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Report Number 12/02
How linear features alter predator movement and the functional
response
by
Hannah W. McKenzie1, Evelyn H. Merrill, Raymond J. Spiteri
and Mark A. Lewis
Oxford Centre for Collaborative Applied Mathematics
Mathematical Institute
24 - 29 St Giles’
Oxford
OX1 3LB
England
How linear features alter predatormovement and the functional response
Hannah W. McKenzie1, Evelyn H. Merrill2, Raymond J. Spiteri3
and Mark A. Lewis1,2,*
1Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences,632 CAB, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
2Department of Biological Sciences, CW 405, Biological Sciences Building, Universityof Alberta, Edmonton, Alberta, Canada T6G 2E9
3Department of Computer Science, 176 Thorvaldson Building, University of Saskatchewan,110 Science Place, Saskatoon, Saskatchewan, Canada S7N 5C9
In areas of oil and gas exploration, seismic lines have been reported to alter the movement pat-terns of wolves (Canis lupus). We developed a mechanistic first passage time model, based onan anisotropic elliptic partial differential equation, and used this to explore how wolf movementresponses to seismic lines influence the encounter rate of the wolves with their prey. The modelwas parametrized using 5 min GPS location data. These data showed that wolves travelledfaster on seismic lines and had a higher probability of staying on a seismic line once theywere on it. We simulated wolf movement on a range of seismic line densities and drew impli-cations for the rate of predator–prey interactions as described by the functional response.The functional response exhibited a more than linear increase with respect to prey density(type III) as well as interactions with seismic line density. Encounter rates were significantlyhigher in landscapes with high seismic line density and were most pronounced at low prey den-sities. This suggests that prey at low population densities are at higher risk in environmentswith a high seismic line density unless they learn to avoid them.
Keywords: encounter rate; mean first passage time; seismic lines;spatial heterogeneity; wolf movement
1. INTRODUCTION
One of the most common functional response models isthe Holling disc equation [1]. In the disc equation, theencounter rate is assumed to be linearly related toprey density by the ‘instantaneous search rate’ [2] or‘area of discovery’ [1]. This leads to the type I functionalresponse, or the type II functional response once hand-lingQ1 time is included. Previous theoretical work suggeststhat this assumption of linearity may not hold if pred-ator movement is partially random, i.e. the new areasearched by the predator per unit time is not constant[3]. The results suggest that, in one dimension andunder certain assumptions regarding prey availability,the encounter rate of predators undergoing a randomwalk has a quadratic relationship with prey density.This quadratic relationship arises because during arandom walk, predators may repeatedly return toregions previously searched. When substituted intothe disc equation, McKenzie et al. [3] showed that aquadratic encounter rate leads to a type III functional
response, thereby demonstrating a link between preda-tor movement modes and the shape of the functionalresponse. McKenzie et al. [3] worked in a simplifiedtheoretical framework, but it is known that predatorssearching for prey in spatially heterogeneous habitatshave both directed and random components to theirmovement. In this case, the encounter rate is expectedto be a more complex function of prey density, perhapsinvolving both linear and quadratic terms, and there-fore possibly leading to some combination of thefamiliar type II and type III functional responses.
Owing to the challenges of reproducing the theoreti-cal arguments used by McKenzie et al. [3] underassumptions of more realistic predator movement inheterogeneous landscapes, here we take a differentapproach to investigate the link between predatormovement and the functional response. We first derivemore realistic predator movement models based onpredator movement data. Then, using these movementmodels, we simulate prey encounter rates over a range ofprey densities and fit the simulated encounter rate datato the encounter rate models. Finally, we substitute theencounter rate function into the disc equation to deter-mine the shape of the resulting functional response.
For the source of spatial heterogeneity, we focus onseismic lines, which are narrow, linear stretches of
*Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rsfs.2011.0086 or via http://rsfs.royalsocietypublishing.org.
One contribution to a Theme Issue ‘Mathematical and theroreticalecology’.
Interface Focus (2012) 00, 1–12
doi:10.1098/rsfs.2011.0086
Published online 00 Month 0000
Received 30 September 2011Accepted 19 December 2011 1 This journal is q 2012 The Royal Society
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forest, cleared for energy exploration ([4] and electronicsupplementary material, appendix figure S1a). Wolves(Canis lupus) have been found to both avoid and uselinear features, depending on the linear feature densityand level of human use [5–8]. Because studies haveshown that wolves moved up to 2.8 times faster onlinear features than in the forest [6], it has been hypoth-esized that seismic lines may benefit wolves if increasedtravel rates result in higher encounter rates with prey[9–11]. Consequently, in landscapes with high densi-ties of seismic lines, species such as caribou (Rangifertarandus) and elk (Cervus elaphus) may be at higherpredation risk owing to higher encounter rates [12–14].
In this paper, we use this example of wolf movementin response to seismic lines to demonstrate the linkbetween animal response to spatial heterogeneity andfunctional response models. We follow the mean firstpassage time approach described McKenzie et al. [3]extended to two dimensions. In the context of preda-tion, we define the mean first passage time as theaverage time required for a moving predator to locatea first stationary prey, given a specific prey density[3]. Therefore, the inverse of mean first passage timeis the encounter rate at that prey density. Using this fra-mework, it is possible to formulate first passage timemodels that reflect the effect of landscape features onpredator movement. After evaluating the influence ofdifferent movement responses of wolves on seismiclines, we use model outputs to assess a set of a priorimodels relating encounter rate to seismic line andprey density, both independently and together. Thefunctional forms of the candidate models are chosento reflect different underlying predator movementmechanisms. Using the best-fit model, we investigatethe implications for wolf functional responses owing toincreased predator mobility in the presence of seismiclines, and discuss how these ideas could be extendedto include other sources of landscape heterogeneity.
