Performance of five statistical methods to infer interactions among moving individuals in a predator–prey system

Rapid development of tracking technologies allow the collection of high‐quality data on multiple simultaneously moving individuals. This, in turn, initiated the development of several methods to infer interactions among moving animals. However, the performance of these methods has not been studied systematically, especially with regard to the factors that are highly relevant for field ecologists, such as duration of the tracking period, its temporal resolution and the proportion of the tracked community. Here, we assessed the performance of three dynamic interaction indices (Coefficient of sociality, Correlation coefficient and Dynamic interaction) and two novel approaches based on step selection functions (SSF‐occurrence‐distribution and SSF‐distance). We tested these methods on the data simulated with a predator–prey system, reflecting three common types of interactions while on the move: avoidance (prey individuals fleeing from the predator), attraction (predator following and chasing prey) and neutral movement (no interactions between predator and prey). We assessed the ability of each method to correctly detect the modelled interaction type by manipulating the perceptual range of the predator, the proportion of tracked prey individuals, the temporal resolution and the duration of the tracking period. We found that the ability to correctly infer interactions increased for all methods with an increase in the perceptual range of the predator and the proportion of tracked prey individuals. In contrast, the duration of the tracking period affected the methods' performance differently: some methods (Cs and SSF‐distance) were insensitive to it, whereas the performance of other methods improved (DI and SSF‐occurrence‐distribution) and worsened (Cr) with the duration of the tracking period. The three dynamic interaction indices and SSF‐distance were rather robust to changes in data resolution. Despite using the predator–prey system as our study case, our findings are applicable to other systems in which two animals on move may interact. We formulate guidelines for field ecologists studying animal movement to select the most suitable method depending on the availability of tracking devices, the duration and resolution of the tracking they can afford, as well as perceptual ranges of study species.

especially with regard to the factors that are highly relevant for field ecologists, such as duration of the tracking period, its temporal resolution and the proportion of the tracked community.
2. Here, we assessed the performance of three dynamic interaction indices (Coefficient of sociality, Correlation coefficient and Dynamic interaction) and two novel approaches based on step selection functions (SSF-occurrence-distribution and SSF-distance).We tested these methods on the data simulated with a predator-prey system, reflecting three common types of interactions while on the move: avoidance (prey individuals fleeing from the predator), attraction (predator following and chasing prey) and neutral movement (no interactions between predator and prey).We assessed the ability of each method to correctly detect the modelled interaction type by manipulating the perceptual range of the predator, the proportion of tracked prey individuals, the temporal resolution and the duration of the tracking period.
3. We found that the ability to correctly infer interactions increased for all methods with an increase in the perceptual range of the predator and the proportion of tracked prey individuals.In contrast, the duration of the tracking period affected the methods' performance differently: some methods (Cs and SSF-distance) were insensitive to it, whereas the performance of other methods improved (DI and SSF-occurrence-distribution) and worsened (Cr) with the duration of the tracking period.The three dynamic interaction indices and SSF-distance were rather robust to changes in data resolution.4. Despite using the predator-prey system as our study case, our findings are applicable to other systems in which two animals on move may interact.We formulate guidelines for field ecologists studying animal movement to select the most

