Modelling the spatial population dynamics ofarctic foxes: the effects of red foxes andmicrotine cycles

Mark D.F. Shirley, Bodil Elmhagen, Peter W.W. Lurz, Steve P. Rushton, andAnders Angerbjorn

Abstract: The Fennoscandian arctic fox (Vulpes lagopus (L., 1758)) population is critically endangered, possibly becauseof increased interference competition from red foxes (Vulpes vulpes (L., 1758)) and fading cycles in microtine rodents,which cause food shortage. It is not known how these factors drive arctic fox population trends. To test their role in arcticfox decline, we developed a spatially explicit and individual-based model that allowed us to simulate fox interactions andfood availability in a real landscape. A sensitivity analysis revealed that simulated arctic fox population size and den occu-pancy were strongly correlated with fecundity and mortality during the microtine crash phase, but also with red fox status.Model simulations suggested that arctic fox population trends depended on microtine cycles and that arctic fox distribu-tions were restricted by red fox presence. We compared the model predictions with field data collected at Vindelfjallen,Sweden. The model recreated the observed arctic fox trend only with the inclusion of arctic fox avoidance of red foxhome ranges. The results indicate that avoidance behaviours can affect population trends and hence that relatively smallnumbers of red foxes can have a strong negative impact on arctic fox population size and distribution.

Resume : La population fennoscandienne de renards polaires, Vulpes lagopus (L., 1758), est serieusement menacee, peut-etre a cause de l’augmentation de la competition d’interference de la part du renard roux, Vulpes vulpes (L., 1758), et del’attenuation des cycles de rongeurs microtines qui cause une penurie de nourriture. On ne sait pas de quelle maniere cesfacteurs controlent les tendances demographiques des renards polaires. Afin de verifier leur role dans le declin des renardspolaires, nous avons mis au point un modele defini spatialement et base sur l’individu qui nous permet de simuler les in-teractions des renards et la disponibilite de nourriture dans un paysage reel. Une analyse de sensibilite montre que la taillede la population et l’occupation des terriers simulees chez le renard polaire sont en forte correlation avec la fecondite et lamortalite durant la phase d’effondrement des microtines, mais aussi avec le statut des renards roux. Les simulations dumodele laissent croire que les tendances des populations de renards polaires dependent des cycles de microtines et que lesrepartitions des renards polaires sont restreintes par la presence des renards roux. Nous comparons les predictions du mod-ele aux donnees de terrain recoltees a Vindelfjallen, Suede. Le modele recree la tendance observee chez le renard polaireseulement si on inclut l’evitement par les renards polaires des domaines vitaux des renards roux. Ces resultats indiquentque les comportements d’evitement peuvent affecter les tendances demographiques et qu’en consequence, de petits nom-bres de renards roux peuvent avoir un important effet negatif sur la taille et la repartition des populations de renards po-laires.

[Traduit par la Redaction]

Introduction

The arctic fox, Vulpes lagopus (L., 1758), was one of theearliest postglacial colonizers of Fennoscandia. It remainedin the tundra habitat of the Fennoscandian mountain chainas the climate turned milder, and peak population sizeswere still in the order of ten thousand foxes in the 19th cen-tury (Tannerfeldt 1997; Linnell et al. 1999b). However, ex-tensive hunting in the early 1900s caused a drasticpopulation decline and the arctic fox has not recovered de-spite complete protection in Sweden, Norway, and Finland

since 1940 (Lonnberg 1927; Hersteinsson et al. 1989). To-day, it is classified as critically endangered and the Fenno-scandian population consists of approximately 120 adultsfragmented into a number of subpopulations (Angerbjorn etal. 2004a; Dalen et al. 2006; Elmhagen 2008). Occasionalimmigrants from the arctic fox population of the Kola Pen-insula and northwestern Russia have kept the genetic varia-tion in Fennoscandia relatively high, but immigrants are toorare to directly impact population size (Dalen et al. 2006;Nystrom et al. 2006).

The failure to recover to former levels after protection has

Received 27 January 2009. Accepted 28 August 2009. Published on the NRC Research Press Web site at cjz.nrc.ca on 25 November2009.

M.D.F. Shirley,1 P.W.W. Lurz, and S.P. Rushton. Institute for Research on Environment and Sustainability, Newcastle University,Newcastle upon Tyne NE1 7RU, UK.B. Elmhagen and A. Angerbjorn. Department of Zoology, Stockholm University, SE-106 91 Stockholm, Sweden.

1Corresponding author (e-mail: m.d.f.shirley@newcastle.ac.uk).

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Can. J. Zool. 87: 1170–1183 (2009) doi:10.1139/Z09-104 Published by NRC Research Press

been related to two environmental factors: increased compe-tition with red foxes, Vulpes vulpes (L., 1758), and poor re-production due to food shortage (Hersteinsson et al. 1989;Linnell et al. 1999b; Elmhagen 2008). Microtine rodents, inparticular Norway lemmings (Lemmus lemmus (L., 1758))but also voles (Microtus Schrank, 1798 and Myodes Pallas,1811 spp.), are the main prey of arctic foxes. These micro-tines are a key food resource, since the reproductive successof the arctic fox, and consequently the size of the arctic foxpopulation, depend strongly on their availability (Angerbjornet al. 1995; Kaikusalo and Angerbjorn 1995; Elmhagen et al.2000). Microtine dynamics is generally cyclic but there havebeen occasional interruptions to the cycles in the 20th cen-tury. These have involved extended periods with irregularand low-amplitude peaks only. It is known that the lastsuch period, occurring between 1982 and 2001, had a nega-tive impact on small rodent specialists such as the arctic fox(Elmhagen et al. 2000; Angerbjorn et al. 2001; Ims and Fu-glei 2005). In addition, it has been suggested that arcticfoxes are limited by interspecific competition with the largerred fox (Hersteinsson and Macdonald 1992). The red fox hasbecome more abundant in Fennoscandia since the 19th cen-tury (Elmhagen and Rushton 2007). Hunting bags indicatethat the population may have tripled in size between 1930and 1960 (Selas and Vik 2007). Red fox presence has alsoincreased in the Fennoscandian mountain tundra, where thered fox was rare before the 20th century (Haglund and Nils-son 1977; Østbye et al. 1978). In parallel with this expan-sion in red fox distribution, which primarily is directedtowards productive mountain tundra at low altitudes, arcticfoxes have retreated to dens situated at the higher altitudesof their former range. Hence, both arctic and red foxes nowbreed in mountain tundra habitat, but dens with arctic foxesare found, on average, 80 m above dens with red foxes (Lin-nell et al. 1999a; Dalerum et al. 2002; Frafjord 2003; Kill-engreen et al. 2007). This appears to be a behaviouralresponse where arctic foxes avoid lower altitudes to reducered fox competition over home ranges as well as the risk ofred fox predation on their cubs (Tannerfeldt et al. 2002; Da-len et al. 2004). However, it is not known whether avoid-ance behaviours in themselves are sufficient to explain theapparent exclusion of arctic foxes from low-altitude habitat,or whether additional interference mechanisms such as intra-guild predation are critical. Furthermore, it is not knownwhether the only effect of these behaviours is to determinewhere the few remaining arctic foxes choose to breed, orwhether avoidance behaviours actually drive arctic fox pop-ulation trends.

