the spatial and temporal variation in
(Abrahams and Dill 1989), modify habitat selection       predation risk that prey face. Lima and Dill (1990)
(Gilliam and Fraser 1987), and change trophic flows by     provide a mechanistic approach to understanding pre-
altering diet selection (Schmitz 1998). In most experi-     dation risk by decomposing Hollings’ (1959) disk
mental studies predation risk is defined as the addition    equation of the predator functional response into its


Accepted 16 February 2005
Copyright # OIKOS 2005
ISSN 0030-1299

                                                               101
OIKOS 111:1 (2005)
two fundamental components: the probability of being     al. 2002). We combine spatial risk of encounter and kill
                               to estimate spatial predation risk following Lima and
encountered (a) and the conditional probability of being
                               Dill’s (1990) derivation of predation risk from Hollings
killed given an encounter (d). Similar decompositions of
                               disc equation. We illustrate our approach for gray wolf
predation risk have been presented by Wrona and Dixon
                               predation on elk (wapiti) during winter in and adjacent
(1991) for an aquatic planarian Á/trichopteran predator Á/
                               to Banff National Park, Alberta, Canada. We address
prey system and by Hebblewhite and Pletscher (2002) for
                               two specific questions: (1) do landscape features asso-
a wolf (Canis lupus ) Á/elk (Cervus elaphus ) system, but
                               ciated with where wolves occur differ from where wolves
neither of these studies focused on the spatial variation
                               encounter and kill elk? (2) In which predation stage
in the components of predation risk.
                               (i.e. search, encounter, kill) do landscape features express
  Several recent studies have explicitly linked predation
                               their greatest effects on predation risk? We discuss
risk to landscape attributes. Kunkel and Pletscher (2000)
                               common sampling and statistical issues for applying
compared landscape features where wolves killed moose
                               this approach to a variety of predator Á/prey systems,
(Alces alces ) to those at random sites, and found moose
                               and conclude with examples of the importance of
were more likely to be killed closer to roads and trails
                               spatially decomposing predation risk, and applications
and farther from forest cover. Thogmartin and Schaeffer
                               to predator Á/prey models in conservation and theoretical
(2000) compared where turkeys (Mellagris gallapavo )
                               settings.
lived in relation to roads to the distance at which they
were killed from roads, and found they were more likely
to be killed farther from roads. Cresswell and Quinn
(2004) showed distance to cover influenced the success of
                               Towards a spatial decomposition of predation risk
predation by sparrowhawk (Accipter nisus ) predation on
redshanks (Tringa totanus ). All three studies attributed
                               Since Hollings’ (1959) seminal work, most workers have
these results to habitat covariates that modified vulner-
                               recognized two main components of predation across
ability to predation. Yet, in the first two examples, these
                               systems. These are the instantaneous risk, or probability,
patterns could have arisen simply because of prey habitat   of encounter, a, and the conditional risk of death, given
use, and not necessarily due to landscape features      an encounter, d (Fig. 1). While some authors decompose
because only kills and available landscape characteristics  these components further, for example splitting a into
were compared. In modeling resource selection of       the probability of detection and evasion (Lima and Dill
caribou (Rangifer tarandus ), Johnson et al. (2002)      1990), we consider a and d to be the basic components
attempted to account for these differences in predation    that include finer divisions (Taylor 1984). Lima and
risk by weighting kill-site locations twice that of      Dill (1990) reformulated Holling’s (1959) functional
predator telemetry locations. However, they had little    response to operationally define predation risk P(k), or
empirical support for these weights. More recently,      the risk of being killed per unit time, as a function of a
Kristan and Boarman (2003) documented the spatial       and d by:
pattern in predation risk to common tortoises (Gopherus
                               P(k)01(exp((adT)                     (1)
agassizii ) from ravens (Corvus corax ) by comparing
attributes of sites where ravens attacked model tortoises   where a is Holling’s (1959) encounter rate or probability
with attributes at random sites, but they also did not    of encounter, d is the conditional probability of the
account for the distribution of prey nor decompose the    attack being successful given an encounter, and T is the
components of predation. In the case of tortoises, it     time interval over which predation risk is being inte-
could be argued that spatial attributes associated with    grated. Equation 1 bounds predation risk, P(k), between
encounters are similar to where they are attacked       0 and 1, even where a and d are relative probabilities
because tortoises have few antipredator strategies once    (below). For the risk of being killed per unit time, we
encountered (Kristan and Boarman 2003). For other       adopt P(k) instead of Lima and Dill’s (1990) P(d) for
prey species, habitats where individuals are most likely to  clarity; P(k) avoids having P(d) and the conditional risk
be encountered may not be where they are most likely to    of being killed, given encounter, d, in the same function.
be killed. For example, deer (Odoicoleus spp.) frequent-   To make P(k) spatially explicit, spatial functions for a
ing open slopes can often evade predators if the slope is   and d need to be derived and substituted into Eq. 1.
steep (Lingle and Pellis 2002, Kunkel et al. 2004).       Data required to estimate the spatial risk of encounter,
Therefore it seems that Lima and Dill’s (1990) critique    a, and conditional risk of death, d, will depend on the
of experimental definition of predation risk is warranted,  predator Á/prey system. Typically, the most difficult
and that an approach to describe spatial and temporal     component will be estimating a not d, for many
variation in predation risk is required.           predator Á/prey systems. Locations of killed prey are
  Here, we demonstrate how to spatially decompose the    often conspicuous and can be readily quantified, and
components of predation risk as a function of landscape    functions describing spatial variation in mortality sites
attributes using resource selection functions (Manly et    have already been developed in many systems (Kunkel