| INTRODUC TI ON
Animal movement, defined as a change in the spatial location of an individual over time, is a key behavioural process of nearly all animals.It fulfils basic requirements of survival and reproduction but also affects the species geographic distribution, thus shaping the structure, dynamics and genetic make-up of populations, communities and ecosystems (Jeltsch et al., 2013;Nathan et al., 2008).
Although most studies, until recently, have neglected potential animal interactions in movement ecology, animals often do not move independently of each other (Nathan et al., 2022;Vicsek & Zafeiris, 2012).Such interactions were relatively little studied until now but the opportunities for such studies arise thanks to highresolution data on multiple simultaneously moving individuals that can be collected due to recent improvements in tracking technologies (Farley et al., 2018;Urbano et al., 2010).However, analyses of such data require suitable methods for assessing interactions among moving individuals.And, although several methods to infer such interactions have been developed recently (Long et al., 2014;Niu et al., 2016;Schlaegel et al., 2019), we still do not know how well they perform under specific conditions, in comparison with other methods.For example, how the proportion of the tracked individuals, the perceptual range and the tracking duration affect the ability of these methods to infer interactions.We close this research gap by focusing on moving individuals of predators and prey and using a simulation to assess performance of three commonly used and two novel statistical methods.
Interactions among moving individuals can be simplified into three broad types: avoidance, attraction, and neutrality (Schlaegel et al., 2019).For instance, social animals that are forming groups move together towards the same destination and individuals in a group often display attraction towards other group members, or courtship during the mating season.On the contrary, prey typically flee from predators, exhibiting avoidance; the same is true for competitors avoiding each other.We investigate such interactions of moving individuals by using predator-prey system as our study system because predator-prey dynamics are well-studied (Banks et al., 2004;Peterson, 1999;Volterra, 1931).Previous studies mainly focused on how predator-prey interactions impact population and community dynamics (Angerbjorn et al., 1999;Drossel et al., 2001;Schneider et al., 2012), but little is known about how predators and prey interact behaviourally while on the move.Although we have chosen to study the interactions of moving individuals in a predatorprey system, our findings are applicable more generally to other systems in which moving individuals may exhibit attraction, avoidance or move independently from each other.
The existing methods to infer interactions of moving individuals differ in their assumptions and the type of the data they require (Calabrese et al., 2018;Langrock et al., 2014;Long et al., 2014;Schlaegel et al., 2019).The absence of guidelines on when to use which method and the fact that some methods are not as straightforward to implement for researchers without strong programming skills hinder their wide use by empiricists.Therefore, the aim of this study is to compare the existing statistical methods in terms of their underlying assumptions, data requirements and performance under a range of conditions.By performance we here mean the power of the method to detect true interactions.To this end, we focused on five statistical methods (Figure 1c): three commonly used indices of dynamic interaction (Cs-coefficient of sociality, Cr-correlation coefficient and DI-dynamic interaction index) that are implemented within the 'Wildlife DI' R package (Long et al., 2014) and two novel methods that are based on stepselection functions (SSF): one uses as a covariate the occurrence distribution of the other moving individual (Schlaegel et al., 2019; herewith referred to as SSF-OD) and the other one uses the distance to the other moving individuals (Roeleke et al., 2022; herewith referred to as SSF-DIST).All five methods are used to estimate interactions from the movement trajectories of individuals.
To test the methods, we simulated movement data with an agentbased model (ABM, Grimm et al., 2006;Tang & Bennett, 2010), suitable method depending on the availability of tracking devices, the duration and resolution of the tracking they can afford, as well as perceptual ranges of study species.

K E Y W O R D S
attraction-avoidance, biased correlated random walk, dynamic interactions, interactions, movement ecology, simulation, step-selection functions, wildlife DI F I G U R E 1 Scheme reflecting the workflow: (a) We simulated movement tracks of individual predator and prey individuals which either interacted with each other or not.The moving individuals may be in one of the two behavioural states: a persistent movement and a nonpersistent movement that is very similar to correlated random walk.(b) We investigated the performance of the statistical methods by varying four factors relevant for field ecologists: four levels of perceptual range of the predator; five levels of proportion of tracked preys; two levels of temporal resolution; four levels of duration of tracking period.(c) The five different statistical methods that were applied to the generated movement tracks.The three on the top (Cs, Cr, DI) are the interaction indices while the two on the bottom are the SSF-based approaches.
because it allowed us to model the data with the known type of interactions and to easily vary multiple factors, which is not possible with observational data.
We investigated the performance of the five methods (Table 1) by manipulating four factors in our agent-based model simulations.One factor reflects the biology of the studied species and represents the perceptual range of predator.The other three factors can be potentially affected by the field researchers and thus will help to optimize data collection if aimed at assessing interactions.The first factor is the proportion of the tracked individuals.Understanding its impact on performance of the methods is crucial because tagging all individuals is expensive and logistically-prohibitive.For example, Liu et al. (2015) found that analysing a large subset of individuals in a group is recommended to yield more accurate group movement characteristics.Next, we examine the effect of temporal resolution and duration of the tracking period on the methods.Given that rapidly developing technologies (Farley et al., 2018) allow tracking individuals at increasingly high temporal resolutions, it is essential to understand what implications temporal resolution has for inferring movement interactions.Indeed, certain movement behaviours and patterns might only be detectable at high temporal resolution (Nathan et al., 2022), and low temporal resolution might obscure or invalidate important biological information (Mannocci et al., 2017;Postlethwaite & Dennis, 2013).Yet, high resolution data can also be a burden, as it often requires more storage and processing power (McCrea et al., 2023) and necessitates fitting complex models, interpreting which is challenging.The period over which an animal is tracked is also important because it may be limited by the lifespan of the tracking device or even animal welfare regulations (Teague O'Mara et al., 2014).
We assume that all methods have their advantages and disadvantages, depending on the context they are used in.We expect that the ability of all five methods to detect true interactions will be lowest (i.e.methods will perform the worst) when the percentage of tracked individuals is low or the predator perceptual range is short, because under these conditions the interaction rates between both individuals will be low.

