Derivation of predator functional responses using a mechanistic approach in a natural system

The functional response is central to our understanding of any predator–prey system as it establishes the link between trophic levels. Most functional responses are evaluated using phenomenological models linking predator acquisition rate and prey density. However, our ability to measure functional responses using such an approach is often limited in natural systems and the use of inaccurate functions can profoundly affect the outcomes of population and community models. Here, we develop a mechanistic model based on extensive data to assess the functional response of a generalist predator, the arctic fox (Vulpes lagopus), to various tundra prey species (lemmings and the nests of geese, passerines and sandpipers). We found that predator acquisition rates derived from the mechanistic model were consistent with field observations. Although sigmoidal functional responses were previously used to model fox-prey population dynamics, none of our simulations resulted in a saturating response in all prey species. Our results highlight the importance of predator searching components in predator-prey interactions, especially predator speed, while predator acquisition rates were not limited by handling processes. By combining theory with field observations, our study provides evidences that predator acquisition rate is not systematically limited at the highest prey densities observed in a natural system. We reinforce the idea that functional response categories, typically types I, II, and III, should be considered as particular cases along a continuum. Specific functions derived with a mechanistic approach for a range of densities observed in natural communities should improve our ability to model and understand predator-prey systems.


Introduction
A long-standing problem in ecology is to measure how the consumption rate of a predator varies with prey availability. The functional response is at the core of predator-prey theory as it establishes the link between predator acquisition rate and prey density (Solomon 1949). Functional 24 response shapes are typically categorised as linear (type I), hyperbolic (type II) or sigmoidal (type III ;Holling 1959a,b). This classification is commonly used by ecologists when incorporating predation into population and community models (Fryxell et al. 2007;Serrouya et al. 2015;Turchin 27 and Hanski 1997), and type II is the most widely applied model (Rall et al. 2012). The shape of the functional response can have major consequences on the outcomes of population and community models. For instance, a type III promotes stability or coexistence whereas a type II destabilizes 30 predator-prey dynamics (Murdoch 1973;Sinclair et al. 1998). Describing the functional response of pairwise trophic interactions is also important to understand higher-order interactions. For instance, the shape of the functional response alone can profoundly change predictions about the 33 outcome of predator-mediated trophic interactions (Abrams et al. 1998;Holt and Bonsall 2017).
Quantifying and determining the shape of functional responses remains an important chal-36 lenge, especially in natural systems. Most empirical research on functional responses has been conducted under controlled laboratory or field enclosure conditions (96%, n = 116 studies, reviewed by Pawar et al. 2012) where prey density is manipulated, predator consumption is 39 recorded, and the functional response models are compared through statistical analysis. In natural systems, our ability to measure functional response is limited by a combination of factors: small sample size, a relatively narrow gradient of observed prey densities, the difficulty to ob-42 serve predator-prey interactions directly, or the difficulty to estimate predator and prey numbers (Ellis et al. 2019;Gilg et al. 2006;Suryawanshi et al. 2017;Therrien et al. 2014). The large variability around predator acquisition rates can also constrains our ability to fully discriminate among 45 functional response shapes, and hence limits our ability to accurately model predator-prey inter-actions in complex and natural ecosystems (Chan et al. 2017;O'Donoghue et al. 1998;Vucetich et al. 2002). Moreover, phenomenological models fail to identify the proximate mechanisms reg-48 ulating predator acquisition rates.
Derivation of functional responses based on measurable features of species behaviour (e.g. speed, attack and success probability) provides several advantages. Compared with phenomenological models, mechanistic models 1) allow assessing the shape of the functional response based on behavioral attributes of the predator, 2) are based on parameters with a direct biological in-54 terpretation, and hence have the potential to reinforce links between theory and data (Connolly et al. 2017). The number of mechanistic models of predator-prey interactions is growing, and most of them aim to predict trophic links based on species traits, especially body size (Gravel 57 et al. 2013;Ho et al. 2019;Portalier et al. 2019). Mechanistic models of functional response further allow the integration of predator-prey pairs to describe trophic links, which can improve our ability to model complex ecological interactions. 60 The main objective of our study is to develop a mechanistic model to characterize and quantify functional responses of a generalist mammalian predator to various prey species. The orig-63 inality of our approach is to assess functional response i) by focusing on four components of predation (searching, chasing, capturing, and handling prey) and ii) by using field experiments and detailed behavioral observations to parameterize each step of the mechanistic model. We 66 evaluated the coherence of our models using data from a long-term field study that estimated prey densities and predator acquisition rates. We also performed sensitivity analyses to identify the main proximate drivers of change in predator acquisition rates. Finally, we modeled the po-69 tential effects of density dependence in components of predation on the shape of the functional responses within the range of prey densities observed in the field.