2. METHODS
2.1. Modelling
Encounter rate is the rate at which predators encounterprey in the landscape (unit: time21). Here, we considera mechanistic model for encounter rate that includes theeffect of predator movement. We modelled encounterrate using mean first passage time. Mean first passagetime, T(x) (unit: time), is the average time requiredfor a predator starting at location x to encounter anynumber of stationary prey, given a specified prey distri-bution and landscape. Although it would be possible toapproximate T(x) from averaging repeated randomwalk simulations from each point x of interest, analternative, computationally efficient approach uses apartial differential equation to describe the surfaceT(x). This approach is based on the Fokker–Planckapproximation for animal movement patterns that isdescribed in depth by Turchin [15] and in particularfor this model by McKenzie et al. [3]. To derive a partialdifferential equation for T, McKenzie et al. [3] begin byencoding aspects of wolf movement believed to beimportant in determining space use, in this case
focusing on wolf response to seismic lines, into a set ofprobabilistic movement rules for individual wolves.These are then translated, using mathematical approxi-mations, into the following partial differential equationfor T.
dxxðxÞ@2T
@x2þ 2dxyðxÞ
@2T
@x@yþ dyyðxÞ
@2T
@y2
þ cðxÞ � rT þ 1 ¼ 0; x [ V;
ð2:1Þ
where the domain V is the landscape of interest. In thismathematical formulation, as is the case in the Hollingdisc equation, the prey are assumed to be stationary andare represented by interior Dirichlet (absorbing disc)boundary conditions. This means that if the predatorstarts within perception radius r of a prey item, thetime required to locate a prey item, T, equals zero.Defining prey in this way allows us to study the behav-iour of the first passage time model for any finitenumber of prey, with spatial locations of our choosing.The particular prey scenarios we considered are furtherdescribed in the §2.3. The coefficients of the partialderivatives are derived mechanistically based on theunderlying predator movement behaviour (electronicsupplementary material, appendix). Equation (2.1)includes diffusive movement in the first three terms(dxx , dxy and dyy), and advective movement in thefourth term (c), which together approximate animalmovement. The dependence of the diffusion andadvection terms on the location x indicates that move-ment terms can vary from one location to other.Directionality in the movement terms can arise eitherfrom the advection term (c), which indicates a direc-tional bias in the movement, or from anisotropicdiffusion (unequal values for dxx , dxy and dyy), whichindicates different levels of random movement in differ-ent directions. For example, in the absence of otherinfluences, far from the landscape features predatorsmay move in a random fashion. However, their move-ment may become more directed as they interact withlandscape features.
The solution to equation (2.1) is a two-dimensionalsurface T(x), where the value of the surface at eachpoint in space is the mean first passage time of a predatorlocated at x. Therefore, the value of the surface at eachpoint in space indicates how long, on average, a predatorstarting x would need to search before locating a preyitem. This surface provides a picture of how the meanfirst passage time varies in space, and can be summarizedby the spatially averaged mean first passage time,
�T ¼
ðV
1
ATðxÞ dx; ð2:2Þ
where A is the area of V. The average mean first passagetime is the mean first passage time assuming that thepredator initially is randomly distributed in the land-scape. For a given landscape and prey density, theaverage mean first passage time can then be related tothe encounter rate by
E ¼1�T :
ð2:3Þ
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It would be possible to account for a non-random initialdistribution of the predator by replacing the uniformweight 1/A in equation (2.2) with a more general prob-ability density function for the initial location of thepredator u(x). For example, the distribution u(x) couldbe a statistical home range model, such as that ofKernohan et al. [16], or a mechanistic home rangemodel, such as that of Moorcroft & Lewis [17].
Based on previous studies of wolf movement in land-scapes with linear features [6,7], wolves are likely toshow varied responses to these features. In the simplestmodel (no response), wolves do not alter their move-ment in response to seismic lines. This model isanalogous to wolf movement in a landscape withoutseismic lines and corresponds to a random walk every-where in the landscape. Although this model may bebiologically unreasonable [5,8], it provides a baselinefor comparison in understanding the effect of increasingprey density on encounter rate in the absence of seismiclines. In the second model (anisotropic diffusion),wolves move faster on seismic lines and are morelikely to continue along them in either direction onceon them. Mechanistically, we considered the anisotropyas arising from higher movement speed in either direc-tion along the seismic lines as well as possiblecorrelations in the random walk when moving alongthe seismic lines. Both these mechanisms individuallywould lead to an enhanced diffusion coefficient [18]but only in the direction of the seismic line. This wasincorporated mathematically by allowing anisotropicdiffusion on seismic lines, where diffusion was increasedalong the seismic line and decreased across it. This
model corresponds to a random walk away from seismiclines and a random walk with anisotropic diffusion onseismic lines. The final model (anisotropic diffusion þbias) was an extension of the previous model where,in addition to being more likely to continue along seis-mic lines when on them, wolves also biased theirmovement towards the seismic line when near them.This model corresponds to a random walk far from seis-mic lines, a biased random walk near seismic lines, anda random walk with anisotropic diffusion on seismiclines. In addition to the terms previously discussed,the equation for this model also includes an advectionterm in regions near seismic lines, pointing in the direc-tion towards the nearest seismic line. Each of the threewolf movement models leads to different forms ofequation (2.1) as summarized in table 1. The detailsunderlying the calculation of the coefficients are givenin the electronic supplementary material, appendix.
2.2. Wolf movements
We studied the movements of four GPS-collared wolvesin the central east slopes of the Rocky Mountains,Alberta, Canada where average daily mean temperatureswere 27.58C in winter (January/February) and 18C inspring (March/April) and total snowfall was 54 cmin winter and 24 cm in spring. This area supportsprey populations of moose (Alces alces), mule andwhite-tailed deer (Odocoilues hemionus and Odocoilesvirginianus), and elk (Cervus canadensis), as well astheir main predator, wolves. Wolf densities ranged from9.7 to 22.3 wolves per 1000 km2 [19]. Seismic line density
Table 1. Summary of the proposed wolf movement models and the corresponding form of the mean first passage time equationfor each. Exact formulae for the coefficients are given in the electronic supplementary material, appendix.
model explanation and form of equation (2.1)
no response wolves do not alter their movement in response to seismic lineswolves move according to a random walk everywhere in the landscape
d @2T@x2
þ d @2T@y2
þ 1 ¼ 0; x [ V
anisotropic diffusion far from seismic lines wolves move randomlywolves move faster on seismic lines and are more likely to continuealong them in either direction once on them
d@2T
@x2þ d
@2T
@y2þ 1 ¼ 0; x off seismic lines
dxxðxÞ@2T
@x2þ 2dxyðxÞ
@2T
@x@yþ dyyðxÞ
@2T
@y2þ 1 ¼ 0; x on seismic lines
anisotropic diffusion þ bias far from seismic lines wolves move randomlywolves bias their movement towards seismic lines when near themwolves move faster on seismic lines and are more likely to continuealong them in either direction once on them
d@2T
@x2þ d
@2T
@y2þ 1 ¼ 0; x off seismic lines
dxxðxÞ@2T
@x2þ 2dxyðxÞ
@2T
@x@yþ dyyðxÞ
@2T
@y2
þcðxÞ � rT þ 1 ¼ 0; x near seismic lines
dxxðxÞ@2T
@x2þ 2dxyðxÞ
@2T
@x@yþ dyyðxÞ
@2T
@y2þ 1 ¼ 0; x on seismic lines
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varied from 0.18 km km22 near the Western border to4.4 km km22 near Rocky Mountain House, with amean of 1.8 km km22. It is likely that seismic linesexperienced a range of human use year-round for hunt-ing, trapping, snowmobiling, off-roading and hiking.