102                                                    OIKOS 111:1 (2005)
and Pletscher 2000, Nielsen et al. 2004). To estimate the  acterized as 1 and unused units characterized as 0. In
spatial risk of encounter, a simple approach could be    designs with used and available units, a resource selec-
based on elementary set logic developed for measuring    tion function (RSF) is estimated using logistic regression
spatial habitat overlap for two species (Minta 1992).    that is proportional to the probability of use (Manly
Extending Minta’s (1992) approach to spatial models,     et al. 2002). The used-available design results in a relative
the product of spatial predator and prey models would    probability because the intercept or b0 coefficient is
represent the joint probability of co-occurrence, which   incorrectly scaled. This problem arises because the true
should be proportional to the probability of encounter    population-sampling fraction is unknown (Boyce and
(Manly et al. 2002). However, the joint probability     McDonald 1999). Because the RSF is only proportional
assumes independence between predator and prey, a      to an RSPF, the odds-ratio is also only a relative
problem identified but not resolved by Minta (1992).     probability ratio. Recent statistical discussion highlights
Assuming independence might be reasonable for alter-     another potential problem with the used-available design
nate prey in a two-prey, one-predator system where the    in logistic models when the ‘contamination’ rate, or false-
predator specializes on the primary prey and predation is  negative rate (units that were used but misclassified as
essentially independent for alternate prey. Assuming     available due to sampling) is high (/20%, Keating and
independence also might be justified for primitive
                               Cherry 2004). However, in geographic information
predators with random or limited searching behavior.
                               system (GIS) applications contamination rate is unlikely
Independence could also be tested for by extending
                               to be large enough to affect logistic models because of
Minta’s (1992) approach. However, for many predator Á/
                               the typically large numbers of available resource units
prey systems, independence may be biologically unrea-
                               (pixels) relative to the sample of used resource units.
listic because of dynamic feedbacks between predator
                               Moreover, Manly et al. (2002, p. 177) show the assump-
and prey.
                               tion that an RSF is proportional to an RSPF is often
  The problem of independence may be circumvented in
                               valid, and used-available designs are useful in a wide
cases where we can estimate spatial encounters directly
                               array of applications (Boyce and McDonald 1999).
from field data. Examples include where encounters
                                While resource selection approaches typically have
between predators and prey can be observed directly
                               been used for predicting the probability of use (Manly
(Fanshawe and Fitzgibbon 1993, Cresswell and Quinn
                               et al. 2002), they also can be used for other spatial
2004), or where they can be estimated indirectly, for
                               events, such as encounters or kills. Predation is a series
example, via snow tracking (Hebblewhite and Pletscher
                               of discrete stages that can be characterized by binary
2002). New technology, such as GPS collars on pre-
                               responses. Searching predators are either successful (a)
dators and prey, may provide additional means to
                               or unsuccessful (1(/ a) at encountering prey, and given
estimate encounters. Encounters also could be estimated
                               an encounter, are either successful (d) or unsuccessful
in experimental settings such as with predation trials on
                               (1 Á/d) in making a kill (Fig. 1). Thus, components of
prey or with artificial baits or nests (Kristan and
                               predation risk can be modeled in a sequential framework
Boarman 2003, Forstmeier and Weiss 2004). Estimating
encounters directly also avoids problems of differential
sampling bias between predator and prey habitat use
(Rettie and McLoughlin 1999). Where the above ap-
proaches provide estimates of spatial locations of
encounters and kills, all that remains is to estimate
spatial functions of a and d, and then substituting these
in Eq. 1 to estimate a spatial predation risk function.




Application of resource selection functions to
modeling predation risk
                               Fig. 1. Schematic representation of the spatial decomposition
Resource selection functions provide an efficient frame-   of the three main stages of predation, search, encounter, and
work for quantifying the spatial probability of encounter  kill, in our wolf Á/elk system over time, T. All elk inside the
                               territory are available to be encountered when wolves are
and kills in ecological landscapes. A resource selection
                               searching for prey, but only some of these at any particular
probability function (RSPF) is defined as any function    time are successfully encountered with probability a. Condi-
that equals the probability of use of a resource unit, and  tional on encounter, elk are killed with probability, d. We
                               randomly sampled wolf search paths with radio telemetry
is easily adapted to spatial data (Boyce and McDonald
                               locations, encounters through snow tracking, kills through
1999, Manly et al. 2002). Logistic regression has become
                               telemetry and snow tracking, and we characterized elk locations
one of the most common statistical approaches to       through telemetry during winters 1997 to 2001, Banff National
estimate habitat selection models with used units char-   Park, Alberta, Canada.