Whilst the availability of microtines, particularly lem-mings, and competition with red foxes may both affect thearctic fox population, their impacts are unlikely to be con-stant over time and space. Firstly, lemming availability de-pends on the phase in the lemming cycle. However, dens inmore productive habitat at lower altitudes offer higher den-sities of alternative prey, such as voles and birds (Oksanenand Oksanen 1992; Elmhagen et al. 2002; Killengreen et al.2007). Access to low-altitude habitats may therefore be im-portant to arctic foxes during low points in the lemmingcycle, and arctic foxes have a higher probability of repro-ducing when they occupy dens at lower altitudes (Linnell etal. 1999a). Secondly, the intensity of red fox competition

changes over time. The main part of the red fox populationis found in forests surrounding the mountain tundra, wherethe rodent community is dominated by voles. Red fox repro-ductive success is related to the forest vole cycle and thusthe number of red foxes that disperse to the tundra and es-tablish home ranges there (Englund 1970; Elmhagen 2003).Since arctic foxes avoid dens situated close to dens occupiedby red foxes, this implies that arctic foxes are at least tem-porarily excluded from parts of the mountain tundra(Tannerfeldt et al. 2002).

Modelling is useful in situations where controlled experi-ments or replicated studies cannot be carried out, for exam-ple on rare, endangered species. The non-recovery of thearctic fox population is clearly a spatial issue. Consequently,there is a need for a spatially explicit model of arctic foxdynamics (Loison et al. 2001). Stochastic simulation model-ling is a tactical modelling approach that attempts to dealwith specific aspects such as population viability or manage-ment in real landscapes. Such models can combine popula-tion modelling with a landscape map (for examples, seeLetcher et al. 1998; Rushton et al. 1999; Shirley et al.2003). Hence, individuals can be located in specific habitatpatches where habitat quality can affect their life histories,and ecological processes can be modelled in a real land-scape. It is therefore possible to study effects on a popula-tion of changes in the landscape (for examples, seeDunning et al. 1992) or the effects of management (Turneret al. 1995). Spatially explicit population modelling has alsobeen used for studies of interspecific competition in caseswhere introduced species have spread and interacted withnative species (Rushton et al. 1997, 2000). Sensitivity analy-ses of individual-based models identify the dependencies ofthe modelled output on life-history parameters and otherprocesses, and thus can be used to explicitly test ecologicalhypotheses (Shirley et al. 2003).

Here we develop a spatially explicit population model forthe arctic fox population in the Vindelfjallen NatureReserve, Sweden. The model simulates the dynamics of arc-tic foxes in relation to microtine cycles and interactions withred foxes. Our hypothesis is that the combined effects of therecent interruption in microtine cycles and the encroachmentof the red fox population are the cause of the decline in thearctic fox population. These two effects are simulated inde-pendently in the model; however, we predict that (i) micro-tine cycles alone are insufficient to explain the decline inarctic foxes, and (ii) avoidance of red foxes by arctic foxesis both necessary and sufficient to explain arctic fox popula-tion trends, against the background of the microtine cycles.

Materials and methods

Study area and empirical backgroundThe study area modelled was located in the nature reserve

of Vindelfjallen in northern Sweden (668N, 168E). The areahas been one of the strongholds of the arctic fox in Fenno-scandia, but the population decreased during the interruptionin the rodent cycle in 1982–2001. Some 20 arctic fox litterswere born in lemming peak years in the 1970s, while only 3litters were born in the lemming peak year in 2001 (Anger-bjorn et al. 1995; Elmhagen 2008). The arctic foxes in Vin-delfjallen were not genetically isolated during the study

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period, i.e., there was at least some migration, but owing tothe natural fragmentation of the Fennoscandian mountainchain, the study area comprised an isolated cluster of arcticfox dens (Dalen et al. 2006; Herfindal et al. 2009).

The western parts of the 100 km � 75 km area were do-minated by mountain tundra intersected by forested rivervalleys, while boreal forests dominated the eastern parts.The treeline was situated 750 m a.s.l. Up to 1150 m a.s.l.,the tundra consisted primarily of dry heaths, while boulderfields covered the bedrock at higher altitudes. The averagearctic fox den was found 915 m a.s.l. (range 760–1160 ma.s.l., N = 83) (Fig. 1; Dalerum et al. 2002). The main preyavailable for foxes above the treeline were Norway lem-mings and, depending on year and altitude, field voles(Microtus agrestis (L., 1761)) (Elmhagen et al. 2000, 2002).Alternative prey such as voles and birds were more availableat low altitudes close to the treeline (Elmhagen et al. 2002).The forest offers even larger quantities of alternative prey,but field voles remain the most important prey for red foxesand the vole cycle is reflected in the population dynamics ofred foxes (Englund 1965).

To parameterize the model regarding prey availability,field data collected from northern Sweden and Vindelfjallenbetween 1985 and 2000 on microtine abundances were cate-gorized into one of four qualitative classes: crash, low, in-crease, and peak years, based on a simplified version of theclasses in Angerbjorn et al. (2001). Lemming abundanceswere derived from Angerbjorn et al. (2001) and Tannerfeldtet al. (2002). Vole abundances in the boreal forest were ob-tained from a small mammal monitoring programme(Swedish Environmental Protection Agency, http://www.emg.umu.se/projects/hornfeldt/index3.html).