| Simulation of movement trajectories
In order to assess performance of the statistical methods used to infer interactions from movement data, we simulated data on individuals that interact while moving, that is perform avoidance (prey fleeing from predator), attraction (predator chasing prey), neutral movement, with no interactions between predator and prey.To this end, we developed an ABM in which one predator and several prey individuals move according to a correlated random walk whose parameters depend on the behavioural state of the individual.An individual can be in one of two states: (i) in a persistent, that is more directed movement state (higher correlation in facing direction), and (ii) a non-persistent movement state that is more similar to a random walk (Figure 1a).The period over which an individual is in each movement state is randomly drawn from a uniform distribution (Supporting Information, Table 2).
The switch between two states is based on a Markov chain process (Howard, 1972).Further the predator and prey switch to a persistent movement, that is hunt or flee, when the prey and predator enter their respective perceptual range (defined distance).Such a behaviour is considered as an interaction in our simulation.The individuals are moving within an area of a fixed size with reflecting borders.The detailed model description follows the Overview, Design concepts, Details (ODD) protocol (Supporting Information).
We investigated how the performance of the methods was affected by four factors: (1) the perceptual range of the predator, (2) the proportion of tracked prey individuals, (3) the temporal resolution at which the data is collected and (4) duration of the tracking period.

TA B L E 1
Overview of the five methods used in this study to assess interactions from movement data.

Reference Description Interpretation
Coefficient of sociality (Cs) Kenward et al. (1993) Differences between mean distance of all simultaneous fixes and mean expected distance obtained with permutations of all fixes  We varied the perceptual range of the predator by one to two and four times that of the prey, while keeping the perceptual range of the prey constant.Regarding the proportion of tracked prey the model was run with one predator (whose movement was tracked) and only one prey individual, then two, three, five and 10 preys, of which only one randomly chosen prey individual is tracked, that is their movement data is collected for the analysis.That means that for simulations with only one simulated prey, 100% of the prey were tracked while for simulations with 10 simulated preys, tracking only one prey equals to 10% of the preys of the actual prey community being tracked.
The temporal resolution at which the data is collected was varied between two different levels: 100% and 2% of the original fixes.
Similarly, we considered four levels for the duration of the tracking period: 2000; 5000; 10,000; and 20,000 time steps.We examined the effects of tracking duration and data resolution by separately varying each of these factors at the above-specified levels and while keeping the other factors at their fixed values (perceptual range at 4; proportion of the tracked prey at 100%), as we did not expect the effects of these factors to depend on one another or on any of the two other studied factors.Since the effect of perceptual range and the proportion of tracked prey may be not independent of each other, we varied them in a full factorial design (i.e.varied one factor across all above-specified levels at each level of the other factor).
In total the investigated combinations of different factors resulted in 180 distinct simulations (for the exact combinations see Table 2), each of which was run with 20 repetitions.Additionally, to assess the Type 1 error (i.e.false positives), we simulated the data under a "null model" of no interactions present, where the prey and the predator move independently of each other according to a two-state correlated random walk (20 repetitions per each investigated factor and, in case of the perceptual range and the proportion of tracked prey, 20 repetition for each level of the proportion of the tracked prey).

| Coefficient of sociality (Cs)
The Cs index is a measure of attraction using the raw distances between fixes (Cs; Poole, 1995).Cs is calculated as:  Values around 0 indicate no correlations between two paths.

| Dynamic interaction index (DI)
The DI index measures the cohesiveness of simultaneous movement vectors with respect to two independent components of movement: distance (also called displacement) and direction (DI; Long & Nelson, 2013).These two components are in contrast to only the movement vector used in Cr.Further, DI does not depend on the path mean vectors as Cr does.DI consists of two measures: it calculates cohesiveness in displacement and direction between two individuals for a defined time period, referred to as di, which is then used to calculate DI.
Here, d is the displacement and the direction of a vector at a time

| SSF-based approaches
The SSFs compare the observed steps of a focal individual to randomly sampled possible steps with certain covariates (in this case the values of the occurrence distribution of other individuals are used as a covariate) and estimate a selection strength.The random The SSF-based approach by (Schlaegel et al., 2019) is based on applying step-selection functions (SSF, Forester et al., 2009) to dynamic occurrence distributions of individuals (Fleming et al., 2015) and is referred to as SSF-OD approach in this study.Animal movement was typically sampled at rather lower temporal resolutions in the past and thus the real path was not known.To deal with this, SSF-OD applies kriging to collected movement paths (e.g.arising from telemetry and GPS data, Fleming et al., 2015), thus allowing to estimate the probability that an animal occurred within an area at a given time period.This probability is referred to as occurrence distribution (henceforth "OD").Kriging yields a dynamic spatial map The second SSF-based approach we use is a modification of SSF-OD that, instead of the occurrence distribution, uses distances between individuals as covariates in SSFs (Roeleke et al., 2022).We refer to this approach as SSF-DIST.SSF-DIST yields a dynamic spatial map that contains the animal's distance to every cell within the given gridded landscape at each given time period.SSFs are then applied to this spatial map to assess whether