                                                            103
OIKOS 111:1 (2005)
using logistic regression (e.g. logit form of [(p)/(1 Á/p]). If  where encounter locations might include a kill location
encounter locations are known (used units), and areas       would represent an RSF design, and would adopt Eq. 2
where prey were either not encountered (unused units) or     above.
areas where prey could have been encountered (available       In some situations, encounters may not be known
units) are known, then an RSPF or an RSF can be used       (Kunkel and Pletscher 2000) and decomposition of
to describe the probability of encounter as a function of     predation risk components will not be possible. Compar-
landscape attributes. Examples of used-unused designs in     ing landscape attributes of kill locations to locations
estimating probability of encounter might include ob-       available to predator and prey using Eq. 2 may
servational studies where a predator is known to have       approximate relative predation risk in Eq. 1 (Nielsen
‘missed’ an encounter, thus representing an unsuccessful     et al. 2004). A spatial function of kill locations is an
encounter. In the example we present below, we consider      improvement over assuming predator locations equal
the more common case of used-available units for         risk but does not permit the derivation from Hollings’
observational studies where encounters are known and       (1959) functional response in Eq. 1. Moreover, by failing
compared to areas in which prey were available to be       to decompose predation, it is not possible to discern if
encountered. Defining availability is crucial to the scope    changes in risk arise at the encounter or kill stage, or are
of inference of RSF models (Manly et al. 2002). We        due to predator or prey effects (Thogmartin and
define availability for estimating the risk of encounter as    Schaeffer 2000). Decomposing predation risk into its
all areas in which either prey or predators occur (Minta     components can reveal valuable insights into the me-
1992). By comparing attributes at sites where prey were      chanisms of predation risk.
encountered to all areas available to predator and prey,       Where we know both the spatial risk of encounter and
                                 death, we can substitute a(x) and d(x) into Eq. 1 to
we estimate the spatial intersection, or overlap for a suite
of landscape covariates (Minta 1992).               estimate a spatial predation risk function, P(k). When
  Using a logistic regression the relative probability of    RSPF models of both encounter and kill can be
encounter, a(x), given availability of landscape covariates    estimated, Eq. 1 yields a true joint probability function
                                 for predation risk. When either a or d are relative
(xi) used by both predators and prey, is equivalent to
Manly et al.’s (2002, p. 100) resource selection function     probability functions, as in our example below, P(k)
(w(xi)), and is proportional to;                 remains a relative measure of predation risk bounded
                                 between 0 and 1 (Johnson et al. 2004, p. 248). In a GIS
      X       
      n
                                 framework, because a(x) and d(x) are functions of
                              (2)
a(x)0exp      bi xi
                                 landscape covariates i 0/1 to n, P(k) is spatially explicit
       i01
                                 and maps of predation risk can be produced (Boyce and
                                 McDonald 1999). A critical assumption is that the
where i 0/refers to landscape covariates 1 through n for
                                 predation risk function applies only over time period T
encounters and available locations. Following Manly
                                 during which data were collected. Therefore, the en-
et al. (2002) we drop the denominator of the logistic
                                 counter and kill functions must be modeled at the same
form and the intercept for this relative function. (Boyce
                                 time scales, i.e. day, month, or season. As a useful
and McDonald 1999, Manly et al. 2002).
                                 extension, time-varying functions could be developed to
 When both kill and encounter locations are known we
                                 test for temporal (seasonal, annual) variation in preda-
can estimate the second term in Eq. 1, d, the conditional
                                 tion risk (sensu Manly et al. 2002, p. 118). Further, RSF
probability of death as a function of landscape attributes
                                 models are scale-dependent (Boyce et al. 2003); for
using logistic regression where 1 0/kill locations (used),
                                 example between selection for location of a home range
and 0 0/encounter locations where no kill occurred
                                 (second-order selection), and selection within a home
(unused). In the case with known encounters, the used-
                                 range (third-order selection, Johnson 1980). Predation
unused design corresponds to a RSPF, or true prob-
                                 risk components from one spatial scale would be
ability function (equivalent to w*(xi) of Manly et al.’s
                                 inappropriate to apply at another spatial scale.
2002, p. 83), and the conditional probability of kill given
encounter is expressed as:

            X
            n
                                 Methods
                 bi xi )
     exp(b0 '
              i0l
                              (3)
d(x)0                               Study area
                X
                n
                   bi xi )
    1 ' exp(b0 '
                                 We illustrate our approach for a wolf Á/elk predator Á/prey
                i0l
                                 system in and adjacent to Banff National Park (BNP,
where i 0/refers to landscape covariates 1 through n for     51815?/116800?), Alberta, Canada, during winters 1997 to
kills and encounters. The intercept b0 is included because    2001. We defined winter as 15 October to 15 April. BNP,
                                 6641 km2 in area, is on the eastern slope of the
the sampling probability is known and a true probability
function is estimated (Manly et al. 2002). Study designs     continental divide in the Canadian Rocky Mountains