The arctic fox dens in Fennoscandia have been used forhundreds of years. They are large structures covered withconspicuous lush vegetation, which makes them easy tofind and survey. Dens in Vindelfjallen have been surveyedevery summer since 1985 by the Swedish Arctic Fox Project(e.g., Angerbjorn et al. 1995; Dalerum et al. 2002;Elmhagen et al. 2002; Tannerfeldt et al. 2002). Hence, for

each year during the study period, we had field data on thenumber of dens occupied by arctic foxes, litter sizes, theproportion of adult foxes breeding, and adult mortality. Thefield data were used to parameterize the model regarding thelife history of the arctic fox during different phases in themicrotine cycle (Table 1).

When red foxes breed above the treeline, they generallyuse arctic fox dens rather than digging their own (Dalerumet al. 2002; Tannerfeldt et al. 2002). Hence, surveys of arc-tic fox dens imply that breeding foxes of both species arefound, although red foxes are more shy of people and conse-quently more difficult to study. To parameterize the modelregarding the life history of red foxes, we used publisheddata on red fox litter sizes and the proportion of vixensbreeding at different phases of the vole cycle in northernSweden (Englund 1970). Summer surveys of arctic fox densin peripheral parts of the study area were focused on denswhere winter surveys had indicated arctic fox presence.Hence, low-altitude arctic fox dens, which are preferred byred foxes, were not always surveyed. To compare modeloutput with real population trends, we compared the outputfor only a core study area, where all arctic fox dens weresurveyed each summer and all breeding red foxes shouldhave been found (Fig. 1).

The interaction between arctic and red foxes was mod-elled in line with empirical evidence of arctic fox behaviourin Vindelfjallen. Adult arctic foxes regularly move up to6 km from their dens during summer (Tannerfeldt et al.2002). The larger red fox should require a 90% larger homerange (Hersteinsson and Macdonald 1992). Assuming circu-lar home ranges, this indicates that red foxes regularly moveup to 8.3 km from their breeding dens. Tannerfeldt et al.(2002) showed that arctic foxes avoided breeding in dens lo-cated within this distance from a den with breeding redfoxes. On the few occasions that arctic foxes did breedwithin 8.3 km of red foxes, there was a high risk of red foxpredation on their cubs. However, red fox predation on arc-tic fox cubs was never observed at dens situated farther than8.3 km from breeding red foxes, and 90% of the arctic foxlitters were born in such dens (Tannerfeldt et al. 2002).Hence, the main mode of interaction between arctic and redfoxes appears to be arctic fox avoidance of potential habitatsurrounding dens inhabited by breeding red foxes.

Overall structure of modelWe developed a spatially explicit population dynamics

model written in the C programming language which simu-lated the life histories of individual arctic and red foxes inthe landscape. The dynamics of the fox populations wasmodelled in response to the availability of den sites, the mi-crotine cycles recorded in the field, and in the case of thearctic fox, avoidance of red fox dens. A yearly time stepwas employed.

Arctic and red foxes show life-long or sequential (social)monogamy and they usually remain in the same home rangeonce they have established (Macdonald 1979; Goszczynski1989; Tannerfeldt and Angerbjorn 1996; Tannerfeldt et al.2002). They can live in social groups with more than oneadult female, but most fox groups in northern Sweden con-sist of one adult pair and their cubs. In cases where there areadditional adults in the home range, these are usually year-

Fig. 1. Study area in Vindelfjallen, northern Sweden. The regionwithin the ellipse is the core area for which field data on inhabiteddens and population sizes were available for all years. Dark greyshading indicates regions below the treeline, light grey shading in-dicates areas with treeline dens, and white indicates areas with tun-dra dens. The approximate locations of known arctic fox (Vulpeslagopus) dens are indicated as black points.

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ling offspring of the breeding pair (Lindstrom 1988;Angerbjorn et al. 2004b). The model mirrored the reproduc-tive system of foxes in northern Sweden. The spatial unitmodelled was the fox social group, which occupied a homerange centred on a breeding den and consisted of a breedingpair of foxes and a variable number of juveniles. Mean mor-tality and fertility rates depended on the phase of the micro-tine cycle (Table 1). The model was stage-structured(Caswell 1989) insofar as discrete life-history stages (adultsand juveniles) were recognised, but the mortality, litter size,and dispersal rates appropriate to each life-history stagewere applied at the level of the individual within each socialgroup.

Reproduction occurred once a year, with only one litterper social group, produced by the dominant female. Thenumber of cubs in each litter was estimated by drawing de-viates from a normal distribution with the mean and stand-ard deviation of the number of cubs varied as a model input(following Akcakaya et al. 1995; Rushton et al. 2000). Thelargest litter sizes observed in Vindelfjallen during the fourmicrotine phases (crash, low, increase, and peak years)were 4, 9, 12, and 18, respectively (unpublished data).Therefore, litter size was truncated to be between zero anddouble the mean litter size (Table 1).

Mortality was modelled at the level of the individual foreach age class (adult and juvenile). Mortality in adult foxeswas assumed to occur after breeding. In juveniles, mortality

occurred as they were recruited into the adult populations. Ifone fox in a pair died, one of the last year’s young was pro-moted to adult status. Last year’s young dispersed if theydid not inherit their natal den through philopatry.

Dispersing foxes moved from natal dens to new dens,where settlement was determined on the basis of a set ofrules. Firstly, a den had to be either empty or occupied byfewer than two adult foxes of the same species. Secondly,an optimal strategy was employed in that dispersing foxesmoved to the closest suitable den. If no dens were availablefor the dispersing fox to move to, the fox was assumed toemigrate from the study area or die (equivalent fates in themodel).

Microtine cycles and fox food availabilityThe area modelled contained forest and tundra habitats,

and the microtine species available for foxes differed be-tween the two habitats. The availability of lemmings andvoles as food for foxes was modelled explicitly, where densites in different habitats were assigned to different classes.

We used the actual arctic fox den locations in our model,and randomly distributed further dens in the forest, forwhich we had no empirical data. Each den in the modelledlandscape was classified as a ‘‘forest’’ den (den in the forest,up to the treeline at 750 m a.s.l.), a ‘‘treeline’’ den (den be-tween the treeline and 915 m a.s.l.), or a ‘‘tundra’’ den (denabove 915 m a.s.l.). Foxes in tundra dens were assumed to

Table 1. Life-history parameters for arctic foxes (Vulpes lagopus) and red foxes (Vulpes vulpes) used in the model.

