Ergodicity breaking and lack of a typical waiting time in area-restricted search of avian predators
Ohad Vilk, Yotam Orchan, Motti Charter, Nadav Ganot, Sivan Toledo, Ran Nathan, Michael Assaf
DD R A F T Ergodicity breaking and lack of a typical waitingtime in area-restricted search of avian predators
Ohad Vilk a, b , Yotam Orchan b , Motti Charter c , Nadav Ganot b , Sivan Toledo d , Ran Nathan b,1 , and Michael Assaf a,1 a Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel; b Movement Ecology Lab, Department of Ecology, Evolution and Behavior, AlexanderSilberman Institute of Life Sciences, Faculty of Science, The Hebrew University of Jerusalem, Jerusalem, Israel.; c The Shamir Research Institute and Department of Geographyand Environmental Studies, University of Haifa, 199 Aba Hushi Boulevard, Mount Carmel, Haifa, Israel.; d Blavatnik School of Computer Science, Tel-Aviv University, Israel.This manuscript was compiled on January 28, 2021
Movement tracks of wild animals frequently fit models of anomalousrather than simple diffusion, mostly reported as ergodic superdiffu-sive motion combining area-restricted search within a local patchand larger-scale commuting between patches, as highlighted by theLévy walk paradigm. Since Lévy walks are scale invariant, superdif-fusive motion is also expected within patches, yet investigation ofsuch local movements has been precluded by the lack of accuratehigh-resolution data at this scale. Here, using rich high-resolutionmovement datasets ( > × localizations) from 70 individuals andcontinuous-time random walk modeling, we found subdiffusive be-havior and ergodicity breaking in the localized movement of threespecies of avian predators. Small-scale, within-patch movement wasqualitatively different, not inferrable and separated from large-scaleinter-patch movement via a clear phase transition. Local search ischaracterized by long power-law-distributed waiting times with di-verging mean, giving rise to ergodicity breaking in the form of con-siderable variability uniquely observed at this scale. This impliesthat wild animal movement is scale specific rather than scale free,with no typical waiting time at the local scale. Placing these findingsin the context of the static-ambush to mobile-cruise foraging con-tinuum, we verify predictions based on the hunting behavior of thestudy species and the constraints imposed by their prey. anomalous diffusion | movement ecology | continuous-time random walks | static-to-mobile foraging continuum | ergodicity breaking M ovement of organisms is of key interest in many scientificfields, playing an essential role in a wide range of biolog-ical and ecological systems. The movement ecology paradigmaims to integrate theoretical and empirical frameworks forstudying the causes and mechanisms underlying movementpatterns, and their consequences, in all organisms (1). Withinthis framework, individual-based modeling of movement is im-perative for understanding the dynamics of ecological systemsat multiple spatial and temporal scales (2). With the increas-ing availability of high-quality tracking data, models can berefined to infer behaviors and to draw causal links betweenobserved phenomena and their underlying mechanisms beyondphenomenological description of the observed patterns. Thiscapacity, however, requires high spatiotemporal resolution oftracking data that is crucial for inferring behavior from move-ment data at sufficiently fine scales in which animals senseand respond to their dynamic environment (1, 3, 4).Movement of animals varies across spatiotemporal scalesdue to resource patchiness, seasonality or other environmentfeatures such as competition, predator-prey dynamics and var-ious other biotic interactions; all typically vary across scales(1, 3, 5–8). Inferring behavior from one scale to another canthus lead to nonrepresentative results (7, 8). In particular,real-life landscapes are typically heterogeneous, and animals routinely alternate between an extensive commuting modeof movement between resource-rich patches, and an inten-sive searching mode of area-restricted search (ARS) for preywithin a local patch (5, 7, 9, 10). Such alternating behav-ior was claimed to increase foraging efficiency under specificconditions (11, 12). Elucidating the drivers of this commonalternating behavior requires