104                                                      OIKOS 111:1 (2005)
(1400 to 3400 m). Vegetation is dominated by closed     ties remain in defining ‘‘true’’ spatio-temporal encoun-
lodgepole pine (Pinus contorta ) forests interspersed with  ters. Resource selection of elk was determined by
riparian Engelmann spruce (Picea engelmanii ) Á/ willow   relocating radiocollared female elk within the Cascade
(Salix spp.), aspen (Populus tremuloides ) Á/ parkland,   territory weekly from the air or ground during winter
and dry grasslands at low elevations. Engelmann spruce-   following standard techniques at the major winter ranges
subalpine fir (Abies lasiocarpa ) forests dominate at    in the Bow Valley and Ya Ha Tinda, and in secondary
higher elevations, interspersed with willow-shrub mea-    winter ranges scattered throughout the Cascade territory
dows, subalpine grasslands, avalanche terrain, and      (McKenzie 2001, Hebblewhite, M., unpubl.).
alpine shrub-forb meadows (Holland and Coen 1983).
Elk are the most abundant ungulate and are the primary
prey of wolves, comprising 40 Á/70% of wolf diet (Heb-    Landscape attributes
blewhite et al. 2004). Wolf predation is equally important
                               We selected landscape attributes known from previous
to elk, accounting for 30 Á/60% of adult female elk
                               studies to influence wolf and/or elk resource selection
mortality (McKenzie 2001). To illustrate our approach
                               and predator Á/prey dynamics (Kunkel and Pletscher
we use data from the Cascade wolf pack, which
                               2000, Roloff et al. 2001, Boyce et al. 2002, Lingle and
established in 1991 inhabiting an area previously unin-
                               Pellis 2002). Landscape attributes measured at encoun-
habited by wolves for up to 30 years (Hebblewhite et al.
                               ter, kills, wolf, elk, and available sites included percent
2004). Winter wolf numbers stabilized around eight
                               slope, aspect classified as eight cardinal directions,
wolves from 1997 to 2001. We used elk telemetry data
                               elevation to nearest 100 m, distance (km) to roads and
from elk within the Cascade pack territory. The Cascade
                               trails, and vegetative cover type in ARCGis 8.2 (ESRI
territory contained two primary elk winter ranges, the
                               Inc.). Topographic variables (slope, elevation) were
Ya Ha Tinda and Bow Valley ranges, and several smaller
                               calculated from a 100 m2 resolution digital elevation
secondary ranges.
                               model for the study area, whereas distance to roads and
                               vegetation layers were measured at a 30 m2 resolution.
                               Roads and trails were derived from the human use

             Á                  atlas of the Central Rockies Ecosystem (Jevons 2001)
Wolf and elk predator       data
             /prey
                               and included active roads used by vehicles, and inactive
Wolf and elk research and capture methods followed      roads and trails used by off highway vehicles, horseback
approved and standard methods (Parks Canada Envir-      riders, or by hikers. We buffered this access layer with
onmental Assessment B-1994 Á/29, Univ. of Alberta      ArcGIS 8.2 to create a distance to human access surface
Animal Care protocol ID# 35112). For more detailed      in km. Because the territory of the Cascade wolf pack
descriptions of wolf and elk monitoring see McKenzie     straddled two mapping jurisdictions for vegetative
(2001) and Hebblewhite et al. (2004). For wolf monitor-   cover type, we merged two landcover maps following
ing, we used systematic (weekly) aerial relocations of    expert advice (D. Zell, Parks Canada). Land cover
radiocollared wolves to characterize the search stage    types were pine, closed conifer, open conifer, avalanche
of predation and to start continuous tracking sessions    path, alpine, grassland, shrub, and rock/ice. We screened
(Fig. 1). During continuous snow tracking of wolves we    for collinearity using tolerance scores following Menard
recorded spatial intersection of the tracks of wolves and  (2002), which resulted in the exclusion of the rock/ice,
elk groups (Hebblewhite and Pletscher 2002) and found    and several aspect categories. We included categorical
elk killed by wolves (Hebblewhite et al. 2004). We      habitat and aspect variables in models using dummy
defined the intersection of wolf tracks with elk tracks   variable coding, excluding the reference category.
as an encounter, and locations of wolf-killed elk as kills
(Fig. 1). Spatial differences between encounters and kills
were evident from the snow tracking sequence with
                               Resource selection function modeling
elk being chased an average of 262m (SD 0/330.2, n 0/96,
range 10 Á/1700 m, unpubl.) after an encounter. Our     We estimated resource selection models for the encoun-
definition of search and encounter assumed (1) wolves    ter (Eq. 2) and kill (Eq. 3) stages of wolf predation, and
always hunted while traveling (Mech and Boitani 2003)    then compared these to overall wolf-search and elk RSF
and (2) spatial intersection of wolf and elk snow      models (using Eq. 2) to test for differences in effects of
tracks represented the spatial encounter location where   landscape attributes between predation stages (Fig. 1).
wolf and elk tracks overlap in space (Fig. 1; Hebblewhite  For the elk RSF, we assessed availability using a
and Pletscher 2002). In reality, our measure of encounter  balanced number of random locations for each indivi-
may not have represented the true spatio-temporal      dual elk in the wolf territory. We calculated variance for
encounter, yet for RSF models we were only interested    beta coefficients in elk models by clustering data by
in spatial encounters. Future extensions of our approach   individual elk in STATA (StataCorp 2001) to reduce
could include spatio-temporal encounters, but difficul-   autocorrelation (Pendergast et al. 1996). For the wolf-