72
The mechanistic model was developed for the arctic fox (Vulpes lagopus), a generalist predator of the tundra ecosystem, using highly detailed empirical observations from a long-term ecological monitoring program in the Arctic (Gauthier et al. 2013). This system offers several benefits 75 to study predator-prey interactions among vertebrates, including a relatively simple food web, an open landscape and the continuous summer daylight allowing direct behavioral observations. The arctic fox is an active hunting predator that travels extensive daily distances within its terri-78 tory in summer (M.-P. Poulin and D. Berteaux, unpublished manuscript). Lemmings and birds (mostly eggs and juveniles) are the main components of the summer diet of arctic foxes in most tundra ecosystems (Angerbjörn et al. 1999;Giroux et al. 2012). Lemmings exhibit population 81 cycles with peak density every 3-5 years (Fauteux et al. 2015), and the arctic fox predation pressure on tundra ground-nesting birds is typically released at high lemming density (Bêty et al. 2002;McKinnon et al. 2014;Summers et al. 1998). Surprisingly, the exact mechanisms driving 84 this well-known short-term apparent mutualism between lemmings and birds are still unclear, but they likely involve fox functional responses (Bêty et al. 2002;Flemming et al. 2016;Summers et al. 1998).

87
A few studies attempted to quantify the functional responses of arctic fox using phenomenological models (Angerbjörn et al. 1999;Eide et al. 2005;Gilg et al. 2006). Relatively low sample 90 sizes reduced the ability of previous studies to fully distinguish between different shapes of functional responses. Moreover, the hoarding behavior of arctic foxes was not considered in previous estimations of functional responses (Angerbjörn et al. 1999;Eide et al. 2005;Gilg et al. 2006). Like 93 many other animals (Vander Wall 1990), arctic foxes can predate more prey than they consume on the short-term, and such behavior can strongly increase prey acquisition rates, e.g. foxes foraging in goose colonies can hoard between 40% and 97% of eggs acquired during the bird 96 nesting period Samelius and Alisauskas 2000). Although type III functional responses were previously used to model fox-prey population dynamics (Gilg et al. 2003(Gilg et al. , 2009, food hoarding may substantially reduce handling time and could therefore make the shape of 99 the functional response linear or slightly convex (Oksanen et al. 1985).

Study system
102 During the summer, the southwest plain of Bylot Island, Nunavut, Canada (73 • N; 80 • W) harbors a large greater snow goose colony (Anser caerulescens atlanticus; ∼20,000 pairs). Insectivorous migratory birds are also nesting in the study area and include the lapland longspur (Calcarius 105 lapponicus), a passerine, and several species of shorebirds (primarily Calidris spp. and Pluvialis spp.). Two species of small mammals are present, the brown (Lemmus trimucronatus) and collared (Dicrostonyx groenlandicus) lemmings. The brown lemming has high-amplitude cycles of abun-108 dance with a 3-5-year periodicity, whereas the collared has low-amplitude cycles (Gruyer et al. 2008). The mammalian predator guild is dominated by the arctic fox and the ermine (Mustela erminea). The arctic fox is the main nest predator of geese (Bêty et al. 2002;Lecomte et al. 2008), Additional details on plant communities and general landscape can be found in Gauthier et al. (2013).