| Evaluating method performance
We assess the method performance by focusing on the power of the methods to detect true interactions and by evaluating type 1 error for wrongly detected interactions.Specifically, to measure the method performance we use the p-values of each method to categorize its outcome as significant (<0.05) or not.We then calculate the proportion of correctly estimated interactions, a metric that is indicative of the method performance.For simulations with existing interactions, we expect the indices Cs and DI to detect attraction (i.e. the movement trajectories of both individuals aligned in the same direction) and thus the coefficients to be significant and positive.We expect the two SSF-based approaches to correctly detect both attraction by predator and avoidance by prey and thus the coefficients to be significantly positive for predator and significantly negative for prey.Under the "Null Model" representing no interactions we expect the methods to not detect any interactions and thus the estimated coefficients to be nonsignificant.As the perceptual range of the predator (1) and the percentage of tracked preys (2) increase, we anticipate that the proportion of correct detections will also increase.Varying the temporal resolution (3) will not affect the proportion of correct detections for any but the SSF-OD method.Lastly, varying the tracking period duration (4) will only affect the outcome of the Cr index.We could only assess the power of the methods for the methods that provide significance of the estimated coefficients, that is Cs and DI indices and both SSF-based methods.Cr is the only index that is not an estimate and thus does not have significance associated with it, meaning we, unfortunately, cannot assess its performance.Additionally, we also looked at the computational time each method requires to calculate an interaction index to guide readers in the choice of methods.

| Perceptual range of the predator and percentage of tracked prey
As expected, the three indices of dynamic interaction and both SSFbased methods yielded values around zero when individuals do not interact (Figure 2).However, Cs had a high rate of false positives as it detected significant interactions in half of the repetitions even though no interactions were present (Table 2).Following Cs, SSF-DIST falsely detected presence of interactions in 20%-23% of simulations (Table 2).Both DI and SSF-OD scored best in their ability to not find any interactions when there really were none (true negatives).
As the predator's perceptual range and the percentage of tracked prey increased, the proportion of correct detections (true positives) increased for Cs, DI and SSF-DIST (Table 2).However, when the predator perceptual range and percentage of tracked prey were low, the methods' ability to detect attraction decreased: both Cs and DI had high proportions of non-significant index values (high rate of false negatives), especially when predator perceptual range was low (Table 2, Figure 2).SSF-OD is likewise characterized by a high rate of false negatives, and the proportion of correctly identified interactions with this method increased only when the predator perceptual range was large (Figure 2; Table 2).SSF-DIST was much more robust in this regard and even for cases where the predator perceptual range was small and percentage of tracked prey low, it correctly detected the appropriate interaction type in approximately half of the cases (Table 2).

| Duration of the tracking period
The duration of the tracking period affected the ability of Cr to detect the appropriate interaction type: when the period duration increased, the absolute values of calculated indices decreased, tending towards zero, that is neutrality (Figure 3a).The proportion of correctly detected interactions with DI and SSF-OD increased with the duration of the tracking period.This effect was even more pronounced for SSF-OD (Figure 3).For Cs and SSF-DIST, the duration of the tracking period did not affect the detection of the appropriate interaction type (Table 2, Figure 3e).

| Time spent in interactions
When no interactions were present, the sampling resolution did not affect the ability of DI and SSF-DIST approach to correctly detect no interactions (Figure S1).However, Cs had a higher rate of false positives when sampling resolution was high (100%) compared to when sampling resolution was low (2%, Table 2).The SSF-DIST approach-when applied to data with high resolution (100%)-yielded coefficients with a much larger absolute magnitude compared to simulations with low resolution (2%) (Figure S1), although the significance did not change and they were all correctly detected as interactions (Table 2).
F I G U R E 2 Results from applying three indices from Wildlife DI (panels a, b, c) and two SSF-based approaches (d1 and d2: SSF-OD; e1 and e2: SSF-DIST) to the movement data simulated while manipulating following factors: presence of interactions, the percentage of tracked prey and the perceptual range of the predator.For the SSF-based approaches the interaction indices are calculated as perceived by the predator (d1, e1) and as perceived by the prey (d2, e2).Values that are significantly different from 0 (at p 0.05 level) are shown with filled points and those that are not significantly different from 0 are shown with a cross.Raw data points for Cr index are shown with empty points because this is the only method that does not allow assessing significance.