Q2 Movement data were obtained from GPS locations ofcollared wolves from four individuals in three packsduring January–April 2005, occupying territoriesacross a gradient of seismic line densities from 1.73 to
Q3 3.60 km km22 [20]. Collars were programmed to collectlocations at 5 min intervals and successfully recordedlocations on 90 per cent of fixed attempts. Data weredownloaded upon retrieval via a remote release mechan-ism (three wolves) or recapture (one wolf). Wolves wereconsidered to be independent units because they wereeither from different packs or the data were collectedduring different time periods.
We described movements via vectors joining the cur-rent and next consecutive wolf location. Each vectorwas characterized by step length and movement direc-tion (with direction North having 08). To investigatethe appropriateness of a diffusion-type model for move-ment, we calculated the mean-squared displacement foreach wolf for time intervals of length 5, 15, 30, 60 and120 min. A simple diffusion model with no correlations
Q2 predicts a linear increase in the MSD as a function oftime interval. We also calculated correlations in succes-sive movement directions using the circular correlationcoefficient (raa)s [21].
Our movement model characterized wolf movementvia step length (r. 0) and relative move direction.Relative move direction (21808 � j � 1808 ) is theangle between the ‘beeline’ move direction of theanimal and the direction of a straight line pointingfrom the current location towards the nearest seismicline (electronic supplementary material, appendixfigure S1). A relative move direction of j+ 908 rep-resents moves along the seismic line j ¼ 08 representsmoves towards the seismic line, and j+ 1808 representsmoves away from the seismic line. Because GPSmeasurement error may result in incorrect inference ofmove direction between locations that are less than 5s.d. of the GPS error kernel apart [22], we consideredonly those relative move directions with correspondingmove distances greater than 55 m [23]. However, weused all of the distances when calculating step lengthsso as to avoid introducing a bias towards longer moves.
To understand how the distance of a wolf to a seismicline affected the step lengths and relative move direc-tions of the wolves, we classified wolf locations intothree groups: on, near and off seismic lines. Seismiclines were assumed to have an average width of 5 mand were buffered by an additional 24.5 m on eachside to account for GPS measurement error in wolflocations [23]. Locations within the GPS error bufferwere classified as ‘on’. Locations between the GPSerror buffer and the distance at which we assumedwolves perceived seismic lines were classified as ‘near’.We arbitrarily chose a distance of 50 m to representthe distance at which seismic lines might be visible towolves. Locations beyond the perceptual range wereclassified as ‘off’. We quantified and compared thedistributions of each near, far and off seismic lines.
Step lengths of canids often follow the exponentialdistribution
f ðrÞ ¼ a exp½�ar�; ð2:4Þ
where a is the mean step length [17]. The maximum-likelihood estimate a is the sample mean, �r. Wecompared the mean step lengths on, near and off seis-mic lines using 90% confidence intervals obtained bynon-parametric bootstrapping [24].
The von Mises distribution is commonly usedto describe animal movement directions ([17] andelectronic supplementary material, appendix). Weassumed that the distribution of relative move direc-tions of wolves on seismic lines followed the bivariatevon Mises distribution
KðjÞ ¼1
4pI0ðkÞexp½k cosðjþ 90WÞ�
þ1
4pI0ðkÞexp½k cosðj� 90WÞ�; ð2:5Þ
where the movement directions are oriented in thedirection of the seismic line. To determine whether ani-sotropic movement was present, we tested the nullhypothesis H0 : k ¼ 0 (no tendency to continue alongthe seismic line in the next move) against the alterna-tive hypothesis Ha : k . 0 (tendency to continue alongthe seismic line in the next move) using the parametricbootstrap likelihood-ratio (PBLR) test [25,26]. Formoves near seismic lines, we assumed that the dis-tribution of relative move directions was univariatevon Mises,
KðjÞ ¼1
2pI0ðkÞexp½k cosðjÞ�: ð2:6Þ
where the movement directions are oriented towards theseismic line. To determine whether bias towardsthe lines was present, we tested the null hypothesisH0 : k ¼ 0 (no tendency to move towards the seismicline in the next move) against the alternative hypoth-esis H0 : k. 0 (tendency to move towards the seismicline in the next move) using the PBLR test [25,26]. Ineach case, the maximum-likelihood estimate k wasfound by numerical maximization of the likelihoodfunction and non-parametric bootstrapped 90%confidence intervals constructed [24].
To investigate the effect that possible correlations insuccessive movement directions would have on themovement patterns, we simulated 1000 wolf movementpaths based on choosing the turning angle and steplength randomly from the measured data values.These simulations were repeated for each individualwolf for situations on and off linear features and themean-squared displacement per day was calculated asa summary statistic.
2.3. Model scenarios
To study the effect of seismic line density and prey densityon encounter rate for wolves, we solved equation (2.1)numerically for various scenarios using COMSOLMultiphysics (COMSOL, Inc. Stockholm, Sweden). Wechose the domain V to be a 25 km � 25 km landscape,
4 Linear features and predator movement H. W. McKenzie et al.
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similar in area to the average home range size of thewolvesin the study area [19]. The edges of the domain were sub-ject to Neumann (reflecting) boundary conditions. Thiscorresponds to the wolf remaining within its homerange. We then computed the encounter rate for eachscenario using equation (2.3). Because wolves remain ata kill site for several hours or longer and typically donot hunt immediately after consuming prey, it is reason-able to assume that between hunting bouts, the preyspecies have time to move. Therefore, the assumption incomputing E from �T that the predator initially is ran-domly distributed with respect to the prey is notunreasonable. Although the movement data of threeout of the four wolves showed evidence of anisotropicdiffusion, only wolf 233 showed evidence of both aniso-tropic diffusion and bias towards seismic lines (see §3).Therefore, in order to compare the effects of the threedifferent movement models on encounter rate, weestimated the coefficients for all the three models (noresponse, anisotropic diffusion and anisotropicdiffusionþ bias) from the wolf movement data for wolf233 using the methods described in the electronicsupplementary material, appendix.