114
The model was parametrized and evaluated using data from Bylot Island, where foxes and their prey have been monitored since 1993. We observed foraging foxes using binoculars and 117 spotting scopes (20 x 60x) from one or two blinds located in the middle of the goose colony during 10 summers between 1996 and 2019. 120 We used the Holling disk equation as a starting point to build the mechanistic model of functional response (Holling 1959a) inspired by the general formalism of Pawar et al. (2012). Predation was broken down into four different processes, which are searching, chasing, capturing, and handling 123 of a prey item by a predator. Acquisition rate of a prey item (species i) by a predator ( f (i)), namely the functional response, takes the following form:

Mechanistic model of functional responses
where α i is the capture efficiency (km 2 /h), N i the prey density (number of i/km 2 ), and h i the 126 handling time of prey (h/i). Capture efficiency is obtained by the product of predator speed (s; km/h), reaction distance (d i ; km), detection (z i ) and attack probability (k i ) of the prey by the predator, and the success probability (p i ) of an attack (table 1): The combination of the time spent chasing the prey once encountered T ci p i and the time spent manipulating the prey once subdued T mi define an overall prey handling time (h i ): α i depends only on prey density, and we assumed that prey are randomly distributed. Satiety 132 was not considered as a potential mechanism limiting acquisition rate. Indeed, foxes can predate more prey than they consume on the short-term; e.g. about 4% (n = 128) and 48% (n = 98) of predated eggs and lemmings are immediately eaten, respectively (Careau et al. 2007). Predator The general model of functional response (eq. 1) allows for a continuum between a linear 141 and a hyperbolic functional response shape. In order to allow the model to extend to a sigmoidal shape, we added density dependence in capture efficiency components that were expected to vary with prey density (i.e. reaction distance and detection and attack probabilities; see below).

Prey specific functional responses
We adapted the general model (eq. 1) to each prey species based on their life history traits and anti-predator behavior ( fig. 1). The specific models for each prey species are provided in table C2. 1B). Geese can actively protect their nests against arctic foxes; therefore, their presence at the nest strongly influences fox foraging behavior (Bêty et al. 2002;Samelius and Alisauskas 2001), which translates into changes in capture efficiency components. Parameter values of capture rates were 159 thus estimated separately for goose nests that were attended or unattended by an incubating female (table C1). When a nest is attended by a highly conspicuous snow goose, we assumed that nest detection probability is 1 within d ( fig. 1A). For unattended nests, we used a detection prob-

171
The general model (eq. 1) was simplified for lemmings as we assumed that an attack is systematically initiated by the fox once a lemming is detected within d ( fig. 1C). The general model was also simplified for passerines and sandpipers as we assumed that once a passerine or 174 a sandpiper nest is detected, the nest is always predated ( fig. 1D).
We incorporated density dependence into the goose and the lemming models within the 177 range of densities observed in our study system. For each parameter in which density dependence was incorporated, the minimum and the maximum parameter values were associated respectively with the minimum and the maximum prey density in order to calculate the slope 180 and the intercept of the density-dependence relationship. In the goose model, we modified attack and success probabilities for attended nests, and reaction distance and detection probability for unattended nests. In the lemming model, we added density dependence in reaction distance and 183 detection probability. The rationale behind these additions is that predators may form search images for abundant prey, which can increase their ability to detect them (Ishii and Shimada 2010;Nams 1997). As predators could also increase their attack rate and success as prey density 186 increases, we added density dependence in attack and success probabilities. We did not incorporate density dependence into the passerine and sandpiper nest models as the range of nest densities observed in our study system is likely too low to influence fox behavior (maximum 189 of 12 nests/km 2 compared to a maximum of 926 goose nests and 414 lemmings per km 2 ). See Appendix C for more details on the incorporation of density dependence.

192
The model was implemented in R v. 3.6.0 R Core Team 2019.

Parameter values
The model was parameterized mostly using data from Bylot Island but also from the literature 195 when data were missing. Parameters were derived from field experiments using artificial nests or estimated using arctic fox GPS tracking data and direct observations of foraging foxes (table   C1). See Appendix C for a detailed description of the method used to extract each parameter.

198
Evaluating the coherence between the mechanistic model and empirical predator acquisition rates Predator acquisition rates at different prey densities were assessed in the field annually using 201 two independent methods. These data did not allow validation of the shape of the functional responses, but they provided a way to evaluate the performance of the mechanistic model in estimating prey acquisition rates at the various prey densities observed in our study system.