F I G U R E 3
Effect of simulation duration (reflecting the duration of the tracking period) on the dynamic interaction indices Cr, Cs and DI (a-c) and the SSF-based approaches (d and e).The simulations were run with only one prey and one predator.The perceptual range of the predator was set to 4 and the temporal resolution was set 100% for Cr, Cs, DI, and SSF-DIST approach.The temporal resolution for SSF-OD was 2%.No results for the SSF-based approaches regarding the perception of the prey are shown, as the result were similar to the results from the perception of the predator.Only the sign of the coefficient changes.

| Proportion of time spent in interactions
The proportion of time that individuals spent interacting in the simulations was linearly related to the magnitude of the calculated Cs, Cr and DI indices (Figure 4a), meaning that indices sufficiently well grasped the interactions.As the ratio of time spent in interactions increased, the magnitude of all three indices also increased, going from zero to one.Similarly, the time spent in interactions correlated positively with the interaction strengths estimated by both SSF-based approaches (Figure 4b,c).For SSF-OD, the interaction strength calculated for predators, as expected, remained positive for the whole range of the time spent in interactions and increased with increasing proportion of time spent interacting.Interaction strength for prey was negative under low of time spent interacting.However, unexpectedly, their absolute magnitude decreased with increasing proportion of time spent interacting, such that the values switched to positive for 100% time spent in interactions, indicating an unexpected attraction of prey to predators.In contrast, for SSF-DIST, as expected, the interaction strengths calculated for the predator remained positive, and for the prey they remained negative, irrespective of the time spent in interactions.As the time spent in interactions increased, the absolute magnitude of the interaction strength coefficients estimated for either predator or prey increased.

| Computational time
All the interaction indices were noticeably faster compared to the SSF-based approaches.For a tracking data set with 500 fixes, it took the Cs, Cr and DI respectively 0.77, 1.09 and 1.96 s to calculate a coefficient.In contrast, SSF-OD and SSF-DIST needed 7.88 and 2.04 min to calculate a coefficient.To put it in perspective, SSF-OD needed approximately 600 times longer than Cs.