We generated simulated landscapes with varyingseismic line densities based on seismic line layers ofwest central Alberta mapped at a resolution of 5 musing Indian Remote sensing satellite imagery [27].A baseline seismic line density of 4.46 km km22 wasused to create landscapes with seismic line densi-ties ranging from approximately 2 km km22 to 9 kmkm22 (electronic supplementary material, appendixfigure S2). Prey were randomly placed in these landscapesat densities of 0.16, 0.5, 1, 1.5, 2, 2.5 and 3 prey km22.We chose to include the density of 0.16 as it is similarto the lower range of density of common prey species inour study area [19]. We assumed wolves encountered aprey when they came within radius r ¼ 100 m of theprey. In the model, this corresponds to a disc inthe domain with radius r ¼ 100 m, centred on the prey,with Dirichlet (T ¼ 0) boundary conditions.
We evaluated the effect of wolf movement responsesto seismic lines based on the first passage time solutionsfrom two sets of simulations. First, for a fixed prey
density of 0.16 prey km22, we simulated the effects ofthe three different movement models outlined intable 1. Simulations were iterated 10 times at each seis-mic line density using a different distribution ofrandomly located prey. Encounter rates were plottedagainst seismic lines densities and visually inspectedto assess the effects of movement responses to seismicline on encounter rates across a range of seismic linedensities. Second, we assumed wolves moved accordingto the anisotropic diffusion model (table 1) and solvedequation (2.1) for 10 replications of each prey densityand seismic line density (i.e. six prey densities � 10replications � 4 seismic line densities ¼ 240 model sol-utions). We chose the anisotropic diffusion movementmodel because we found that it reflected the greatesteffect of seismic lines on encounters with prey for ourinvestigation. In each case, we used T(x) to computethe encounter rate using equations (2.2) and (2.3). Weused the outputs of these simulations to evaluate a setof a priori candidate models (table 2) relating encoun-ter rates to seismic line or prey densities or both usingthe nonlinear regression analysis package nls in R andcomparing the fit of the models based on AIC [28] Q2.
The functional forms of the candidate models(table 2) were proposed based on the results obtainedearlier [3], which suggested that different underlyingpredator movement mechanisms lead to differentrelationships between encounter rate and prey density.McKenzie et al. [3] found that in one dimension, Q4theencounter rate for predators undergoing only advectivemovement was a linear function of prey density. This isconsistent with the assumption of a constant areasearched per time made by Holling [1]. In Q5contrast,McKenzie et al. [3] found that the encounter rate forpredators undergoing only random movement was aquadratic function of prey density. Depending on thedensity of seismic lines, predators in our model undergosome combination of anisotropic diffusion along seismiclines and random searching off seismic lines via isotropicdiffusion. Although the anisotropic diffusion model dif-fers mathematically from the advective movementmodel, there is a similarity. Anisotropic diffusionbiases movement in relation to both directions along
Table 2. Candidate models for encounter rate (E) of wolves as a function of prey density (N) and seismic line density (S).
model explanation form
single variate models: seismic line density
A1 linear E ¼ b0 þ b1S
A2 quadratic E ¼ b0 þ b1S þ b2S2
A3 exponential E ¼ AebS
single variate models: prey density
B1 directed search E ¼ b1N
B2 random search E ¼ b1N2
B3 combination of directed and random search, using a sum E ¼ b1N þ b2N2
B4 combination of directed and random search using an intermediate power E ¼ ANb
multivariate models
C1 no effect of seismic lines E ¼ ANb
C2 linear interaction between seismic lines and prey density E ¼ ANb þ b1NS
C3 nonlinear interaction between seismic lines and prey density (same power) E ¼ ANb þ b1NbS
C4 nonlinear interaction between seismic lines and prey density (different power) E ¼ ANb1 þ b1Nb2S
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the seismic line, whereas advection biases movement be-haviour in relation to a single direction. Based on thissimilarity, we consider the anisotropic diffusion todescribe a particular form of directed motion. There-fore, we proposed several model forms for theindependent and combined effects of prey density andseismic lines that have a mechanistic basis for describ-ing encounter rates when the search is a combinationof directed and random motion by analogy with theearlier results [3].
2.4. The functional response
The functional response f(N) describes the per capitakill rate as a function of prey density N. Here, weinvestigate the potential variation in the functionalresponse owing to increased predator mobility in thepresence of seismic lines. To model the functionalresponse, we use the Holling disc equation,
f ðN Þ ¼EðNÞ
1þ EðNÞTh; ð2:7Þ
where the encounter rate E depends on prey density andTh is the constant handling time. The handling time isdefined by Holling to be the sum of the attack time(including evaluating, pursuing and catching the prey)and the handling time (including processing and con-suming the prey). Traditionally, the encounter rate isassumed to be proportional to the prey density. Instead,we used the best multivariate model for encounter ratefrom table 2. Using the best model for encounter rate,we asked the question: how does the functional responsechange as the proportion of directed and random move-ment changes owing to increasing seismic line density?To answer this question, we compared the functionalresponse in landscapes with seismic line densities of 0,4.46 and 8.91 km km22 by computing the ratio of thefunctional response at each seismic line density tothat when there are no seismic lines present. If theratio is 1, then seismic lines do not alter the kill rate.If the ratio is greater than 1, the presence of seismiclines leads to an increase in the kill rate. The largerthe ratio, the larger the difference between the encoun-ter rates in landscapes with and without seismic lines,and the stronger the effect of seismic lines. To see ifthe magnitude of the effect of seismic lines on the func-tional response depended on the handling time, wecompared results assuming handling times for small-bodied (Th ¼ 10.6 h) and large-bodied (Th ¼ 20.4 h)prey [19].
3. RESULTS
3.1. Wolf movements
Visual inspection of the mean-squared displacements asa function of measurement time interval showed varia-bility between wolves but exhibited approximatelylinear growth in the MSD as a function of time,except at short time intervals (electronic supplemen-tary material, appendix figure S3). When the 5 minmove directions were constrained to include only thosewith step lengths of at least 55 m so as to remove
errors (§2), we found statistically significant positivecorrelations in wolf movement directions with positivecircular correlation coefficients of 0.20, 0.61, 0.33 and0.43 for wolves 230, 232, 233 and 234. Movement pat-terns of wolves in landscapes with seismic lines werenot consistent among individuals. All wolves had alonger mean step length on seismic lines than off seismiclines (figure 1a, p, 0.1). Wolves 230, 232 and 234 hada shorter mean step length off seismic lines than nearseismic lines, while wolf 233 had a shorter mean steplength near seismic lines. On seismic lines wolves 230,233 and 234 had distributions of relative move direc-tions that differed from the uniform distribution, withmoves along seismic lines occurring more often thanmoves in other directions (figure 1b, p, 0.001). Wolf232 moved randomly with respect to seismic lines. Incontrast, near seismic lines, only wolf 233 had a non-uniform distribution of relative move directions(figure 1c, p, 0.001). When off seismic lines, allwolves had uniform distributions of relative movedirections (figure 1d). We chose to use the move-ment parameters from the data for wolf 233 in oursimulations because wolf 233 showed evidence of allthe movement behaviours of interest (electronicsupplementary material, appendix table S1). The maxi-mum-likelihood fit of wolf 233 step lengths to theexponential probability density function yielded mean-squared displacement values of 0.017 km2 5 min21
off-lines and 0.043 km2 5 min21 on-lines (electronic sup-plementary material, appendix table S1) scaling up to4.90 km2 d21 off-lines and 12.4 km2 d21 on-lines. Thesimulations of movement patterns yielded estimates forthe off-line mean-squared displacement as 4.96 km2 d21
and the on-line mean-squared displacement of18.83 km2 d21 for wolf 233, indicating that successivecorrelations in move direction have the effect of increas-ing the on-line mean-squared displacement per unit timeby about 50 per cent. Other simulatedmean-squared dis-placements for wolves were 5.86, 9.88 and 7.92 km2 d21
(off-line) and 10.93, 24.08 and 32.26 km2 d21 (on-line)for wolves 230, 232 and 234, respectively.