detailed information on animalmovement both within and between patches at high spatialand temporal accuracy, for multiple individuals and over asufficiently long time, which are difficult to obtain with stan-dard wildlife telemetry tools (7). Furthermore, establishedscaling laws in animal movement (see below) commonly en-gage averaging across conspecific individuals (13, 14), whereasintraspecific variation has only recently received significantattention (14–16). While many studies have distinguished anextensive commuting mode from an intensive searching (ARS)mode, movement within ARS has seldom been analyzed, over-looking a critical scale of foraging movement essential forelucidating the patterns and mechanisms underlying variationin movement behavior.Animal movement has often been modelled by anomalousdiffusion, and considered scale-free by incorporating the Lévy-walk formalism (e.g. 17–19). Lévy walks – random walks (RWs)with jump lengths drawn from a fat-tailed distribution (20, 21)– have been hypothesized to reflect an optimal foraging strategyin landscapes with sparsely distributed resources (22, 23). Yet,
Significance Statement
The scale-free nature of animal movement patterns underlies along-standing question in ecology, physics, and related fields.In this study, we show that contrary to the scale-free foraginghypothesis, movement patterns of wild animals are highly scalespecific. Analyzing high-resolution movement data of threespecies of avian predators and employing theory of anomalousdiffusion and continuous-time random walks reveals a phasetransition between local, within-patch search and larger-scaleforaging behavior. In particular, local area-restricted search dis-plays subdiffusive and nonergodic behavior, while large-scalecommuting is superdiffusive and ergodic. Importantly, avianpredators in our study remain stationary during local searchfor highly variable durations including exceedingly long times,suggesting multiple hunting tactics and behavioral modes.
O.V., R.N. and M.A. designed the model and performed the data analysis. Y.O., M.C., N.G., S.T.and R.N. established and maintained the tracking systems, and collected the data. O.V. and M.A.performed the analytical and numerical calculations. O.V., R.N. and M.A. wrote the manuscript.Authors declare no competing interests. To whom correspondence should be addressed. E-mail: [email protected],[email protected]
January 28, 2021 | vol. XXX | no. XX | a r X i v : . [ q - b i o . P E ] J a n R A F T this hypothesis remains debatable (5, 15, 24–26), since anoma-lous diffusion may also arise from alternative mechanisms (19).Here local movement within ARS are referred to as jumps,while movement between ARS is referred to as commuting.Furthermore, due to the scale-free nature of these power-lawdistributions, the Lévy-walk hypothesis implies that patternsare similar at different scales (21). This stands in contrast withthe growing evidence for multiphasic movement (5–8), thoughanalyses based on high-resolution tracking of wild animalsacross multiple scales are still too scarce to generalize whetheranimals mostly move in a scale-free or a scale-specific manner.Besides a fat-tailed jump distribution, Lévy walks can alsopossess long waiting times (WTs) (27), reported for ambush(sit and wait) predators such as marine molluscs and fish(28), mobile free-ranging foragers such as seabirds (29, 30),insects (9) and cattle (31) in experimental arenas, and humans(32). The underlying reasons for long WTs span from longrests (33), pauses to more effectively search for hidden preyand to organize attacks, or due to intra- and inter-specificinteractions such as mating, territorial guarding and predatoravoidance (34, 35). Overall, animals exhibit a wide range ofintermittent foraging behaviors, from ambush with very longstops and short or no moves, through saltation (stop and go)alternating between stops and moves of intermediate length, tocruising (widely ranging) with constant movement and a fewstops (34, 35). It has been recently suggested that scale-freetemporal dynamics are the equivalent of the Lévy-walk scale-free spatial patterns for ambush predators (36). We propose adifferent conjecture, see below, that there exists a shift fromnonergodic to ergodic and possibly from sub- to super-diffusivedynamics along this static-to-mobile foraging continuum.In this study, we use high-resolution data from a reverse-GPS wildlife tracking system (37) to characterize within-patchARS movements of three species of avian predators: barnowls ( Tyto alba , hereafter “owls”), black-winged kites (