| DISCUSS ION
In this study we assessed an issue of high practical relevance in ecological studies of animal movement: how five statistical methods perform when assessing interactions of moving individuals under a range of scenarios based on study design and species' ecology.Based on our findings we formulate guidelines for field ecologists studying animal movement to select the most suitable method depending on their research question, the number of tracking devices available, the duration and resolution of the tracking data, as well as perceptual ranges of study species.
We showed that when the perceptual range of the predator increases, the present interactions are more easily detected by all methods.This is because the predator can detect the prey from far away and heads towards it, resulting in a more persistent movement towards their prey and more time spent in interactions.However, when the perceptual range was low, the interaction rates between predator and prey were also low, resulting in a lower absolute value of interaction strength.Predators indeed do differ in the distance from which they may perceive their prey.Many predator species estimate the preys´ location through visual, olfactory or auditory cues left by the preys that could already have moved far away.For example, wolves can smell their prey from more than one mile away (Macdonald & Sillero-Zubiri, 2004) and procellariform seabirds like albatrosses, petrels and shearwaters use olfactory cues to locate foraging spots within large areas of up to thousands of square kilometres (Nevitt, 2000).Kestrels can track and assess the number of voles in large areas over a short period of time by detecting vole scent marks in ultraviolet light (Viitala et al., 1995).We recommend that for predators utilizing such cues all five methods can be used, since the predator can detect their prey from further away and thus move in a more persistent way towards their prey resulting in reliable interaction estimates.
In contrast, we suggest that these methods should be applied carefully to movement data of animals with restricted perceptual ranges.For example, the hawk-eagles in the Argentinian rain forest only detect their prey-monkeys from an average distance of 31.9 m due to landscape features obscuring their vision (Janson et al., 2014).
When prey abundance is sparse and predator species have no sensory guidance to where to find their food, studies observed that some species revert to random search strategies observed in wandering albatrosses or basking sharks (Bartumeus et al., 2005;Humphries et al., 2012).Similarly, these methods are less applicable to predators that rely on ambush strategies instead of tracking their preys (MacArthur & Pianka, 1966).Ambush predators such as rattle snakes (Clark, 2004), forest owls (Zulla et al., 2022) and leopards (Karanth & Sunquist, 2000) remain often motionless hiding within burrows or by camouflage and wait for the prey to come within their range.Since predators relying on ambush strategies will have lower active encounters with their prey, the methods tested here might not be able to correctly estimate these interactions.In such cases, the SSF-based approaches provide the possibility to vary the assumed The relation between the percentages of time spent interacting in the simulation and the interaction strengths estimated by the three dynamic interaction indices (a) and two SSF-approaches (b: SSF-OD, c: SSF-DIST).Negative values indicate avoidance, while positive values indicate attraction.The output of the SSF-based approaches is split into how the predator responds to the movement of the prey (b1, c1) and how the prey responds to the predator (b2, c2).The simulations were run for 10,000 steps with one prey and one predator.The perceptual range of the predator was set to 4. The proportion of time spent in interactions is zero when the predator and prey did not interact during their movement (neutrality) and this proportion is one when the predator and prey interacted during the complete duration of the simulation.The black regression line (b, c) passes only through the coefficient estimates that were significant.Crosses represent nonsignificant interaction estimates while points in black represent significant estimates.temporal awareness of a focal individual towards its social environment, by changing the duration of the time interval within which the covariates (occurrence estimates/distance) are calculated.This time interval should thus be chosen based on the species' biology, hunting strategy and the environmental structure of their surroundings.
Many field ecologists face the challenge of using a limited number of tracking devices due to their high costs (Matthews et al., 2013), influencing the number of preys to be tracked.Furthermore, not all wildlife tracking devices are guaranteed to function and many fail due to construction errors or damage sustained from the carrier, resulting in fewer individuals tracked eventually than initially intended (Cumming & Ndlovu, 2011;Uno et al., 2010;Zucco & Mourão, 2009).
Our study shows that tracking a low percentage of animals is associated with the risk of false negatives, thus limiting the inference of interactions among individuals.For example, when faced with few tracking devices, solitary animals or those forming small social groups like roe deer (Hewison et al., 1998) could be successfully analysed.However, when applying a small amount of tracking devices on species in large aggregations like gnu or zebras, the outcome of the methods should be interpreted cautiously.
Interestingly, the SSF-DIST approach seems to be more sensitive towards detecting significant interactions when the percentage of tracked prey individuals is low.This makes this approach especially suitable for studies with limited number of tracking devices, though one must be aware of the high rates of false positive for this method.
It must be noted that in our model prey individuals do not exhibit any collective behaviours, that is do not interact with other prey or form social groups.Yet, the formation of social groups often occurs in nature and serves as a defence mechanism in response to predator pressure (Sumpter, 2006;Ward & Webster, 2016).For such groupforming species having a higher percentage of tracked individuals per group might be less relevant since all individuals move together when encountering a predator and do not act on their own.Indeed Silk et al. (2015) suggested that social networks are quite robust to subsets of the population and that sampling only 30% of a population can generate reliable estimates of an individual's position in a social network.
All three dynamic interaction indices (Cr, Cs, DI) correctly assessed the interactions across both sampling resolutions, corroborating the previous findings of Long et al. (2014).We thus conclude that these methods that were initially designed for low-resolution data are also suitable for inferring interactions from modern high resolution movement data, although limited to inferring a cohesiveness between two animals.Applying the SSF-OD approach to high resolution data did not yield meaningful results as the approach has been specifically designed for data with low sampling resolutions (Schlaegel et al., 2019).For empiricists who deal with high resolution data and aim to estimate individual specific interactions, we recommend using the SSF-DIST approach (Roeleke et al., 2022).While the SSF-OD approach relies on a specific distance threshold that decreases as the sampling resolution increases due to the nature of how occurrence distributions are calculated, the SSF-DIST approach does not need such thresholds.This finding highlights the fact that high-resolution data that is increasingly being collected nowadays is not always advantageous.
In regard to the duration of the tracking period, most of the methods were insensitive to it, except for the index Cr, which tended towards zero with longer tracking durations (number of time fixes) even though the modelled interaction strength stayed the same.
Such behaviour indicates that this index is not suitable for long tracking periods.