204
First, we obtained goose eggs and lemming acquisition rates by conducting direct observa-tions of foraging foxes for 10 summers between 1996 and 2019 during the goose incubation period 207 (details on behavorial observations can be found in Bêty et al. (2002) and Careau et al. (2008)).
For each year, the acquisition rate was calculated as the total number of prey acquired (goose eggs or lemmings) divided by the total length of the observation bouts of individual foxes. The 210 acquisition rate of a clutch of eggs was estimated by dividing the acquisition rate of goose eggs by the annual average clutch size. For the years where information was available, we also calculated the acquisition rate for attended and unattended nests. We estimated annual goose nest 213 density either by visual counts of the nests located in the observation zone (range: 0.5-3 km 2 ) during the incubation period (1996)(1997)(1998)(1999)2019) or over a fixed 0.2 km 2 plot within the intensively monitored core area of the goose colony (2004)(2005)(2015)(2016). We estimated lemming density ). An estimation of the acquisition rate is obtained by: The daily nest survival rate was modeled using the logistic exposure method (Shaffer 2004).

231
Additional details on daily nest survival rate calculations and nest monitoring methods can be found in Royer-Boutin (2015). Density estimates for all prey species were standardized as the number of nests per km 2 .

234
Uncertainty and sensitivity analysis We quantified how uncertainty in parameter values affected estimation of predator acquisition rates by using the Latin hypercube sampling technique (an efficient implementation of the Monte 237 Carlo methods; Marino et al. 2008). This analysis allowed us to investigate the uncertainty in the model output generated by the uncertainty in parameter inputs. Each parameter was represented by a probability distribution (uniform or normal truncated) based on the distribution of empiri-240 cal data (table C1). For some parameters, the biological information was limited, so we assigned a uniform distribution allowing for a large range bounded by minimum and maximum values.
Latin hypercube sampling was then applied to each distribution (N = 1000 iterations). This 243 method involved dividing a probability distribution into N equal probability intervals that were then sampled without replacement, resulting in N iterations of the model using each combination of parameters values. This method allowed us to explore the entire range of each parameter 246 and most of them encompass various environmental conditions (e.g. weather conditions, prey availability). We computed the median, the 90, the 95, and 99 percentiles of the model output by using the empirical cumulative distribution. 249 We also conducted a local sensitivity analysis to identify key parameters of the mechanistic models within the range of prey densities observed in our study system. We modified each 252 parameter value by ± 100% while holding others constant, and we assessed how this variation affected the predator acquisition rate (expressed as % of change).

255
From 1996 to 2019, we observed foraging foxes in the goose colony for 124 hours. Average goose nest density was 409 nests/km 2 (range: 100-926 nests/km 2 ) and lemming density was 193 ind./km 2 (range: 11-414 ind./km 2 ; The uncertainty analysis revealed that varying all parameters simultaneously generated considerable variation in model output ( Figure 3: Sensitivity of predator acquisition rates to changes in parameter values of the mechanistic models used to assess the functional response of arctic fox to goose nests (at 100 (A1), and 1000 nests/km 2 (A2)), to lemmings at 250 ind./km 2 (B) and to passerine and sandpiper nests at 10 nests/km 2 (C).
Predator speed was an influential parameter of the functional response of all prey species ( fig.  3). Goose nest acquisition rate was generally more affected by parameters associated with unat-282 tended nests than attended nests ( fig. 3A). The magnitude of change in goose nest acquisition rate related to the changes in manipulation time increased slightly with nest density. Lemming acquisition rate was not affected by chasing and manipulation time, whereas detection distance, 285 and detection and success probability had an influence equivalent to predator speed ( fig. 3B).
Similarly, functional response models of passerine and sandpiper nests were not sensitive to change in manipulation time, whereas detection distance and detection probability had an influ-288 ence equivalent to predator speed ( fig. 3C).
Even though the shape of the functional response changed slightly between models without 291 or with density dependence in capture efficiency components (allowing for a gradient between type I and type III), a maximum difference of 1.4 nests/fox/h at 1000 goose nests/km 2 and 1.3 lemmings/fox/h at 450 lemmings/km 2 were found between models ( fig. 4).