Elanuscaeruleus , “kites”), and common kestrels (
Falco tinnunculus ,“kestrels”), all common residents in Israel. These species rep-resent a rather restricted part of the static-to-mobile foragingcontinuum and have been classified, based on diet and foragingtechnique, into the same guild of “carnivore ground hawkers”(38). Focusing on ARS, these three species combine multiplesearch tactics including short cruising flights, static hoveringand perching while pursuing prey (39–42). However, sincehovering and especially long perching have been frequentlyobserved at this scale in all three species, we hypothesized thatlocal search (ARS) will be characterized as subdiffusive andnonergodic due to relatively long WTs. We combined track seg-mentation tools and models of anomalous diffusion to test thishypothesis, and to examine the notions of scale-specific move-ment, ergodicity breaking and ageing. While ageing indicatesa tendency to decrease diffusivity over time (43), in the contextof ecology, ergodicity breaking is manifested by a discrepancybetween long-time averaging over an individual search patternand an ensemble average of searches. Importantly, this bringsabout a large variability in search patterns of an individualleading to the absence of a typical search. Overall, we studywhether foraging behaviors are inferable from models account-ing for stochastic processes without explicitly incorporatingcomplex interactions among individuals/species or differentialresponses to environmental heterogeneity in space and time.
Theoretical model
In physics, ergodicity breaking is defined as a disparity be-tween the mean square displacement (MSD) and time-averagedsquare displacement (TASD), which can cause time-averagedmeasurable quantities to be irreproducible (43, 44). The MSDis defined as the ensemble averaged square displacement of anindividual’s position with respect to a reference position. Inanomalous diffusion, the MSD satisfies (cid:10) x ( t ) (cid:11) ∼ t α , whereangular brackets denote ensemble averaging and t is the timeof measurement. Here, the dynamics is superdiffusive for α > α <
1, whereas α → δ (∆) = 1 t − ∆ Z t − ∆0 [ x ( t + ∆) − x ( t )] dt , [1]where an overline denotes time averaging. For simple Brownianmotion [e.g. Pearson’s RW (45)] and ∆ (cid:28) t one obtains δ (∆) ∼ ∆ ∼ (cid:10) x (∆) (cid:11) . Moreover, the TASD does not dependon the total measurement time t . In contrast, if the TASD andMSD scale differently, the underlying process is, by definition,nonergodic; that is, the ensemble averaging is different from the time averaging (20). Here, the process is said to display weakergodicity breaking, to be distinguished from strong ergodicitybreaking where phase space is separated into non-accessibledomains. Note that while not all subdiffusive processes displayergodicity breaking (44), we show that our system is bothsubdiffusive and nonergodic. Importantly, in many realisticcases, the TASD is a more convenient experimental measurethan the MSD as the former provides good statistics in thelimit ∆ (cid:28) t and does not require a reference position or time,which is often arbitrary or unknown in ecological systems (43).To assess anomalous diffusion within ARS, we apply thecontinuous-time random walk (CTRW) formalism, frequentlyused in physical, biological, and ecological systems (18, 32,46, 47). Originally introduced by Montroll, Weiss, and Scher(48), CTRW is a generalization of Pearson’s RW, defined interms of the WT τ between successive jumps, which is arandom variable drawn from probability distribution function ψ ( τ ). When the average WT h τ i diverges, the process displayssubdiffusive dynamics and weak ergodicity breaking (20, 43).We assume power-law distributed WTs: ψ ( τ ) ∼ τ − (1+ α ) . [2]which, for 0 < α <