Life-history parameterRodentphase Arctic fox Red fox Source

Mean litter size (no. of pups) Crash 2.5 3.0 Arctic: Angerbjorn et al. (1995) and unpublished data; red:calculated from Englund (1970)

Low 5.1 4.5Increase 7.0 4.5Peak 10.1 5.8

Mean proportion breeding Crash 0.05 0.25 Arctic: estimated from Angerbjorn et al. (1995), Macpherson(1969), and unpublished data; red: calculated from Englund(1970)

Low 0.10 0.57Increase 0.40 0.57Peak 0.70 0.63

Juvenile annual mortality Crash 0.95 0.90 Estimated from Macpherson (1969) and Meijer et al. (2008)Low 0.6 0.6Increase 0.5 0.5Peak 0.5 0.5

Adult annual mortality Crash 0.60 0.60 Arctic: unpublished data; red: no reliable data available (seetext)

Low 0.44 0.44Increase 0.44 0.44Peak 0.35 0.35

Critical distance between redand arctic foxes (m)

8300 na Tannerfeldt et al. (2002)

Red fox settlementcoefficient

na –0.0087 Gradient of binary logistic regression

Note: Available data on litter sizes originated from observations of cubs at weaning for arctic foxes but from placental scars for red foxes. The latterwere reduced by 0.5 to adjust for early mortality. For both species, mean proportion breeding was calculated from data on the proportions of vixens ofdifferent ages that breed during different microtine phases. Data on yearly juvenile mortality for red foxes in northern boreal and tundra habitats do notexist. We assumed the same mortalities as for arctic foxes, but with a slightly lower mortality during the crash phase, since Englund (1970) found lowermortality in red fox juveniles during their first summer than Meijer et al. (2008) found in arctic foxes. For adult red foxes, we assumed the same yearlymortality rates as for adult arctic foxes. na, not available.

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have lemmings as their main food source, foxes in forestdens were assumed to have voles as their food source, andfoxes in treeline dens had access to both microtine species(Englund 1965; Elmhagen et al. 2002). Vole and lemmingcycles were considered separately for forest and tundradens, respectively. Since field voles are the most importantmicrotine species for foxes in forest, data on this specieswere used to represent the vole cycle. For dens classified astreeline dens, the highest phase of either voles or lemmingswas assigned to that den, as it was assumed that both wereavailable as food and the most numerous microtine specieswould determine fox breeding success and survivorship. Ateach time step, the microtine phase for each den type in themodelled landscape was an input to the model.

Den use: The interaction between arctic and red foxesThe model tested the hypothesis that scramble competi-

tion for den sites was the main mode of interaction betweenthe two species. All dens in the modelled landscape wereconsidered to be available for occupation by foxes of bothspecies. However, dens were not equally suitable for bothspecies. Red foxes prefer productive habitats, for exampleforests over tundra, and there is considerable evidence thatthey prefer low-altitude dens in more productive habitatswhen they establish in the tundra (Linnell et al. 1999a;Dalerum et al. 2002; Frafjord 2003; Killengreen et al.2007). We therefore explicitly modelled den preference byred foxes by altitude, but determined arctic fox den usagesolely by proximity to dens occupied by red foxes. Foxhome ranges can encompass more than one den, but onlyone is occupied.

To model den preferences we used the occupancy of densat the study site from 1985 to 1999 to determine a settle-ment function, which described the preference that red foxeshad for dens on the basis of their altitude. The presence orabsence of red fox litters observed at a den at any point dur-ing the study period (based on 16 observations) was fitted toa binary logistic regression model with altitude as the pre-dictor variable. Altitude was a significant predictor of redfox presence, with 69.6% concordance between recordedand predicted presence for red foxes. The regression coeffi-cient was used to derive probabilities of den occupancy byred foxes at each den site in the model (Fig. 2), and the gra-dient of the regression — the red fox settlement coefficient— was varied in the sensitivity analysis. Data after 1999were not used, since efforts to control red fox numbersthrough hunting were initiated as part of an arctic fox con-servation programme, and this might have affected red foxsettlement.

Simulated den use by arctic foxes was determined byproximity of a red fox; a dispersing arctic fox would not oc-cupy a den that was within 8.3 km of a den occupied by redfoxes. In all cases it was assumed that once occupied, an in-dividual den site could not be taken over by either speciesunless it was vacated by reason of mortality.

Parameterization of fox life historiesThe model required parameters for births, deaths, and dis-

persal as well as the settlement probability described above.These data were derived from field studies and the literature

(Table 1). Litter size, proportion of pairs breeding, and mor-tality were all varied according to the microtine phase.

Fox life-history patterns are typified by high juvenilemortality and lower adult mortality (Harris 1977; Macdonaldet al. 1998; Heydon and Reynolds 2000). We used unpub-lished data on adult mortality of arctic foxes in Vindelf-jallen. Macpherson (1969) estimated the average juvenilemortality of arctic foxes to be around 55%, and field dataindicate crash phase mortalities close to 100%. Juvenilemortality in a Swedish subpopulation 300 km from Vindelf-jallen was 92% in the crash phase, but this subpopulationwas the target of conservation actions that may have low-ered juvenile mortalities (Meijer et al. 2008). Most cubs areborn in good years and we therefore assumed that juvenilemortalities would be between 50% and 60% during allphases but the crash phase. Published data on red foxes gen-erally originate from populations in far more productivehabitats than northern Sweden, and we found no comparabledata on mortality. We therefore used the same mortalities asfor arctic foxes, but with a slightly lower mortality duringthe crash phase, since red foxes tend to have access to moretypes of alternative prey.

Foxes are considered good dispersers, and dispersal dis-tances up to 300 km for red foxes and above 1000 km forarctic foxes have been recorded (Pulliainen 1965; Wrigleyand Hatch 1976; Allen and Sargeant 1993). The largest dis-tance between dens in the modelled area was less than100 km, and so we did not limit dispersal distance of thefoxes. Arctic and red fox juveniles usually disperse duringthe first autumn or winter, and dispersal was simulated lastin each year (Andrews et al. 1973; Strand et al. 2000). Foreach year the processes of birth, death, and dispersal weresimulated for each den. Dens were selected in a random se-quence, regardless of the species of fox resident in that den.

Fig. 2. Parameterization of the red fox settlement function from abinary logistic regression of den occupancy by red foxes (Vulpesvulpes) at Vindelfjallen, 1985–1999: P(occupancy) =e6.304 – 0.009 � altitude / (1 + e6.304 + 0.009 � altitude). For this regression,G = 5.96, p = 0.015. The dotted lines indicate the boundaries be-tween the three habitat types.