                                                          105
OIKOS 111:1 (2005)
search RSF, we compared wolf telemetry locations to      20%, or five subsets. Predictive capacity of partitioned
                                models were evaluated against the withheld training data
1000 random locations within the wolf territory. Wolf
                                using Spearman rank correlations (rs) between training
relocations were screened ]/24 h between locations to
                                and test data grouped within ten bins (Fielding and Bell
reduce autocorrelation (Otis and White 1999). Analyses
                                1997, Boyce et al. 2002). We conducted all statistical
with multiple packs should use an appropriate grouping
                                analyses in STATA 7.0 (StataCorp 2001).
by pack (Pendergast et al. 1996) to account of lack of
independence among individuals within a pack. We
estimated a(x), the encounter RSF, by comparing all
elk encounters obtained from snow tracking with the
                                Results
same 1000 random locations as for describing resource
availability within the wolf territory. Finally, we used all  Data and model assessment
elk encounters and kill locations of elk in the final d(x),
                                During four winters from 1997 to 2001 we collected 119
or probability of death, RSPF model. Our analyses
                                telemetry locations ]/24 h apart on six different wolves
correspond to the third-order (Johnson 1980) or within
                                in the Cascade pack, snow backtracked wolves approxi-
home range scale, but our approach could easily be
                                mately 1250 km finding 77 groups of elk tracks
applied at different spatial scales (Boyce et al. 2003).
                                encountered by wolves, and sampled 119 elk that were
  Because our objective was to compare how the
                                killed by wolves. Elk killed by wolves included 29 adult
different stages of predation (search, encounter, and
                                female, 52 adult male, 7 yearlings, 9 calves, and 31
kill) and elk distribution were affected by landscape
                                unknown female age classes. We collected 4890 telemetry
attributes, we used a constrained model selection
                                locations on 104 radiocollared adult female elk in both
approach to select a constant set of parameters to
                                the Bow Valley and Ya Ha Tinda winter ranges between
compare across models. Without consistency among
                                1997 and 2001, with an average of 29 telemetry
variables, coefficients would not be comparable because
                                locations/elk/winter.
the covariance matrix adjusts coefficients differently
                                 Six covariates, distance to human access, slope,
with different combinations of covariates (Hosmer and
                                elevation, and the cover types of grassland, pine, and
Lemeshow 1989, McCullough and Nelder 1989). There-
                                open conifer were consistently retained in top-ranked
fore, we first created candidate sets of hypothesized
                                models for all predation stages (Table 1, 2). Where model
models and then fit RSF or RSPF models for each
                                selection uncertainty arose, such as for the kill stage
component (wolf search, encounter, kill; elk), and used
                                (Table 1, 2), DAICc scores suggested a close tie between
AICc to rank models based on Akaike weights, wi for
                                the lower ranked, full six covariate model, and other
each model (Burnham and Anderson 1998). We then
                                top models (Table 1, 2). With the exception of the kill-
used the sum of all Akaike weights for each covariate to    stage RSPF, most models containing the top six
rank covariates in order of importance following Burn-     covariates were consistently ranked first or second out
ham and Anderson (1998, p. 140). We selected a         of all competing models (Table 1, 2), despite some
consistent set of landscape attributes to build compara-    model selection uncertainty (i.e. wi from 0.15 to 0.38).
tive RSF models for each predation stage from this       Therefore, we felt justified in comparing the importance
ranked set of variables in the top models. In addition to   of these six covariates among predation stages. Using
assessing overall model fit, we compared estimated beta    these six covariates, RSF models for elk occurrence, wolf
coefficients for each covariate for effect size and      search, and wolf-encounter a(x) had good model fit
significance using 90% confidence intervals (we set      (Table 2, all likelihood ratio-test x2 P-values B/0.0005,
P0/0.10) to determine at which predation stage land-      and high Nagerleke’s R2 / 0.22). By contrast, the RSPF
scape features had the largest effect. Using these models   predicting d(x) had poor model fit, and many of the
we mapped a(x) and d(x) using Eq. 2 and 3 in ArcGIS      predictor variables were not selected for (Table 2,
8.2 at the resolution of 30 m2 pixel (1. We then        likelihood ratio-test x2 P-value0/0.12, Nagerleke’s
substituted the spatial RSF probabilities a(x) and d(x)    R2 0/0.10). Similarly, the Spearman rank correlation
into Eq. 1 using map calculator in ArcGIS 8.2 to        from the k-fold cross-validation was lower (0.529/0.05,
estimate the relative predation risk surface for our study   SE) for d(x) than the other models (0.679/0.08 to 0.869/
area. All figures are displayed with a histogram        0.01, Table 2). Based on Akaike weights, simpler models
smoother in ArcGIS 8.2.                    for the kill stage with less than the six covariates did
  Evaluating performance of RSF models based on        not greatly improve model fit (unpubl., Table 1).
normal logistic regression diagnostics (i.e. ROC, R2, etc)
are flawed in use-availability designs (Fielding and Bell
1997, Boyce et al. 2002). Therefore, we evaluated the
                                Elk and wolf occurrence
predictive performance of all models using k-folds cross-
validation (Boyce et al. 2002), where k-partitions of the   Probability of elk occurrence within the territory of the
dataset are made following a test to training ratio of     Cascade wolf pack decreased with increasing distance to

106                                                   OIKOS 111:1 (2005)
Table 1. Akaike weights (wi) for covariates in the four predation stage RSF models: for elk, wolf-search, wolf-encounter, and wolf-
kill in Banff National Park, Alberta, winters 1997 Á/2001. Shown are the Akaike weights for each covariate, along with the averaged
weight across all four predation stages and the average rank of covariate importance.