3.2. Model scenarios
Examples of the solution to mean first passage timeequation (equation (2.1)) are shown in figure 2. Forany point x ¼ (x,y) in the domain, the value of themean first passage time surface is the average time fora wolf starting at that location to encounter a preyitem. Changes in prey density or seismic line densitycause local differences in the mean first passage timesurfaces. For example, surface height, the presence ofpeaks and the steepness of gradients differ betweenthe example surfaces. These local differences translateinto different average mean first passage time values,�T (see equation (2.2)), which we investigated furtherwith statistical models.
For a fixed prey density, encounter rate was constantwhen wolves did not alter their movement (no responsemodel) with response to seismic lines (figure 3a).Encounter rates increased linearly with seismic line den-sity when wolves followed a movement pattern resultingfrom faster movement on seismic lines and a tendency
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on near off on near off on near off on near off
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Figure 1. Mean step lengths and distributions of relative move directions of wolves calculated from 5 min GPS data. (a) Meanstep lengths of wolves on, near and off seismic lines. Bars are 90% non-parametric bootstrapped confidence intervals. Relativemove direction of wolves (b) on, (c) near, and (d) off seismic lines. Solid lines are the maximum-likelihood fit of the bestmodel chosen using the PBLR test.
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and the seismic line densities are 0, 4.46 and 8.91 km km22, respectively. As seismic line density increases, the values of the meanfirst passage time surface decrease, leading to a decrease in the average mean first passage time, �T . (d), There are no seismic lines,but prey density is increased to three prey km22. Increasing prey density also leads to a decrease in the mean first passage timeand the average mean first passage time.
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to continue along seismic lines once on them (anisotro-pic diffusion model). When an additional bias towardsseismic lines when wolves were near seismic lines wasincluded (anisotropic diffusion þ bias model), encoun-ter rates increased linearly but not as rapidly whencompared with the model without the bias.
Given that seismic lines had the greatest effect onencounter rates under the anisotropic diffusion move-ment model, we examined the joint effects of seismicand prey densities for this movement mode(figure 3b,c). The model providing the best fit for theeffects of seismic lines on encounter rate (electronic sup-plementary material, appendix table S2:A1) wasconsistently the linear model at all prey densities, butthe slope of the relationship increased with prey density(figure 3b) indicating an interaction between seismiclines and prey density under this movement mode. Apower model provided the best fit for relating encounterrate to prey density (electronic supplementary material,appendix, table S2:B4), where the constant scaling expo-nent (b), but variable A coefficient, also suggested anonlinear interaction (electronic supplementary
material, appendix table S3). Indeed, when encounterrate was modelled as a function of both prey and seismicline density, the best-fit model included a significantinteraction term between seismic line and prey density(p, 0.001, electronic supplementary material, appendixtable S3). The positive coefficient for the interactionterm showed that seismic line density enhanced therate at which the predators encountered prey.
3.3. The functional response
The observed increase in encounter rate due to increasingseismic line density translates into a functional responsethat increases more quickly to saturation (figure 3d,e). Inall cases, the ratio of the functional responses in land-scapes with and without seismic lines is greater than 1,meaning that the presence of seismic lines leads to anincrease in kill rates (figure 3f ). For handling times forboth small-bodied and large-bodied prey, the ratio islarger at low prey densities than at higher prey densities.This suggests that the effect of seismic lines on the func-tional response is larger when prey density is low. In
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Figure 3. Simulated encounter rates (equations (2.1)–(2.3)) and the effect of seismic line density on the encounter rate and func-tional response. (a) Encounter rates for the different movement models as a function of seismic line density when prey density isfixed at 0.16 prey km22 (squares, no response; circles, anisotropic diffusion; asterisks, anisotropic diffusion þ bias). (b) Encounterrate at each prey density for the anisotropic diffusion model. Lines show the fit of the best model for encounter rate, which was thelinear model in each case. (c) Encounter rate at each seismic line density for the anisotropic diffusion model. The dotted lineindicates a density of 0.16 prey km22 (compare with (a)). Lines show the fit of the best model for encounter rate, which wasthe power model in each case. (d) Predicted encounter rate for three seismic line densities. Encounter rate was predicted bythe model C3: E ¼ ANb þ b1 Nb S, where the coefficients are A ¼ 7.43 � 102 2, b ¼ 1.15 and b1 ¼ 2.74 � 1023 (solid lines, noseismic lines; dashed lines 4.46 km km22; dotted lines, 8.91 km km22). (e) The functional response, assuming handling timesfor small-bodied (black lines) and large-bodied (grey lines) prey. ( f ) The ratio of functional responses in landscapes with andwithout seismic lines, assuming handling times for small- and large-bodied prey.
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addition, across prey densities, the ratio of functionalresponses in landscapes with and without seismic linesis larger for small-bodied prey than for large-bodiedprey. Therefore, the effect of seismic lines is moreapparent when handling time is shorter.
4. DISCUSSION
All four wolves in this study, as well as wolves in theboreal forests of Alberta [6], have demonstratedincreased movement rates when travelling on seismiclines. Reduced debris and snow crusting on open seismiclines in winter compared with the forest may facilitatemovement along the lines [6,29]. Three of the fourwolves studied also exhibited a higher probability ofcontinuing along seismic lines once on them. The pro-pensity to remain on seismic lines is also reflected inhabitat selection studies, where wolves were found toselect seismic lines more than expected by chance [12].The rapid movement and use of linear features havecontributed to reports that wolves use them as travelroutes [5,7,9,10,12]. The strong directional persistenceof wolves while travelling on the seismic lines that wereport may have been shaped, in part, by the straight-ness of seismic line across the landscape relative toother linear features like roads and trails [30]. Wolf233 showed biased movement towards seismic lineswhen near them, which is the first quantification ofsuch a bias, despite other reports of wolves changingtheir direction in order to move directly to adjacentcompacted trails in nearby montane areas [7].