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Sensitivity analysisThe sensitivity of the model to life-history parameters was

investigated by analysing how population size and popula-tion persistence varied in response to variations in model in-puts. Latin hypercube sampling (LHS) following themethods of Vose (1996) was used to select input parameters.LHS uses stratified sampling without replacement to selectsuites of input parameters from known distributions of life-history variables. In practical terms, the probability distribu-tion for each variable is split into n intervals of equal proba-bility, where n is the number of sets or suites of inputvariables selected. Once selected, an interval cannot be usedagain. The aim was to provide a range of input values foreach variable that could potentially occur under field condi-tions (sensu Bart 1995). A full discussion of the methodol-ogy for the sensitivity analysis can be found in Shirley etal. (2003).

Where insufficient data were available to identify the dis-tribution function for a parameter, a uniform distributionwas assumed with upper and lower limits derived from theliterature. The use of uniform distributions for all of the in-put variables leads to the selection of values for parametersthat are near the extremes of their distributions more fre-quently than would be expected in reality. However, it alsoensures that all potential values (within the known range ofobserved behaviours for each variable) are sampled. Onethousand sets of the input parameters were selected and 100replicates of each parameter set were simulated, each simu-lation lasting 15 years. The starter population for each runconsisted of the populations of animals present in 1985. Foreach fox species, seven outputs from the simulation modelwere generated at the end of a 15 year simulation. Each ofthese outputs consisted of the median of the 100 replicates.These were the number of foxes and the number of occupieddens for each of the three den types, and the number of in-dividuals that failed to settle in the modelled area and thuswere considered to have emigrated. These data were thencorrelated with the input variables, and partial correlationcoefficients were calculated to assess the impact of the indi-vidual life-history parameters on the dynamics of the foxpopulation in the landscape as simulated by the model.

Comparing model outputs with observed populationdynamics of the two fox species

A habitat and altitude map of the study area (100 km �75 km) was stored in GRASS, a geographical informationsystem (Fig. 1). Within this landscape, the locations of 83arctic fox dens above the treeline were known from fieldwork in the area. We assumed that red fox dens in the forestwould occur at a density comparable to that of arctic foxdens on the tundra. Based on habitat availability (3000 km2

of forest), 130 dens were randomly distributed below thetreeline. The simulation model was run for the whole of thearea in Fig. 1, but we retrieved model output data only fromwithin the smaller ellipse, leaving the rest as a ‘‘bufferzone’’ providing a source of arctic and red foxes. Extensivemonitoring of arctic fox dens had been carried out withinthis smaller area in 1985–1999 (Angerbjorn et al. 1995),providing field data with which we could compare model re-sults.

The model was used to simulate the dynamics of the arc-

tic fox population in the absence of red foxes in response toreal microtine cycles. The model was used to simulate45 years of microtine cycles, with the 15 years of microtinecycles recorded at Vindelfjallen repeated three times to givea 45 year data set. Twenty replicate sets were performed andthe median den occupancy was calculated. We then simu-lated the dynamics of the arctic fox population in responseto the observed microtine phase in the presence of the redfox. This scenario of 15 years was replicated 20 times, andthe median number of dens occupied by arctic foxes wascalculated.

The model predictions were compared with observed arc-tic fox dynamics from the Vindelfjallen region between1986 and 2000. During this period, red foxes bred in thetreeline and tundra regions of the study area. Observed mi-crotine phases for each year were used. This scenario of15 years was replicated 100 times, and the median numberof arctic foxes, along with the lower and upper quartiles,was calculated.

ResultsThe five most significant variables in the sensitivity anal-

ysis have been ranked for each model output (Tables 2 and3 for arctic foxes and red foxes, respectively). For arcticfoxes, the most important microtine phase determining pop-ulation size was the crash phase for all arctic fox life-historyparameters. The highest partial correlation coefficients (Ap-pendix A) for arctic fox population size and den occupancywere adult and juvenile mortality in crash years, mean littersize in crash years, and the red fox settlement coefficient.The critical distance between red and arctic foxes was alsoa significant correlate with population size and den occu-pancy.

Adult mortality in low and crash phases was a statisticallysignificant correlate with den occupancy and population sizein tundra dens, less so in treeline and forest dens, respec-tively. This reflects the fact that adult arctic foxes occupyhome ranges for life, and no active aggression of red foxesdriving out arctic foxes was included in the model. In in-crease and peak years, arctic fox litter size was ranked amore important predictor in the lower altitude dens than inthe higher altitude dens. In these years, arctic foxes aremore likely to be found at lower altitudes owing to higherlevels of recruitment.

For arctic fox emigration, mean litter size in the crashyears and juvenile and adult mortality in low and crashyears were the most significant correlates. All of these pa-rameters influenced the number of arctic fox subadults try-ing to find home ranges.

Overall, red fox life-history parameters were significantcorrelates only with arctic fox population size and den occu-pancy of treeline and tundra dens. The settlement coefficientfor red foxes also followed this pattern. Red foxes and arcticfoxes are likely to interact only in dens above the treeline,and in general, parameters resulting in a higher red fox pop-ulation at high altitudes result in lower population sizes forarctic foxes.

Red fox population size and den occupancy were largelyunaffected by the presence of arctic foxes, but no processwhereby the arctic foxes could affect red foxes was included

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Table 3. Summary of the sensitivity analysis for red foxes.

Den occupancy Population size

Rank Forest Treeline Tundra Forest Treeline Tundra Emigrants1 Mean litter size,

crashRed fox settlement

coefficientRed fox settlement

coefficientMean litter size,

crashRed fox settlement

coefficientRed fox settlement

coefficientMean litter size,

crash2 Adult mortality,

crashMean litter size, crash Mean litter size, crash Adult mortality,

crashMean litter size, crash Mean litter size,

increaseAdult mortality,

crash3 Mean litter size,

increaseMean litter size,

increaseMean litter size,

increaseJuvenile mortality,

crashMean litter size,

increaseMean litter size, crash Juvenile mortality,

low4 Juvenile mortality,

crashAdult mortality, crash Adult mortality, crash Mean litter size,

increaseJuvenile mortality,

crashAdult mortality, crash Juvenile mortality,

crash5 Mean litter size,

lowJuvenile mortality,

crashMean litter size, low Mean litter size,

lowAdult mortality, crash Mean litter size, low Mean litter size,

increase

Note: The significant life-history variables are listed in order of the ranks of their partial correlation coefficients. The data can be found in Appendix A (Table A2).