Covariate            Elk      Wolf-search  Wolf-encounter    Wolf-kill   Average Akaike    Average rank
                                                  weight, wi

Elevation           0.952       1.000       0.983      0.996       0.983         1
Distance to human access    0.952       0.921       0.983      0.996       0.963         2
Slope             0.614       1.000       1.000      1.000       0.903         3
Grassland           0.925       0.917       1.000      0.740       0.895         4
Open conifer          0.710       0.530       0.226      0.870       0.584         6
Pine              0.520       0.936       1.000      0.572       0.757         5
Closed conifer         0.483       0.233       0.440      0.459       0.404         7
Shrub             0.338       0.081       0.296      0.137       0.213         8
Avalanche           0.027       0.115       0.000      0.000       0.036         11
Deciduous           0.044       0.081       0.000      0.563       0.172         9
South aspects         0.000       0.000       0.247      0.000       0.062         10

roads (b0/(/0.377), and increased at lower elevations         was about a third more likely than being encountered
(b0/ (/0.219) and on shallower slopes (b0/ (/0.017)          (1.07), while the odds of being killed in open conifer
during winter (presented as relative odds ratios in          stands (0.35) was about half that of being encountered
Table 3, and as beta coefficients in Fig. 2). Probability       (0.66). Similarly, elk were about fifth again more likely to
of elk using grasslands (b0/2.05) and open conifer          be killed in high elevation areas (0.83) than being
(b0/1.65) was high while probability of elk use of pine        encountered (0.71). There was no difference in the
forests (b0/(/0.125) was low. The probability of a          odds of elk being encountered and killed near roads or
resource unit being used by wolves at the search stage        on slopes of varying steepness (Table 3).
decreased with increasing elevation (b0/(/0.380), slope
(b0/ (/0.021) and distance to roads (b0/(/0.703). Wolf
use was higher in grasslands (b0/1.255) and pine forests
                                   Predation risk
(b0/0.44), but decreased in open conifer (b0/ (/0.204,
Table 3).                               The spatial functions of relative probability of encounter
                                   (Fig. 3a) and probability of kill (Fig. 3b) from the
                                   equations in Table 2 illustrate the spatial application of

   Á                                predation risk in real landscapes (Fig. 3c). Wolves
Elk      encounters and kills
   /wolf
                                   avoided high elevation and steep slope areas more than
In our study area there were important statistical          elk (Table 3, Fig. 2), concentrating elk Á/wolf encounters
differences in the effects of landscape attributes on         in valley bottoms (Fig. 3a). The strength of the topo-
wolf predation risk stages for elk (Table 3, Fig. 2). The       graphic effect on encounter overwhelmed effects of other
relative odds of an elk being killed in grasslands (1.34)       variables once translated onto the real ecological land-
were about five times less than the odds of elk being         scape in Fig. 3a. However, given an encounter, habitat
encountered in grasslands (7.65). Once an elk was           appeared to have the strongest effects on risk of being
encountered, odds of being killed in pine stands (1.45)        killed (Fig. 2, 4b, Table 3), with risk of death being

Table 2. Model fit for predicting the relative probabilities of resource use by elk, p(elk), wolves, p(wolf), encounters between wolves
and elk, a(x), and the true conditional probability of elk being killed by wolves, given an encounter, d(x), within the Cascade pack
wolf territory during winters 1997 Á/2001, Banff National Park, Alberta. Models are shown with corresponding number of
parameters ki, DAICc, Akaike weight (wi), Nagerleke’s R2, and the Spearman rank correlation (rs) obtained from k-folds cross
validation (see text) as a means of evaluating model predictive performance shown with SE. See text for statistical modeling details.

           Na        DAICb    wb            R2d
Model             ki              Model         Likelihood    Likelihood     k-folds cross
                      c    i
                              rankc         ratio X2              validation re
                                                 ratio P-value          s

Elk         9780   7    0.61    0.15    2     0.22     139.12               0.869/0.01
                                                  B/0.0005
Wolf         1190   7    0.60    0.20    2     0.32     295.70               0.829/0.05
                                                  B/0.0005
Encounter0/a(x)   1077   7    0.00    0.38    1     0.28     144.10               0.679/0.08
                                                  B/0.0005
Killf 0/d(x)     189   7    0.02    0.11    4     0.10     11.41      0.0700      0.529/0.05

a- Sample size including 4890 random locations for elk, 1000 for wolf and encounter, and 77 encounters and 119 kills for the
probability of kill model.
b- AICc and weights are reported for each model of the top model set for each predation stage, and are not comparable across the
different stages. They are presented to allow evaluation of their strength.
c- Model rank among candidate models set based on DAICc, from Table 1.
d- Nagerleke’s pseudo Á/ R2 value.
e- Spearmans rank correlation coefficient for k-folds procedure averaged from five partitions. See text for details.
f- Conditional on encounter.