Incorporating these movement data into an advec-tion–diffusion framework, we assessed the implicationsof varied movement strategies using the first passagetime models on two movement components. We intro-duced anisotropy in the diffusion components of firstpassage models. At the same time, the possibility ofmovement bias towards seismic lines led us to introducea bias term via advection. Both of these movementresponses to seismic lines increased encounter rateswith prey over when seismic lines were absent, consistentwith the results of previous spatially explicit, individual-based models [31]. However, we also expected that thebias towards the seismic line, i.e. wolves being morelikely to get on seismic lines than to leave them, wouldresult in the highest encounter rates. This was not thecase. In fact, although increased mobility was advan-tageous in covering more area to find prey, a movementbias towards seismic lines resulted in wolves findingfewer prey that were positioned away from the seismiclines. Unless prey are attracted to seismic lines or existat very high densities (figure 3c), our results indicatethat it would not be advantageous for predators to biastheir movements towards seismic lines, explaining poss-ibly why the bias was not commonly observed amongthe wolves we studied.
The purpose of fitting statistical encounter ratemodels to the first passage time simulation results wasto assess how well the model parameters would reflectthe underlying movement mechanisms being modelled.Under the anisotropic diffusion model, the statisticalencounter rate models showed that wolf movement
did not follow either a purely directed search (modelB1) or a purely random search (model B2), even whenno effect of seismic lines was included. Instead, thebest model fit was intermediate, reflecting a mixtureof both movement modes. This model exhibited amore than linear increase with respect to prey density,which is consistent with type III functional response.This is consistent with the results of McKenzie et al.[3], who showed that the encounter rate of randomlydiffusing predators in one dimension was related toprey density by a power law.
Encounter rates increased linearly with seismic linedensity, but this increase was contingent on the distri-bution of prey with respect to the seismic lines [32].Simulations were based on real landscapes from thestudy area. For this study area, seismic lines havebeen shown to be distributed randomly across the land-scape, unlike some other linear features such as roads(E. H. Merrill, unpublished data) Q6. Because prey werealso assumed to be randomly distributed, the linearincrease directly reflects that wolves spent more timein directed movement in landscapes with higher seismicline densities than when seismic lines were not pre-sent. If prey were to avoid seismic lines, as James &Stuart-Smith [12] have reported for caribou (Rangifertarandus), encounter rates are expected to decline.However, as seismic line density increases, prey areless able to avoid seismic lines [33].
Our assumption that the ungulate prey did not moveis a simplification that we needed to make to apply firstpassage time analysis to a complex spatial environment.Clearly, prey can move during the search period,although typically at a slower rate than the wolves,and this would have some effect on the results. However,it is unlikely that the wolves were following the prey onthe seismic lines because their rates of movement weresubstantially higher than those of ungulate prey [13].We assumed that all prey encountered were killed andthat prey were encountered when wolves were within100 m of the prey. While the success in killing a preyonce detected can be highly variable, we simplified themodel for the purpose of understanding the influencemovements during search on functional response.
The presence of the interaction between prey densityand seismic line density in the best-fit model indicatesthat the effect of prey density on encounter rate is modi-fied by seismic line density. Therefore, it is not the effectof increasing seismic line density alone that leads toincreased encounter rate, but the interaction betweenseismic line density and prey density. We interpret thepositive coefficient of the interaction term to meanthat increasing seismic line density will have a strongereffect on encounter rate at high prey densities than atlow prey densities. Despite this interaction term leadingto a larger increase owing to seismic lines in the magni-tude of the encounter rate at high prey densities, usingthe Holling disc equation, we show that seismic lineshave the greatest relative influence on potential killrates in environments with low prey densities. This fol-lows because predator mobility constrains searchingsuccess more acutely at low prey densities. As prey den-sity increases, predators shift from being search limitedto being handling time limited, and the benefit of
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increased mobility diminishes. However, even at highprey densities increased search efficiency may stillalter predation rates when handling times are short.Although search time theoretically may not be the lim-iting process at high prey densities [1], wolves do notalways invest the time in consuming the full prey and,at the extreme they have demonstrated surplus killing[34], which is consistent with short handling time.Additionally, in multi-prey systems, high search effi-ciency in environments with seismic lines may increaseencounters with rare prey, that if more preferred or vul-nerable, may result in dietary shifts [35], particularly fora coursing predator like the wolf, whose broad scalemovements may homogenize spatial heterogeneity inprey [36]. Altered predation pressures within a preycommunity have implications of apparent competitionand prey persistence ([37–39], see also [40]).
The approximately linear relationship betweenmean-squared displacement and measurement timeinterval (electronic supplementary material, appendixfigure S3) indicates that a diffusion-based model maybe appropriate for modelling the movement of wolves.However, the issue of positive correlations in the set ofmovement directions retained for analysis has impli-cations for both the bootstrapping methods, whichimplicitly assume independence of data, and for the cal-culation of the diffusion coefficient, which assumes nopersistence in movement direction. This is a difficultissue to deal with in a satisfactory manner. Oneapproach for reducing correlations in move directionsis to subsample data by taking it over less frequenttime intervals [17]. However, we chose not to use thissubsampling approach because it would have removedthe important detailed information required to seehow wolves move in relation to roads over short spatialscales. A second approach is to rescale the estimate ofthe diffusion coefficient based on the simulations ofwolf movement on- and off-lines. This would have leftthe off-line diffusion coefficient the same but wouldhave increased the on-line diffusion coefficient byapproximately 50 per cent. An alternative method forcorrecting the diffusion coefficient, which leads to simi-lar results, is based on the work of Patlak (seeappendix C of Moorcroft & Lewis [17]). This attemptsto approximately correct for persistence in move-ment direction by rescaling the diffusion coefficient by1/(1 2 c), where c is the mean cosine of the turningangle, and the turning angle is the difference in consecu-tive movement directions. We measured that thequantity c averaged 0.18 off seismic lines and 0.43 onseismic lines, suggesting that correlations could makethe diffusion coefficient 22 per cent higher off seismiclines and 75 per cent higher on seismic lines. WhilePatlak’s approach can deal with spatial heterogeneity,it has not yet been extended to the case of anisotropy,which is a central element in our model. Regardless,we also chose not use either of these methods to rescalethe diffusion coefficient, noting that the variation calcu-lated between the different wolves is larger thancorrection term that such rescaling would entail. How-ever, had we made such a rescaling, the overall firstpassage times would be somewhat reduced, and theeffect of seismic lines on first passage times would be
enhanced. A third mathematically interesting approachto deal with correlations is to use a non-diffusive model,based a velocity jump process where there is persistencein movement direction. This non-diffusive Q7transportequation framework has only just recently been extendedto deal with anisotropic movement patterns [41] and theconnection with first passage time analysis has not yetbeen developed. Finally, a full analysis of correlationsmust also include autocorrelations over periods longerthan a single time step, which have the potential tofurther increase the mean-squared displacement per unittime (electronic supplementary material, appendixfigure S3), as well as interactions from measured cor-relations with GPS errors. We suggest that these areimportant avenues for future research.