Discussion
Most models of predator-prey interactions make assumptions regarding the shape of the func-297 tional response expressed by predators. By quantifying the characteristics of a predator (e.g. speed and detection distance), we developed a mechanistic model of arctic fox functional response to various prey species. Our model derives the shape of the functional response along a 300 gradient from linear to sigmoidal. We benefited from extensive empirical data on all prey species to parameterize and evaluate the coherence of the model. Predator acquisition rates derived from the mechanistic model were consistent with field observations, and the main proximate mecha-303 nisms driving predator acquisition rates were also identified. In all prey species, predator speed was an influential parameter, while handling time was not a limiting process. Although type III functional responses were previously used to model fox-prey population dynamics (Gilg et al. 2003(Gilg et al. , 2009), our simulations indicate that predator acquisition rate was not systematically limited at the highest prey densities observed in our study system. Our model allows for a mechanistic interpretation of the functional response of predator-prey pair and could be extended to more 309 complex modules involving multiple predators and prey species.
Holling's functional response models (type II and III), which are commonly used in popula-312 tion dynamics models (Gervasi et al. 2012;Serrouya et al. 2015;Turchin and Hanski 1997), predict that predator acquisition rates should eventually saturate at high prey densities. In our study system, we found no evidence of arctic fox saturation at the highest prey densities observed. Sev-315 eral factors may explain this result. First, the hoarding behaviour of arctic fox may substantially reduce handling time by limiting the constraints associated with digestion and satiety, which can make the functional response shape linear or slightly convex even at high prey densities (Oksa-318 nen et al. 1985). Second, while predator acquisition rates must theoretically become constrained by handling and/or digestion at high prey densities, the prey densities required to reach a saturation point could be rarely observed in natural systems. Indeed, empirical support for saturating 321 functional response in the wild is relatively rare and comes mostly from controlled laboratory experiments in which the range of prey densities may exceed the range observed in nature (99% of all type II functional response were derived from controlled laboratory experiments (n = 61 stud-324 ies); reviewed by Rall et al. 2012). Such an issue can be avoided when mechanistic approaches are used to derive functional responses. One particularity of our system is the presence of a large goose colony where prey density can be quite high (up to ∼900 nests/km 2 ). Interestingly, 327 even in this context, we found no evidence of predator saturation. Hence, our results add to a growing body of research indicating that predators may not become systematically satiated at the highest densities of prey observed in nature (Chan et al. 2017;Novak 2010;Preston et al. 2018).

330
Historically, a categorical approach was adopted by ecologists to define functional responses.
A linear functional response was traditionally attributed to filter feeders (Jeschke et al. 2004), a hyperbolic shape (type II) to invertebrates and a sigmoidal shape (type III) to vertebrate predators (Holling 1965). Although, this categorization has some heuristic value in introductory texts and can be useful in some aspects of research where categorization is necessary, types I, II, and 336 III should be considered simply as particular cases along a continuum. Instead of using a priori shapes to describe functional responses, our study illustrates how mechanistic models can generate functions linking prey density and predator acquisition rates that are specific, and hence 339 more relevant, to the range of densities observed in a food web. Considering the strong effect of functional responses on the outcome of predator-prey models (Abrams et al. 1998;Sinclair et al. 1998), such specific functions should improve our ability to adequately simulate and quantify the 342 strength of direct and indirect species interactions in natural communities.
We did not incorporate predator dependence in the functional response model, despite a 345 growing body of studies indicating that some mechanisms (e.g. facilitation, interference) are likely to occur in functional responses (Novak et al. 2017). However, arctic foxes maintained summer territories (averaging 9.6 km 2 ) with low overlap (A. Grenier-Potvin and D. Berteaux, 348 unpublished manuscript), which prevents potential interference within territories. We are thus confident that variation in predator density should not affect our main conclusions. Nonetheless, the mechanistic model could be extended to more complex predator-prey systems, including 351 predator interference.
Habitat characteristics could affect several parameters of the mechanistic model, hence the 354 functional response shape and magnitude could be modulated by the structural complexity of the landscape (Barrios-O'Neill et al. 2015;Toscano and Griffen 2013). For instance, the detection distance of a nest by arctic foxes could be lower in dense vegetation (Flemming et al. 2016), 357 the attack probability could be lower for nests located in wetlands and islets only accessible by swimming Lecomte et al. 2008), and the success probability of an attack could be modulated by the presence of complex networks of lemming tunnels offering refuges.
Exploration of the effects of structural complexity on functional responses remains rare (but see Barrios-O'Neill et al. 2015;Lipcius and Hines 1986;Toscano and Griffen 2013), and more empirical research is needed to integrate these sources of variation in mechanistic models.