                                                                 107
OIKOS 111:1 (2005)
                                                   predation risk in real landscapes. For example, encoun-
                   1.5
                                                   ters were driven largely by topographic variables slope
                           Elk
                                                   and elevation (Fig. 3a), whereas habitat covariates had
                    1
Es tim a te d b e ta coe ffic ie nt




                           Wolf
                                                   the greatest effects on the risk of death, given an
                           Encounter
                                                   encounter (Fig. 3b) for our study area. Important trends
                           Kill
                   0.5                               in predation risk between stages were also revealed. For
                                                   example, grasslands and open conifer consistently re-
                                                   duced risk as predation escalates from search to
                    0
                                                   encounter to kill, while the opposite occurred in pine
                                                   stands and with decreasing elevation (Fig. 3). Broader
                   -0.5
                                                   analyses will be required to determine whether these
                                                   patterns hold across different wolf territories. Never-
                    -1                               theless, in this example if we had defined predation risk
                                                   simply as those areas used by wolves, we would have
                                                   overestimated risk by 60% in grasslands and open
                   -1.5
                                                   conifer, and underestimated risk by 20% in pine and at
                       Dist.    Elev.    Grass  Pine    O.conf.
                                                   higher elevations (Fig. 2, Table 3). Thus, predation risk
Fig. 2. Beta-coefficients for distance to human access (km),
                                                   for elk was a function of not only where wolves were, but
elevation (in 100 m intervals), grassland, pine, and open conifer
                                                   of landscape attributes that rendered elk more or less
habitat’s from logistic regression resource selection function
models for elk, wolves, wolf encounters with elk, and wolf-killed                   vulnerable to predation once encountered. Studies that
elk in Banff National Park, Alberta. Estimates are presented
                                                   assume predation risk is equivalent to predator habitat
with 90% confidence intervals, and non-overlapping significant
                                                   use may be misleading. Similarly, if a landscape attribute
differences are also indicated in Table 3. Slope is not shown
because of the large Y-axis scale relative to the estimates of                    decreased the encounter risk but increased vulnerability
effect sizes for slope (Table 3).                                   once encountered, studies that do not distinguish
                                                   between encounter and kill may not uncover the
reduced in grasslands and open conifer relative to other                       attributes that influence predation because the compo-
habitats. Combined in Eq. 1, the conditional nature of                        nents negate each other.
the risk of death on encounter is clearly illustrated (Fig.                       By distinguishing between components of predation,
3c). For example an elk’s risk of death in pine forests is                      mechanisms driving the observed statistical patterns may
modified by its topographic position in Fig. 3c because                        be hypothesized and tested using field experiments or
of the dominant effect of topography on risk of                            further analyses. For example, the safety afforded to elk
encounter.                                              in grasslands may result from increased predator detec-
                                                   tion and vigilance in open habitat (Dehn 1990) or larger
                                                   group sizes. Elk group sizes are typically greater in
                                                   grasslands, and while wolf encounter and attack rates
Discussion                                              may increase for large herds, individual elk predation
                                                   risk declines with increasing group size because of
Our example clearly illustrates the importance of care-
                                                   dilution effects (Hebblewhite and Pletscher 2002). In-
fully defining predation risk for prey, and demonstrates
                                                   deed, our decomposition of predation risk results
the utility of our approach to spatially decompose
                                                   suggests elk reduce predation risk the most in grasslands,
predation risk for revealing the behavioral aspects of

Table 3. Parameter estimates presented as relative odds-ratios, standard errors (SE), and associated p-values for independent
variables in RSF models for elk, wolf search, wolf encounter, and wolf kill models. Statistically significant differences (at P 0/0.10)
between different predation stages (elk, wolf, encounter, kill) within a covariate are marked with different letters (a, b, c, etc).

Variable                        Distance to road   Elevation       Slope     Grassland     Pine     Open conifer
                                 a          a           a       a         a
                                                                          1.635 a
Elk                            0.685        0.803        0.983      7.655      0.882
SE                             0.1257       0.0479       0.0136     2.5212      0.3175     0.5202
P-value                          0.005                  0.203              0.728      0.122
                                       B/0.0006              B/0.0005
                              0.495 a       0.684 b       0.979 a     3.508 b     1.553 a     0.815 a
Wolf
SE                             0.1154       0.0375       0.0116     1.2333      0.3919     0.2485
P-value                          0.003                  0.077              0.081      0.503
                                       B/0.0005              B/0.0005
                              0.485 a       0.711 b       0.965 b     1.862 b     1.066 a     0.661 a
Encounter
SE                             0.1627       0.0530       0.0173     0.9681      0.3599     0.2620
P-value                          0.031                  0.047      0.232      0.85      0.297
                                       B/0.0005
                                   a
                                        0.838 a       0.971 b     1.346 b     1.451 b     0.350 b
Kill                            0.4833
SE                             0.4770       0.0970       0.0153     0.7556      0.3941     0.1458
P-value                          0.43        0.071        0.058      0.597      0.265      0.012


108                                                                       OIKOS 111:1 (2005)
Fig. 3. Spatial maps of the decomposition of the components of predation, illustrated using wolf predation on elk in the Cascade
valley portion of the Cascade pack wolf pack territory from 1997 Á/2001 in Banff National Park (light boundary). Predation risk
decomposes into (a) probability of encounter, given availability, (b) probability of kill, given encounter, and the product of (a) and
(b) equal the relative probability of death, (c) P(k) following Lima and Dill (1990) where P(k)0/1(/exp((adT). Shown are
wolf telemetry locations (m, n 0/119), encounters with elk (', n0/77), and wolf killed elk (, n0/118). Relative probabilities were
derived using resource selection functions (Manly et al. 2002).