| Guidance for field ecologists
Based on our findings we derive guidelines on what methods should be best used (Figure 5) depending on the tracking duration, sampling resolution, and expected interaction rate among individuals of studied species (Figure S2), which is complementing existing guidelines on data collection and sampling design optimization (Silva et al., 2023).When interested in asymmetrical behavioural responses of individuals, that is when one individual is responding stronger than others (between male and female individuals) or reciprocal interactions such as the attraction towards the prey for the predator and avoidance of the predator for the prey, we suggest to use the SSF-based approaches as they allow to effectively identify individual-specific responses.Unexpectedly, we found that the interactions of prey towards predator (modelled as avoidance) were detected as neutral by the SSF-OD approach if individuals spent more than 90% of the time in interactions.We assume that this might be due to the step selection function not being able to detect avoidance in such cases due to larger confidence intervals when non-interaction cases become rare in the dataset.A similar finding of avoidance interaction indices having much larger confidence intervals compared to attraction was also reported by Schlaegel et al. (2019).Besides this specific case, both SSF-approaches are especially useful in studying asymmetric or reciprocal responses, which are vastly present in nature either between individuals of two different species or between conspecifics of different sex and age (Roshier & Carter, 2021;Schlaegel et al., 2019;Ward et al., 2021).In contrast, the dynamic interaction indices are not able to assess such asymmetric responses.Another advantage of the SSF-based approaches over the interaction indices is that they allow analysing the interaction of one individual towards multiple other individuals at once, while the interaction indices are limited to dyadic movement analyses.However, it is important to keep in mind that both SSF-based approaches, and especially SSF-OD, are considerably more computationally costly than the three dynamic interaction indices.Furthermore, SSF-OD is best suitable for the data collected at coarse resolution, a condition for which it was originally developed.
When interaction rates are low, (Figure S2; first compartment) for example, when a predator has low hunting rates (e.g.Eurasian lynx hunts roe deer on average every 5 days; Okarma et al., 1997) or a researcher assesses interactions between two territorial conspecifics (Schuelke & Kappeler, 2003) or concurring species (Vanak et al., 2013) that fear interference competition, we recommend using the SSF-DIST, irrespective of the duration of the tracking period and the sampling resolution.Other cases of low interaction rates might result from a low number of tracked individuals and thus increasing the risk of analysing two individuals that might coincidentally not have interacted within the tracking period.Therefore, should that be the case and the field ecologists be limited in the number of tracking devices they can afford, SSF-OD should be interpreted carefully and SSF-DIST should rather be the method of choice.However, when the interaction rates are high, such as between conspecifics that form groups or heterospecifics that might migrate together or share the same landscape (e.g.zebras and wildebeest; Schmitt et al., 2014), SSF-OD will perform best, as the individuals move close enough together that their occurrence estimates overlap.For example, Schlaegel et al. (2019) was able to accurately detect attraction, avoidance and neutrality between male and female bank voles that have overlapping ranges.
The battery life of the tracking device is one important criterion that field ecologists use to base their decisions about for how long and at what sampling resolution to collect movement data.When working with long tracking periods we would suggest avoiding using Cr as its reliability seems to decrease with the tracking duration.On the other hand, if animals can only be tracked for a short time period, this might lead to only observing few interaction events, in which cases we would recommend to not to use the SSF-OD approach, as it seems to struggle when the movement paths do not overlap, but instead to use SSF-DIST.
Many animals express different behavioural states depending on season, their biological rhythm or in response to environmental factors affecting animal movement and interactions (Gorini et al., 2012;Nathan et al., 2008).For studies that track individuals over relatively long periods, we would advise, regardless of the method used, to subset movement paths into shorter meaningful bouts as this might allow to reveal several types of animal interactions, which would otherwise be hidden.For example, the interactions between hyenas and other predators (Périquet et al., 2021) or between female and male bears (Long & Nelson, 2013) changed across seasons.Yet, it can be challenging to identify temporal changes in interactions within a behaviourally heterogeneous movement data set without prior knowledge.In such cases, segmentation methods are used to subset movement paths into bouts that are assumed to reflect different underlying behaviours (Edelhoff et al., 2016;Gurarie et al., 2009).An aggregated behaviour summary of the movement path is generated by gliding an analysis window over the entire path, allowing to define an appropriate length of time periods.Alternatively, variograms can be used to detect different movement behaviours within movement paths and to decompose them into segments (Fleming et al., 2014).Interestingly, time series of the local DI statistic can be plotted, which allows to graphically investigate temporal phases of behavioural states, without the need of additional methods (Long et al., 2014).
An approach that is increasingly used in movement ecology to identify different behavioural states is Hidden Markov models (HMM).It allows identifying major movement states in telemetry data based on the changes in the animal's step length and turning angle (Langrock et al., 2012;Patterson et al., 2009).HMMs have already been used to investigate group movement dynamics (Langrock et al., 2014) by assessing the tendency of simultaneously moving individuals to be attracted to a so-called "group centroid".
Although this method was originally used for group-moving individuals, the authors highlighted that it is a rather flexible tool and can F I G U R E 5 Decision tree diagram visualizing the choice of each of five methods depending on the conditions.When faced with low interaction rates between the individuals of interest we advise to use the SSF-DIST approach regardless the tracking period and sampling resolution.When the interaction rates are high, any of the five methods perform well.For studies with long tracking periods, we advise to use Cs, DI, and both SSF-based approaches.When the tracking period is too short or the sampling resolution is fine, we advise against the SSF-OD approach.It should be noted that the ideal tracking period and sampling resolution are dependent on the studied animal species.Individuals of some species might interact several times with con-or heterospecifics over few hours and thus a tracking period of several days might be sufficient, whereas other species might only interact once every week with con or heterospecifics and therefore a tracking period on order of months might be needed.be used more broadly to investigate attraction, repulsion or neutral behaviour among moving individuals more generally.Therefore, we demonstrate a further application of HMMs in inferring interactions from movement data when working with animals that are not moving in groups (Figure S2).Briefly, we tested whether HMMs can be used to identify a switch of movement state resulting from the distance between individuals.We expect that especially in predatorprey scenarios, in which interaction state is often sufficiently long (compared to the tracking data resolution), HMMs might be practical to identify these interaction states.Our suggestions are that future research is needed to assess how comparable HMMs are with other existing methods and to reveal the pros and cons of using HMMs in inferring interactions from movement data.
To sum up, we assessed the performance of five statistical methods to infer interactions from movement data, by using a relatively simple simulation model of moving predator and prey individuals.
We expect our study to motivate theoretical ecologists who develop and test statistical methods for movement data analysis to critically scrutinize the methods under a set of factors relevant in the field.
Similarly, we hope that the guidelines we provide here will aid field ecologists to optimize their study design, data collection, and selection of the appropriate method.