363
The outputs of the mechanistic model were generally consistent with field observations. However, adding more complexity could improve its performance and our ability to identify the main 366 drivers of predator acquisition rates. For instance, group defence and mutual vigilance are additional factors that may reduce predator acquisition rates at high prey density (Clark and Robertson 1979). Although there is no evidence of group defence in geese (Bêty et al. 2001), the snow 369 goose could benefit from the vigilance and early warning provided by neighbors nesting nearby (Samelius and Alisauskas 2001). Beyond a threshold of goose nest density, such anti-predator behavior could reduce the proportion of unattended nests with increasing nest densities. Nest 372 attendance probability was an influential parameter of the goose model and mutual vigilance may partly explain why acquisition rates observed in the field at moderate-high nest densities were under the model median ( fig. 2A).

375
One mechanism often advanced for explaining the apparent mutualism between two prey sharing a common predator is predator saturation or satiation (Abrams and Matsuda 1996;Holt 378 1977). Our results showed that the arctic fox is not satiated at the highest lemming densities observed in our study system. This suggests that the underlying mechanism for the short-term positive effect of high lemming density on arctic bird reproductive success (Bêty et al. 2002;381 Blomqvist et al. 2002) is likely not predator satiation nor saturation, and it reinforces the need to adopt a mechanistic approach to fully understand predator-prey interactions. Short-term positive indirect effect could arise from changes in various components of the functional response.

384
For instance, attack probability of an attended goose nest could be inversely dependent of lemming density, or daily distance traveled by the predator (speed) could vary according to prey availability. As indicated by our sensitivity analyses, attack probability was not a strong driver of prey acquisition rates while predator speed was an important parameter affecting all prey species. Hence, lemming-induced changes in predator speed due to changes in reproductive state and activity budget of foxes could be an alternative hypothesis explaining the apparent 390 mutualism between lemmings and arctic birds. To fully identify the main proximate drivers of indirect interactions in natural communities, we need to identify the components of capture rate for a given prey that change according to variation in densities of other prey and evaluate the 393 impact of such changes on prey mortality using mechanistic models.

Conclusion
Previous studies of functional responses typically tried to discriminate between predetermined

Appendix C: Additional Methods
Derivation of the mechanistic model of functional response 420 The area searched (A; km 2 ) by a predator is expressed by the product of predator speed (s, km/h), the reaction distance to a prey item i (d i , km), and the time spend searching (T s , h): A potential encounter occurs when the predator comes within the distance (d i ) at which one can 423 detect and react to the other. As not all prey within this area may be detected, attacked and subdued by the predator, we introduced the detection probability (z i ), the attack probability (k i ), and the success probability of an attack (p i ). Capture efficiency of a prey item i by the predator 426 is expressed by: The number of prey captured (V αi ) during a search duration T S for a density N i is: The time spent searching (T S ) is defined as: where T t is the time available for feeding; T ci V αi p i is the time spent chasing prey once they are encountered; and T mi V αi is the time spent manipulating prey if they are subdued.

432
By simplifying Eq. C4 we have: We can combine the chasing and manipulation time to produce an overall prey handling time Substituting T s from Eq. C5 into Eq. C3, we arrive at: The functional response of a predator ( f (i)) is the number of prey captured per predator per unit of time. This is expressed by dividing Eq. C7 by T t :

Estimation of parameter values Predator speed
A total of 16 foxes (7 females and 9 males) were equipped with a GPS collar (Radio Tag-  Average predator speed (km/day) was estimated by adding linear distances between successive locations, using the adehabitatLT library in R (R Core Team, 2019). Predator speed was extracted from June 5 to July 9 to focus on the incubation period of most birds. We removed 453 from analyses all fixes obtained <48 hours after capture and days where the number of fixes was insufficient (<75% of all fixes). Predator speed was converted per hour and average speed was 1.52 km/h (sd = 0.59 km/h, n = 123 fox-days).