consistent with earlier non-spatial work on predation        actions on the predation stage that had the greatest effect
                                  or flexibility to management.
risk (Hebblewhite and Pletscher 2002). In contrast to
                                    Our objective was to illustrate the importance of
grasslands, dense cover in pine forests may render elk
                                  decomposing predation risk. In applications where
more vulnerable to predation by wolves because detec-
                                  predicting predation risk for management or conserva-
tion distance may be reduced or woody deadfall may
                                  tion is the main priority, the need to model predation
slow escape of fleeing prey (Kunkel and Pletscher 2000).
                                  risk with a consistent set of covariates would not
Because our analysis was conducted at a resolution of a
                                  unnecessary. One would simply select the best model
30-m pixel we did not measure deadfall, but resource
                                  for each predation stage, or where model selection
attributes at the microsite level ( B/30 m) could be
                                  uncertainty arose, adopt a model averaging approach
measured to test more mechanistic hypotheses (Kunkel
                                  (Burnham and Anderson 1998). Total predation risk
and Pletscher 2004).
                                  over the landscape would then result from the combined
  Insights into the mechanisms of predation may have
                                  effects of the best encounter and kill models. Further-
important management and conservation implications.
                                  more, if information about encounters were unavailable,
For example, endangered mountain caribou (Rangifer
                                  predation risk could be approximated using a direct
tarandus tarandus ) are thought to spatially separate from
                                  comparison of kills and availability of habitats to
predators by migrating to high elevations to reduce the
                                  predators and prey. One important caveat is that we
risk of encountering wolves (Seip 1992). Recent caribou
                                  focused on a single wolf territory and ignored density in
declines are hypothesized to arise from the combined
                                  our example because wolf numbers over the study were
effects of a numeric response in wolves from increasing
                                  relatively constant. Thus our relative spatial predation
alternative prey (moose) density as a result of early seral
                                  risk function would be valid for the Cascade pack we
habitats from forestry (Terry et al. 2000) and/or in-        modeled. Yet, total predation risk is not only related to
creased encounter rates by wolves due to human-           the spatial predation risk function, but to the numeric,
modified trails (Seip 1992). Cause-specific survival data      or spatial density of predators (Messier 1994, Kristan
support the hypothesis that predation plays a key role in      and Boarman 2003). This is similar to non-spatial
caribou decline (Seip 1992, Kinley and Apps 2001).         predator-prey dynamics where total predation rate is a
However, demonstrating whether roads increase encoun-        function of both the functional and numeric responses
ter rates, or that roads or attributes associated with roads    (Messier 1994). Thus, while the predation risk function
themselves make prey more susceptible to predation         (Eq. 1) will identify risky habitats, for example, pine for
once encountered may require different mitigating          elk, predation risk may vary across pack territories
actions. Without the decomposition of predation risk,        dependent on the number of predators, wolf pack size in
these and other mechanistic hypotheses about landscape       our example. To incorporate the effect of density on
effects on predation risk would be difficult to test.        predation risk, Kristan and Boarman (2003) weighted
Decomposition of predation risk can focus conservation       predation risk for tortoises by raven density to estimate

                                                                 109
OIKOS 111:1 (2005)
total predation risk. Therefore, for situations of multiple  ecology and landscape ecology in real ecological land-
                               scapes (Lima and Zollner 1996, Lima 2002).
predators or packs, different packs should be weighted
according to pack size, although this assumes predation
                               Acknowledgements Á/ Funding was provided by: Parks Canada,
risk responds linearly to predator density, which may not   the Central Rockies Wolf Project, Univ. of Montana, Univ. of
always be the case (Messier 1994). Assessing the func-    Alberta, Alberta Sustainable Resource Development, Alberta
                               Conservation Association, Alberta Enhanced Career
tional response of predation risk as a function of
                               Development, Paquet Wildlife Fund, Rocky Mountain Elk
predator density (sensu Mysterud and Ims 1998) using     Foundation, Foothills Model Forest, Patagonia, and the Canon
this approach would provide a considerable advance in     Á/ National Parks Science Scholarship for the Americas (MH).
                               Elk data were provided by Parks Canada. We thank the dozens
our understanding of spatial predation risk.
                               of field assistants who made this study possible, safe and
  We believe the most exciting opportunities for this    exceptional fixed-wing aircraft support by Mike Dupuis, and
approach is in their application to spatial models of     strong logistical support from Tom Hurd, Dave Dalman, Cliff
                               White and Dave Norcross, Parks Canada. We thank Mark
predator Á/prey dynamics. Predator Á/prey dynamics have
                               Boyce, Stan Boutin, Mark Lewis, Cormack Gates, Paul Paquet,
been modeled spatially using lattice-networks (Tobin     Dan Pletscher, Jacqueline Frair and Hawthorne Beyer for
and Bjornstad 2003), heuristic simulation models       helpful discussion. Reviews by Dan T. Haydon and Nathan
                               Varley greatly improved the manuscript.
(Donalson and Nisbet 1999), and individually-based
models (McCauley et al. 1993), to name a few ap-
proaches. Most models simplify landscape structure into
a few patch types, and model predator Á/prey dynamics in
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                                                     Subject Editor: Esa Ranta




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