Cs
functions applied to distances between two individuals SSF ~ + 1 Attraction SSF ~ − 1 Avoidance SSF ~ 0 Neutral TA B L E 2 For all levels of manipulated factors shown are proportion of cases in which Cs, DI, SSF-OD and SSF-DIST methods correctly detected the appropriate interaction type when the interaction is present.

Figure
FigurePerceptual range of predator d 0 is the mean spatial distance between temporally simultaneous fixes of two individuals within a time threshold and d E is the expected mean distance based on n 2 permutations of the temporally simultaneous fixes.The index ranges from −1 to 1 where positive values mean attraction, negative values mean avoidance and values close to 0 mean independent movement.
2.2.2 | Correlation index (Cr)The Cr describes dynamic interactions by calculating the differences in movement vectors between two individuals at simultaneous time fixes while taking into account the mean movement vector of the whole trajectory of each individual (Cr;Shirabe, 2006).Cr is calculated as: where v t and w t correspond to movement vectors of two individuals at given time fixes t within the trajectory of n number of fixes.v and w are mean vectors of the individual's overall movement paths.Vectors are represented by their speed and movement direction.Values for Cr range from −1 to 1 where negative values correspond to negative correlation (paths of two individuals are in opposite direction) and positive values indicate positive correlation (both paths are in same direction).
period t for both individuals and within the trajectory of n number of fixes.The localized di is calculated by incorporating displacement and direction of both individuals for each time period.Thus, the localized di can distinguish between cohesiveness in distance di d and cohesiveness in movement direction di .Values for DI also range from −1 to 1 where negative values correspond to repulsive movement paths and positive values indicate cohesive movement paths.Values around 0 indicate neutral movement.
steps are obtained using the distribution of turning angles (usually following a von Mises distribution) and step length distribution (gamma or exponential distribution) of the actual individual's movement path.If the estimates of selection strength are positive, we interpret this as an attraction of a focal individual to the other one; if selection strength is negative, this means avoidance.Estimates around zero would suggest neutral behaviour of the focal individual towards the other one.These methods can distinguish, differently to the three methods outlined above, between attraction, avoidance and neutrality from the perspective of each single individual.Further, compared to the indices of dynamic interaction which can only be applied to movement data of two individuals at a time, the SSF-based approaches can be used to analyse the response of a focal individual towards multiple individuals.Important to address is that the coefficients are not bound between −1 and +1, but their scale depend on the scale of the covariates used.This means that the coefficient should only be interpreted with their corresponding p-value.
of the animal's occurrence within given time periods where values close to one indicate locations that the individual has most likely visited and values close to zero indicate locations where the animal most likely did not occur.SSF are then applied to assess how such occurrence distribution of one individual affects the movement of another (focal) individual.In our study we applied SSF-OD to data with low temporal resolution only (1%), as applying it to data with high resolution would yield meaningless results.
the focal individual moves closer to or further away from the other individual compared to randomly sampled locations.In this case, larger distances among two individuals result in positive estimates of selection strength, meaning that such positive values reflect avoidance.Shorter distances, on contrary, result in negative estimates of selection strength, which are interpreted as attraction.In our study, for a better comparison of the results obtained with both SSF-methods, we multiplied the selection strength estimates obtained with SSF-DIST by −1, so that positive values indicate attraction and negative values indicate repulsion, exactly like in SSF-OD approach.