Goose parameters
Nest attendance probability. During goose incubation period, the time spent on the nest by females average 93% (µ = 93.6, se = 1.6%, n = 7 females; Poussart et al. 2000 and µ = 93, n = 41 459 females; Reed et al. 1995). During incubation recesses females usually remained close to their nests, and 90% of all records (n = 183) were within 20 m (Reed et al., 1995). Since there is uncertainty in the proportion of females within 10 m, we used 90% as the maximum and 50% as 462 the minimum. By combining this information, we can estimate a minimum probability of nest attendance at 96.5.% and a maximum probability of nest attendance at 99.3%.
Detection probability. We used artificial nests to assess experimentally the detection probability size is proportional to the number of observations. The curve represents the average detection probability and the gray band represents the 95% confidence interval of the regression.
Reaction distance. The reaction distance on attended nests was defined as the distance at which an attack can be initiated by the predator. This parameter was estimated with direct observations of foraging foxes in summer 2019 (µ = 0.0328, se = 0.007, n = 25 attacks). On unattended nests, 480 the reaction distance was defined as the maximum distance at which the predator can detect an unattended nest. We used a combination of direct observations combined with artificial nests (same experience as for the detection probability) to estimate the reaction distance on unattended 483 nests (µ = 0.0365 km, se = 0.009, max = 0.1 km, n = 13). We used 0.1 to 0.12 km as a range of maximum distance as the detection probability was still around 30% at 0.1 km ( fig. C1). As our sample size was limited for attended and unattended nests, we assigned a uniform distribution 486 for both parameters. Complete predation probability. Based on direct observations of foraging foxes, the probability of a complete predation once an egg is subdued on unattended nests was 69% (se = 0.12, n = 16) 510 and was 47% (se = 0.13, n = 15) on attended nests.

Lemming parameters
Reaction distance and detection probability. Based on direct observations, foxes generally initiate 513 their attacks on lemmings within 5 m (on 29 attacks recorded in 1996-1999 and 2019). We assumed that the maximum reaction distance was twice that distance (10 m) and that detection probability followed a decreasing sigmoid function. Hence, detection probability was considered fairly high 516 within 5 m radius (100-80%) and declined sharply between 5 and 10 m.
Chasing time. By conducting direct observations of foraging foxes, we estimated the average chase time per lemming attacked (µ = 88 sec, se = 7.1, n = 246 attacks).

519
Success probability. By conducting direct observations of foraging foxes, we estimated the probability of a successful attack on a lemming at 51% (se = 0.03, n=268 attacks).
Manipulation time. Based on direct observations of foraging foxes, we estimated the average 522 manipulation time per lemming captured, which includes consumption and hoarding time (µ = 37 sec, se = 3.0, n = 93).

525
Reaction distance. The reaction distance was defined as the maximum distance between an observer (a simulated predator) and the nest when the bird flushes the nest. To measure the reaction distance, a human observer approached nests from a random bearing at normal walking speed 528 (∼4 km/h). The observer made several approaches until the bird leaves the nest. At each approach the distance between the observer and the nest was noted. Flush distance was recorded for 45 different nests in summer 2019 and ranged from 0 to 20 m (µ = 2.8 m, sd = 3.6 m, n = 77)

531
The distance was measured either by pacing or with a GPS unit.
Detection probability. Detection probability was estimated by following the same method as reaction distance. Besides recording flush distances, the distance between the observer and the 534 nest was noted even if the bird didn't flush. We used a linear model with a binomial distribution to model detection probability (0; the bird didn't flush, 1; the bird flushed) in relation to the minimum distance between the observer and the nest (n = 167, fig. C2).

Sandpiper parameters 543
Reaction distance and detection probability. Reaction distance was estimated by Smith and Edwards (2018) following the same method as described for passerines reaction distance. The mean reac-

Parameter name Value Distribution
Predator speed µ =1.52 km/h, sd = 0.59, n=123 fox-days Normal-truncated Table C2: List of equations to derive the functional response of arctic fox to each prey species.

Exploration of density dependence in capture efficiency components
We incorporated density dependence into the goose and the lemming models within the range 558 of densities observed in our study system. For each parameter in which density dependence was incorporated, the minimum (p min ) and the maximum (p max ) parameter values were associated respectively with the minimum (N min ) and the maximum (N max ) prey density in order to calculate 561 the slope and the intercept of the density-dependence relationship: In the goose model, we modified attack and success probabilities for nests attended, and reaction distance and detection probability for nests unattended (e.g. Fig. C3). In the lemming 564 model, we added density dependence in reaction distance, detection, attack and success probabilities (e.g. Fig. C4). Literature Cited ---. 1959b. The components of predation as revealed by a study of small-mammal predation