Table 2. Summary of the sensitivity analysis for arctic foxes.

Den occupancy Population size

Rank Forest Treeline Tundra Forest Treeline Tundra Emigrants1 Red fox settlement

coefficientJuvenile mortality,

crashJuvenile mortality,

crashJuvenile mortality,

crashJuvenile mortality,

crashJuvenile mortality,

crashMean litter size,

crash2 Juvenile mortality,

crashMean litter size, crash Mean litter size,

crashRed fox settlement

coefficientMean litter size, crash Adult mortality,

crashJuvenile mortality,

crash3 Mean litter size, crash Red fox settlement

coefficientAdult mortality,

crashMean litter size, crash Adult mortality, crash Mean litter size,

crashJuvenile mortality,

low4 Adult mortality, crash Adult mortality, crash Distance to red fox

denAdult mortality, crash Mean litter size, low Mean litter size,

lowAdult mortality, low

5 Distance to red foxden

Distance to red foxden

Adult mortality,low

Mean litter size, low Red fox settlementcoefficient

Distance to red foxden

Mean litter size, low

Note: The significant life-history variables are listed in order of the ranks of their partial correlation coefficients. The data can be found in Appendix A (Table A1).

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in the model. Adult mortality and mean litter size, alongwith juvenile mortality in crash years, were the most impor-tant correlates with red fox population size. The red fox set-tlement coefficient was vital for den occupancy at high-altitude tundra and treeline dens, but was less important forforest dens. The number of red foxes emigrating from themodelled area was significantly affected by the same param-eters that affected overall population size of red foxes,although juvenile mortality in low years had a higher partialcorrelation coefficient for emigration than for populationsizes (Appendix A, Table A2).

In the absence of red foxes, predicted den occupancy byarctic foxes increased steadily (Fig. 3), increasing by 50%in between peak den occupancies (from year 7 to year 22and year 12 to year 27). This process continued until birthsand deaths in the tundra dens reached an equilibrium of 27occupied dens.

When red foxes were included in the model (Fig. 4), thenumber of dens occupied by arctic foxes was greatlyreduced owing to the exclusion of den sites caused by thecritical distance parameter of the model. The first 15 yearswithout red foxes (from Fig. 3) are plotted alongside thesedata for comparison. Without the presence of red foxes inthe model, arctic foxes established themselves in all the tun-dra dens, spreading to dens at lower altitudes whenever themicrotine cycle was sufficient to produce an excess of dis-persing foxes.

The model successfully predicted arctic fox populationsize (Fig. 5). The observed data nearly always fell withinthe range of the upper and lower quartiles of the model pre-dictions, and in most years had a good match to the pre-dicted data. The correlation coefficient for the medianmodel result and the field data was 0.767 (n = 15, p < 0.01)for population size (Fig. 5a) and 0.815 (n = 15, p < 0.01) forden occupancy (Fig. 5b).

DiscussionOverall, this spatially explicit modelling approach was

able to recreate the observed dynamics of the arctic fox atthe study site in northern Sweden (Fig. 5). Critically, the in-clusion of both microtine cycles and the avoidance of redfox denning sites by arctic foxes was necessary to achieve a

match between the simulated and observed population dy-namics. Microtine cycles on their own were insufficient toexplain the observed population decline (Fig. 4). These find-ings illustrate the success of the approach and its potentialutility to the conservation of arctic foxes. Spatially explicitmodels are increasingly used in conservation and wildlifemanagement because they attempt to link population model-ling with real landscapes where the management is to becarried out. Whilst these models have an intuitive appeal,they are invariably complex because they attempt to simu-late life-history processes in heterogeneous landscapes, afeature that otherwise precludes the use of analytical model-ling approaches. The use of spatially explicit populationmodels (SEPMs) has been criticised because they attempt tosimulate processes that are often poorly understood(Wennergren et al. 1995; Lima and Zollner 1996) or haverequirements for data that are difficult to obtain (Ruckel-shaus et al. 1997). On the other hand, SEPMs have beensuccessfully used in a tactical sense to investigate the impactof individual processes such as predation and dispersal onmodelled populations of mink and water voles (Rushton etal. 2000), interspecific competition in tree squirrels (Rushtonet al. 1997; Lurz et al. 2001), and socially mediated diseasetransmission in badgers (Shirley et al. 2003).

The model recreates the life history of both arctic and redfoxes in the study area. The key processes were mortality,litter size, and interspecific competition. Mortality and littersize of arctic foxes are well understood and were based on15 years of field research carried out at Vindelfjallen. Thesedata are lacking only with regards to juvenile (cub) mortal-ity in some phases of the microtine cycle, because of thepractical difficulty of recapturing these individuals in thefield (Meijer et al. 2008). In contrast, the life histories ofred foxes in these habitats are poorly understood, and it wasnecessary to make assumptions based on arctic fox life his-tories. This assumption may have led to underestimation ofthe impact of red foxes.

Fig. 3. Effect of rodent cycle on arctic fox (V. lagopus) den occu-pancy in the absence of red foxes (V. vulpes). Each point is themedian of 20 replicates.

Fig. 4. Effect of the presence of red foxes (V. vulpes) on den occu-pancy by arctic foxes (V. lagopus). Observed rodent phases wereused. Each point is the median of 20 replicates, except for the redfoxes, where the maximum of the 20 replicates is shown.

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Our first hypothesis was that microtine cycles alone werethe cause of the observed decline in arctic fox numbers atVindelfjallen. The sensitivity analysis of the model revealedthat microtine phase had an influence on the importance ofcertain life-history parameters. In particular, litter size andmortality in crash years were important drivers of both denoccupancy and population size, and low-phase litter size wasalso of high significance. These results support the conclu-sions of field studies by Angerbjorn et al. (1995) and Strandet al. (1999). They highlight the importance of food avail-ability to the population dynamics of arctic foxes, and indi-cate that further research into the impact of microtine cycleson fox life histories is required. Red fox demography anddistribution were also significantly affected by crash phasesof the microtine cycles. However, red foxes prefer low-altitude dens where voles predominate, and crashes in volenumbers may not be in synchrony with crashes in lemmingnumbers (Oksanen and Oksanen 1992).

The second aim of this study was to investigate whethersimple avoidance of red foxes by arctic foxes could explainthe observed decline in arctic foxes. We found that the ma-jor significant predictors of arctic fox population sizes were

associated with den choice and the mechanism of interactionbetween the two species.