Our understanding of the influence of predatorresponse to spatial heterogeneity on search behavioursand its implications for predator–prey interactions isnow emerging [3,42–45]. Because details on animal move-ments are now readily available with GPS technology [46],it may be possible to quantify distinct modes of movement[47–50] even if an understanding of motivation for themovement behaviours is less clear [51]. In this paper, wehave shown that a new alternative model structure forthe encounter rate component of the functional responseis appropriate when predators alter their movement inresponse to landscape heterogeneity. We illustrate thepoint for wolves in real landscapes where seismic line den-sities alter the directional component of search. In termsof conservation, our results indicate that increasing seis-mic line density or other linear features associated withland development that affects wolf movement will havea relatively large impact on single prey systems whenthe prey is at risk (i.e. low density) and small (i.e. smallhandling times) or on multi-prey systems where there ispreference for the rare species.
We wish to acknowledge support from Alberta Ingenuity,NSERC C-GSM, the University of Alberta and the MITACSMobility Fund (H.W.M.), an NSERC CRO-261091-02(E.H.M.), NSERC Discovery and Accelerator Grants and aCanada Research Chair (M.A.L.). This publication was basedon the work supported in part by Award no. KUK-CI013-04made by King Abdullah University of Science andTechnology (KAUST; M.A.L.). We thank N. Webb forproviding helpful feedback, H. Beyer for assisting with theGIS analysis and J. Berger for field expertise.
REFERENCES
1 Holling, C. 1959 The components of predation as revealedby a study of small mammal predation of the Europeanpine sawfly. Can. Entomol. 91, 293–320. (doi:10.4039/Ent91293-5)
2 Hassell, M. 1978 Arthropod predator–prey systems. No. 12in monographs in population biology. PrincetonUniversity Press Q8.
3 Mckenzie, H. W., Lewis, M. A. & Merrill, E. H. 2009 Firstpassage time analysis of animal movement and insightsinto the functional response. Bull. Math. Biol. 71,107–129. (doi:10.1007/s11538-008-9354-x)
4 Timoney, K. & Lee, P. 2001 Environmental managementin resource-rich Alberta, Canada: first world jurisdiction,third world analogue? J. Environ. Manag. 63, 387–405.(doi:10.1006/jema.2001.0487)
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5 Thurber, J., Peterson, R., Drummer, T. & Thomasma, S.1994 Gray wolf response to refuge boundaries and roads inAlaska. Wildl. Soc. Bull. 22, 61–68.
6 James, A. 1999 Effects of industrial development on thepredator–prey relationship between wolves and caribouin Northeastern Alberta. Universtiy of Alberta, Canada.
7 Callaghan, C. 2002 The ecology of gray wolf (Canis lupus)habitat use, survival, and persistence in the central rockymountains. University of Guelph, Canada.
8 Whittington, J., Clair St, C. C. & Mercer, G. 2005 Spatialresponses of wolves to roads and trails in mountain valleys.Ecol. Appl. 15, 543–553. (doi:10.1890/03-5317)
9 Edmonds, E. & Bloomfield, M. 1984 A study of wood-
land caribou (Rangifer tarandus caribou) in west central
Alberta, 1979 to 1983. Edmonton, Alberta, Canada: AlbertaEnergy and Natural Resources, Fish and Wildlife Division.
10 Bergerud, A. & Elliot, J. 1986 Dynamics of caribou andwolves in northern British Columbia. Can. J. Zool. 64,1515–1529. (doi:10.1139/z86-226)
11 Jalkotzy, M., Ross, P. & Nasserden, M. 1997 The effectsof linear developments on wildlife: a review of selectedscientific literature. Arc Wildlife Services LtdQ8 .
12 James, A. & Stuart-Smith, A. 2000 Distribution of caribouand wolves in relation to linear corridors. J. Wildl. Manag.
64, 154–159. (doi:10.2307/3802985)13 Frair, J. L., Merrill, E. H., Allen, J. R. & Boyce, M. S. 2007
Know thy enemy: experience affects elk translocationsuccess in risky landscapes. J. Wildl. Manag. 71,541–554. (doi:10.2193/2006-141)
14 McPhee, H. M. 2009 Kill site selection and attack rates of
wolves in heterogeneous environments in West Central
Alberta. University of Alberta.Q815 Turchin, P. 1998 Quantitative analysis of movement.
Sinaur Associates, Inc.Q816 Kernohan, B. J., Gitzen, R. A. & Millspaugh, J. J. 2001
Analysis of animal space-use and movements. In Radio
tracking and animal populations (eds J. J. Millspaugh &J. M. Marzluff), pp. 126–166. Academic Press.Q8
17 Moorcroft, P. & Lewis, M. A. 2006 Mechanistic home
range analysis. Princeton University Press.Q818 Turchin, P. 1991 Translanting foraging movements in het-
erogeneous environments into the spatial distribution offoragers. Ecology 72, 1253–1266. (doi:10.2307/1941099)
19 Webb, N. F. 2009 Density, demography, and functional
response of a harvested wolf population in west-central
Alberta, Canada. University of Alberta.Q820 Webb, N. F., Allen, J. R. & Merrill, E. H. 2011 Demogra-
phy of a harvested population of wolves (Canis lupus) inwest-central Alberta, Canada. Can. J. Zool. 89, 744–752. (doi:10.1139/z11-043)