Assessing the validity of the model of interaction betweenthe two species is more problematic. Spatial modelling ofcompetitive interaction between species has been undertakeninfrequently. This is because the processes underlying suchinteractions are often complex, and largely behavioural innature. Behaviour is notoriously difficult to model (Limaand Zollner 1996), in contrast to other interspecific interac-tions such as predation, where there is a clear consequenceof the interaction in the mortality of the individual prey.Studies that have included consideration of competitionhave either explicitly limited the population of the animalin question owing to the presence of a superior competitor(e.g., Vucetich and Creel 1999), or have allocated resourcesto the superior competitor first and then allowed the otherspecies to use what remained (e.g., Rushton et al. 1997).The former mode of action does not accurately reflect theinteraction between red and arctic foxes in our study area,and the second results in competitive exclusion. In thismodel, the interaction between the two fox species was con-sidered to be interference competition, and the mechanismmodelled was scramble competition for den sites, with densite choice by arctic foxes determined by the proximity ofdens occupied by red foxes. Is this a realistic mechanismfor explaining the interaction between the two species? Thisform of intraguild interference competition has been ob-served in a number of canid species. The extinction ofwolves in North America was followed by an increase incoyote abundance, while coyotes were extirpated on IsleRoyale after the arrival of wolves in the 1950s and declinedin Yellowstone National Park after wolf reintroduction inthe 1990s (Mech 1970; Berger et al. 2008). Likewise, re-duced coyote populations in North America were followedby increased abundance of red foxes (Linhart and Robinson1972). Avoidance by the inferior species appears to result incompetitive exclusion from parts of the habitat and habitatsegregation. Coyote densities were higher in areas wherewolf densities were low, and coyote home ranges werefound outside or near wolf pack territory borders (Fullerand Keith 1981; Thurber et al. 1992). Red foxes establishedhome ranges outside or close to coyote territory borders, andwhere coyote and red fox home ranges overlapped, the areaof overlap was used less than expected or at different times(Voigt and Earle 1983; Major and Sherburne 1987; Sargeantet al. 1987; Harrison et al. 1989). On this basis, the mecha-nism modelled seems plausible. However, the model wasconservative in estimating the effects of red foxes on arcticfoxes because other interactions have been observed be-tween the two fox species. There is evidence that red foxeskill both adult and juvenile arctic foxes (Frafjord et al.1989). We did not include such interactions, and as such wemay have underestimated the impact of red foxes on arcticfoxes.

Despite the model’s success in recreating the overalltrends in arctic fox populations over time, the model failedto emulate the magnitude of the peaks of arctic fox num-bers, owing to the high extinction rate of arctic foxes in themodel. Of the 100 replicates that comprise the median datashown in Fig. 5, 21 replicates became extinct within 5 years.If only those replicates that survived the first 5 years are

Fig. 5. Comparison between the observed arctic fox (V. lagopus)population and the predictions of the model: (a) population size,(b) den occupancy. The solid line indicates the data at Vindelfjallen,the dotted line the median of 100 replicates. The shaded portion ofthe graphs indicates the region between the lower and upper quar-tiles.

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taken into consideration, the range in peak size is far morecomparable with that observed at Vindelfjallen (the meansquared difference between observed and simulated dropsfrom 252.7 to 162.8). The extinction rate of 21% foundhere is comparable with that predicted by Loison et al.(2001). In their population viability analysis, the probabilityof extinction for observed microtine cycles (at comparableadult survival rates) varied between 0% and 40%. Ourmodel showed that these high extinction rates were due tomassive fluctuations in fecundity. It is likely that the fecund-ity measured in the field will be more tightly linked to envi-ronmental predictors than either model has been able toinclude.

The model underpredicted the peaks in arctic fox numbersfollowing an increase year of microtines; such peaks werepresent in all replicate model runs, but the peak numberswere not as high as those observed in the field data fromVindelfjallen. Only the results for a core area within theoverall modelled system are presented, with the rest of thearea forming a buffer zone from which to provide immi-grants. However, immigrants into the whole modelled areafrom nearby populations were not considered, and this couldhave contributed to the reduced magnitude of the peakscompared with the field data. As a next step, there is a needto model Scandinavian populations as an interlinked meta-population system, allowing for dispersal between them(Loison et al. 2001), and this model is a first step towardsthat goal.

Without the presence of red foxes, arctic fox numberswere predicted to increase until the equilibrium was reached.During this increase, den occupancy of arctic foxes in-creased from 6 dens in the first year to a median of 10.5dens after 15 years and a median of 17.5 dens after 30 years.When red foxes were included in the model, arctic foxes de-clined, despite the fact that a maximum of only two dens peryear were occupied by red foxes in the tundra zone duringthis entire period. This level of red fox presence is similarto the one suggested by den surveys in the study area(Tannerfeldt et al. 2002). In the absence of red foxes in themodel there was greater stability in the model and popula-tion extinction was virtually nonexistent. When red foxeswere present, the dynamics of the arctic fox populationswas more variable, with greater fluctuation in populationsize and more predicted extinctions. Given that the modelledinteraction between the species was conservative in its ef-fects on arctic foxes, the results highlight that arctic foxpopulations are very sensitive to the presence of red foxes.This has considerable implications for any managementaimed at enhancing arctic fox, particularly if the manage-ment does not include the red fox.

We conclude that avoidance of red foxes may be themechanism behind recent changes in the distribution of arc-tic foxes in Fennoscandia as they respond to competitionfrom an expanding red fox population (Linnell et al. 1999a;Dalerum et al. 2002; Tannerfeldt et al. 2002; Frafjord 2003;Killengreen et al. 2007). Nonetheless, microtine cycles alsohave a significant effect on arctic fox demography, and evenwithout the presence of red foxes, the species may be threat-ened owing to the recent disruption of microtine cycles(Angerbjorn et al. 2001; Ecke et al. 2002; Ims and Fuglei2005). It is clear that, based on the predictions of the simu-

lation model, any efforts to control red foxes in the tundramust be very efficient, and should extend below the treelinebeyond the avoidance zone of arctic foxes reported byTannerfeldt et al. (2002). The Fennoscandian arctic fox pop-ulation was the target of an EU LIFE-Nature conservationprogramme, SEFALO+, from 1999 to 2007. The main ac-tions were red fox control and supplemental feeding. The re-sults of the programme support our model results. Only themost efficient method of red fox control, hunting fromsnowmobiles during winter, had a positive impact on localarctic fox populations. In the future, our model could beused to predict the outcome of different management scenar-ios suggested in a new action plan (Elmhagen 2008), and isthe first step towards an integrated spatially explicit metapo-pulation model of arctic fox population dynamics as sug-gested by Loison et al. (2001). A concerted conservationeffort that includes modelling is needed to avoid local ex-tinction of the arctic fox in Fennoscandia.

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Appendix AAppendix A begins on the following page.

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Table A1. Results of the sensitivity analysis for arctic foxes (V. lagopus).

Den Population

Life-historyparameter Rodent phase Forest Treeline Tundra Forest Treeline Tundra Emigrants

Arctic foxesMean litter size Crash 0.375 (3) 0.466 (2) 0.578 (2) 0.318 (3) 0.39 (2) 0.45 (3) 0.193 (1)

Low 0.222 (6) 0.237 (7) 0.273 (7) 0.263 (5) 0.332 (4) 0.378 (4) 0.103 (5)Increase 0.217 (7) 0.212 (8) 0.23 (8) 0.172 (8) 0.157 (9) 0.162 (9)Peak 0.065 (15) 0.088 (14) 0.102 (17) 0.088 (17) 0.081 (7)

Mean proportionbreeding

Crash 0.068 (21) 0.07 (23)

LowIncreasePeak

Juvenile mortality Crash –0.417 (2) –0.562 (1) –0.646 (1) –0.414 (1) –0.579 (1) –0.641 (1) –0.159 (2)Low –0.153 (10) –0.165 (12) –0.203 (9) –0.136 (10) –0.147 (12) –0.162 (8) –0.131 (3)Increase –0.088 (14) –0.126 (16) –0.111 (16) –0.072 (15) –0.104 (16) –0.091 (15) –0.068 (8)Peak

Adult mortality Crash –0.357 (4) –0.413 (4) –0.548 (3) –0.314 (4) –0.375 (3) –0.483 (2) –0.095 (6)Low –0.211 (8) –0.276 (6) –0.352 (5) –0.178 (7) –0.226 (7) –0.266 (6) –0.113 (4)Increase –0.173 (9) –0.181 (10) –0.169 (12) –0.146 (9) –0.154 (10) –0.133 (12)Peak

Red fox settlementcoefficient

0.436 (1) 0.434 (3) 0.341 (6) 0.34 (2) 0.326 (5) 0.207 (7)

Red foxesMean litter size Crash –0.097 (12) –0.146 (15) –0.123 (15) –0.108 (11) –0.146 (13) –0.131 (13) –0.064 (10)

Low –0.091 (19) –0.07 (19) –0.078 (21) –0.064 (21)Increase –0.175 (11) –0.149 (13) –0.152 (11) –0.119 (14)Peak –0.068 (22)

Mean proportionbreeding

Crash

LowIncreasePeak

Juvenile mortality CrashLow 0.064 (24)Increase 0.102 (18) 0.095 (17) 0.081 (19) 0.068 (20)Peak 0.097 (13) 0.085 (20) 0.09 (13) 0.088 (18) 0.07 (19)

Adult mortality Crash 0.103 (11) 0.21 (9) 0.173 (11) 0.092 (12) 0.181 (8) 0.144 (10)Low 0.109 (17) 0.085 (18) 0.079 (20)Increase 0.151 (14) 0.132 (14) 0.117 (15) 0.089 (16)Peak

Red fox settlementcoefficient

–0.156 (13) –0.182 (10) –0.124 (14) –0.135 (11)

Critical distancebetween red andarctic foxes

–0.229 (5) –0.385 (5) –0.376 (4) –0.188 (6) –0.318 (6) –0.306 (5)

Note: Each significant partial correlation coefficient (rpartial) has been given, along with the rank of the coefficient in the set of 35 parameters. For n =1000, the critical value for rpartial is 0.062 at the 5% level, 0.081 at the 1% level.

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Table A2. Results of the sensitivity analysis for red foxes (V. vulpes).

Den Population

Life-historyparameter Rodent phase Forest Treeline Tundra Forest Treeline Tundra Emigrants

Arctic foxesMean litter size Crash

LowIncreasePeak

Mean proportionbreeding

Crash

LowIncreasePeak

Juvenile mortality CrashLowIncreasePeak

Adult mortality Crash 0.081 (11) 0.064 (11)LowIncreasePeak

Red fox settlementcoefficient

Red foxesMean litter size Crash 0.487 (1) 0.444 (2) 0.32 (2) 0.424 (1) 0.387 (2) 0.267 (3) 0.232 (1)

Low 0.274 (5) 0.255 (7) 0.212 (5) 0.29 (5) 0.305 (6) 0.235 (5)Increase 0.346 (3) 0.427 (3) 0.317 (3) 0.294 (4) 0.377 (3) 0.275 (2) 0.085 (5)Peak 0.064 (14) 0.114 (11) 0.071 (12) 0.077 (10) 0.115 (11) 0.072 (7)

Mean proportionbreeding

Crash

LowIncrease –0.089 (12) –0.07 (12)Peak

Juvenile mortality Crash –0.341 (4) –0.334 (5) –0.21 (6) –0.34 (3) –0.371 (4) –0.226 (6) –0.133 (4)Low –0.201 (7) –0.177 (9) –0.155 (9) –0.175 (7) –0.163 (9) –0.144 (8) –0.136 (3)Increase –0.104 (9) –0.162 (10) –0.122 (10) –0.069 (11) –0.117 (10) –0.093 (10)Peak –0.08 (12)

Adult mortality Crash –0.441 (2) –0.407 (4) –0.271 (4) –0.389 (2) –0.369 (5) –0.242 (4) –0.166 (2)Low –0.241 (6) –0.239 (8) –0.161 (8) –0.192 (6) –0.204 (8) –0.139 (9) –0.079 (6)Increase –0.2 (8) –0.28 (6) –0.192 (7) –0.157 (8) –0.237 (7) –0.161 (7)Peak –0.085

(11)Red fox settlement

coefficient0.095 (10) 0.489 (1) 0.424 (1) 0.391 (1) 0.351 (1) –0.068 (8)

Critical distancebetween red andarctic foxes

0.067 (13)

Note: Each significant partial correlation coefficient (rpartial) has been given, along with the rank of the coefficient in the set of 35 parameters. For n =1000, the critical value for rpartial is 0.062 at the 5% level, 0.081 at the 1% level.

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