21 Webb, K. 1975 Statistics for directional data. J. R. Stat.Soc. Ser. B (Methodological) 37, 349–393.
22 Jerde, C. L. & Visscher, D. R. 2005 GPS measurementerror influences on movement model parameterization.Ecol. Appl. 15, 806–810. (doi:10.1890/04-0895)
23 Mckenzie, H. W., Jerde, C. L., Visscher, D. R., Merrill, E.H. & Lewis, M. A. 2009 Inferring linear feature use in thepresence of GPS measurement error. Environ. Ecol. Stat.16, 531–546. (doi:10.1007/s10651-008-0095-7)
24 Efron, B. & Tibshirani, R. 1993 An introduction to the
bootstrap. Chapman and Hall/CRC.Q825 Barnard, G. A. 1963 Discussion of: the spectral analysis of
point processes (Barlett, MS). J. R. Stat. Soc. Ser. B
(Methodological). 25, 294.26 Dennis, B. & Taper, M. 1994 Density-dependence in time-
series observations and natural population: estimation andinference. Ecol. Monogr. 64, 205–224. (doi:10.2307/2937041)
27 Frair, J., Merrill, E., Visscher, D., Fortin, D., Beyer, H. &Morales, J. 2005 Scales of movement by elk (Cervus ela-
phus) in response to heterogeneity in forage resourcesand predation risk. Landscape Ecol. 20, 273–287.(doi:10.1007/s10980-005-2075-8)
28 Burham, K. P. & Anderson, D. R. 1998 Model selection
and multinomial inference, 2nd edn. Springer. Q829 Huggard, D. J. 1993 Effect of snow depth on predation
and scavenging by gray wolves. J. Wildl. Manag. 57,382–388. (doi:10.2307/3809437)
30 Whittington, J., St Clair, C. C. & Mercer, G. 2004 Pathtortuosity and the permeability of roads and trails towolf movement. Ecol. Soc. 9 Q9.
31 McCauley, E., Wilson, W. & De Roos, A. 1993 Dynamicsof age-structured and spatially structured predator–preyinteractions: individual-based models and population-level formulations. Am. Nat. 142, 412–442. (doi:10.1086/285547)
32 Cain, M. 1985 Random search by herbivorous insects: asimulation model. Ecology 66, 876–888. (doi:10.2307/1940550)
33 Frair, J. L., Merrill, E. H., Beyer, H. L. & Morales, J. M.2008 Thresholds in landscape connectivity and mortalityrisks in response to growing road networks. J. Appl.
Ecol. 45, 1504–1513. (doi:10.1111/j.1365-2664.2008.01526.x)
34 DelGiudice, G. D. 1998 Surplus killing of white-tailed deerby wolves in northcentral Minnesota. J. Mammal. 79,227–235. (doi:10.2307/1382858)
35 Garrott, R. A., Bruggeman, J. E., Becker, M. S., Kali-nowski, S. T. & White, P. J. 2007 Evaluating preyswitching in wolf-ungulate systems. Ecol. Appl. 17,1588–1597. (doi:10.1890/06-1439.1)
36 Holt, R. D. 1984 Spatial heterogeneity, indirect inter-actions, and the coexistence of prey species. Am. Nat.
124, 377–406. (doi:10.1086/284280)37 Holt, R. D. 1977 Predation, apparent competition, and
structure of prey communities. Theor. Popul. Biol. 12,197–229. (doi:10.1016/0040-5809(77)90042-9)
38 Noonburg, E. & Byers, J. 2005 More harm than good:when invader vulnerability to predators enhances impacton native species. Ecology 86, 2555–2560. (doi:10.1890/05-0143)
39 DeCesare, N. J., Hebblewhite, M., Robinson, H. S. &Musiani, M. 2010 Endangered, apparently: the role ofapparent competition in endangered species conservation.Anim. Conserv. 13, 353–362. (doi:10.1111/j.1469-1795.2009.00328.x)
40 McLellan, B. N., Serrouya, R., Wittmer, H. U. & Boutin,S. 2010 Predator-mediated Allee effects in multi-prey sys-tems. Ecology 91, 286–292. (doi:10.1890/09-0286.1)
41 Hillen, T. & Painter, K. 2012 Transport and anisotropicdiffusion models for movement in oriented habitats. InDispersal, individual movement and spatial ecology: a
mathematical perspective (eds M. A. Lewis, P. K. Maini &S. Petrovskii). Springer. Q10
42 Cuddington, K. & Yodzis, P. 2002 Predator–preydynamics and movement in fractal environments. Am.
Nat. 160, 119–134. (doi:10.1086/340611)43 Mols, C., van Oers, K., Witjes, L., Lessells, C., Drent, P. &
Visser, M. 2004 Central assumptions of predator–preymodels fail in a semi-natural experimental system.Proc. R. Soc. Lond. B 271(Suppl. 3), S85–S87. (doi:10.1098/rsbl.2003.0110)
44 Travis, J. & Palmer, S. 2005 Spatial processes can deter-mine the relationship between prey encounter rate andprey density. Biol. Lett. 1, 136–138. (doi:10.1098/rsbl.2004.0293)
Linear features and predator movement H. W. McKenzie et al. 11
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45 McPhee, H. M., Webb, N. F. & Merrill, E. H. 2011Time-to-kill: measuring attack rates in a heterogeneouslandscape with multiple prey types. Oikos. (doi:10.1111/j.1600-0706.2011.20203.x)Q11
46 Cagnacci, F., Boitani, L., Powell, R. A. & Boyce, M. S. 2010Theme issue ‘challenges and opportunities of using GPS-based location data in animal ecology’ preface. Phil.
Trans. R. Soc. B 365, 2155. (doi:10.1098/rstb.2010.0098)47 Morales, J., Haydon, D., Frair, J., Holsiner, K. & Fryxell,
J. 2004 Extracting more out of relocation data: buildingmovement models as mixtures of random walks. Ecology85, 2436–2445. (doi:10.1890/03-0269)
48 Fryxell, J. M. et al. 2008 Multiple movement modes bylarge herbivores at multiple spatiotemporal scales. Proc.
Natl Acad. Sci. USA 105, 19 114–19 119. (doi:10.1073/pnas.0801737105)
49 Van Moorter, B., Visscher, D., Benhamou, S., Boerger, L.,Boyce, M. S. & Gaillard, J. M. 2009 Memory keeps youat home: a mechanistic model for home range emergence.Oikos. 118, 641–652. (doi:10.1111/j.1600-0706.2008.17003.x)
50 del Mar Delgado, M., Penteriani, V., Nams, V. O. &Campioni, L. 2009 Changes of movement patterns fromearly dispersal to settlement. Behav. Ecol. Sociobiol. 64,35–43. (doi:10.1007/s00265-009-0815-5)
51 Nathan, R. 2008 An emerging movement ecology para-digm. Proc. Natl Acad. Sci. USA 105, 19 050–19 051.(doi:10.1073/pnas.0808918105)
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