1, give rise to a diverging mean and ergod-icity breaking. In contrast, Pearson’s RW is a Poisson processwith exponentially-distributed WTs, ψ ( τ ) = τ − exp( − τ /τ ),where τ is the mean WT, resulting in ergodic dynamics. No-tably, in CTRW the jump length can also be taken as a randomvariable; we have numerically verified that our main resultsare independent of the jump length choice (see SI).To properly quantify the subdiffusive process at hand, oneoften employs the so-called averaged TASD (43): (cid:10) δ (∆) (cid:11) = 1 /N N X i =1 δ (∆) , [3]where TASD is averaged over an ensemble of N ARS segments;averaging is necessary here due to the irreproducible natureof the process (i.e. large diversity across trajectories). Fur-ther assuming that the movement is confined to a spatially et al. R A F T bounded subregion (strictly speaking, due to an external con-fining potential), and using the fractional differential equationformalism and Eq. (2) (see SI), one can show that (49): (cid:10) δ (∆) (cid:11) ∼ (∆ /t ) − α , [4]which is valid for long measurement times ∆ (cid:28) t . This standsin contrast to Pearson’s RW and most ergodic processes forwhich the TASD saturates δ (∆) ∼ (cid:10) x (∆) (cid:11) ∼ ∆ , uponreaching a length scale comparable to the system size.Below, we show that the observed nonergodic and subdiffu-sive movement patterns in ARS indeed stem from power-lawdistributed WTs and infer the subdiffusion exponent α viaEq. (4). Importantly, we show that during ARS the distri-bution of the TASD around the mean is relatively broad,characteristic of ergodicity breaking and ageing. Results
We tracked 60 owls (18 adults and 42 fledglings), 21 kites (6adults and 15 fledglings), and 15 kestrels (all adults) in twodistinct locations in Israel: the Hula Valley and the JudeanPlains. Individuals were tracked using ATLAS, an innovativereverse-GPS system that localizes extremely light-weight, low-cost tags (37). Each ATLAS tag transmits a distinct radiosignal which is detected by a network of base-stations dis-tributed in the study area. Tag localization is computed usingnanosecond-scale differences in signal time-of-arrival to eachstation, allowing for real-time tracking and alleviating the needto retrieve tags or have power-consuming remote-downloadcapabilities (50). The tracking frequency of individuals was be-tween 0.125 and 1 Hz. We collected additional information forthe adult birds: location of the nest, sex, breeding status, andbrood size (when relevant). Only individuals who had > a-priori , as in general they do not correspond towell-defined, discrete spatial units (5). These are thus definedvia the segmentation procedure and are robust with respectto changes in the segmentation parameters (see SI). We em-phasize that movement data were segmented into two naturalscales without a-priori assuming that movement patterns are Fig. 1. (a) Seventy-five minutes of tracking at a frequency of 0.5 Hz were segmentedinto 4 ARS and 3 commuting segments. These consisted of 2,280 localizations ofa female owl (tag 4782) on the 4th of May 2018, starting from 7:50 pm local time.The arrows show the direction of motion. The 4 ARS segments are drawn in redand are 4, 2, 4 and 57 min long, while the 3 commuting segments are in blue, eachapproximately 1 min long. The inset shows one local search of size × m ,57 min long. This ARS consists of 5 local clusters with WTs of 14, 36, 2, 3, and 1min, each represented by a different color and highlighted by a red circle. In movingbetween the local clusters, the animal performs local jumps ranging from to m.The non-stationary behavior of the animal (inset’s upper right corner) lasts for lessthan a minute. (b) The averaged TASD ( Eq. (3) ) of ARS segments (red triangles)and commuting segments (blue dots) of a female owl. The dashed and dash-dottedlines are power laws with ∆ . and ∆ . respectively, clearly separating betweensubdiffusive ARS and superdiffusive commuting mode. Inset shows the distribution ofTASD around the average TASD (see Eq. (5) ) for both cases, at a specific time lag ∆ that is much smaller than each segment’s total time. either scale specific or scale free, but rather a-posteriori testedthe scale-free prediction of the Lévy walk foraging hypothesis.In Fig. 1(b), we compare the measured diffusivity of com-muting and ARS segments by calculating the averaged TASDin both cases for an individual female owl. While the com-muting segments show an averaged TASD that resemblesballistic motion ( (cid:10) δ (∆) (cid:11) ∼ ∆ . ) ∗ , the averaged TASD ofARS segments is qualitatively different and is clearly subd-iffusive ( (cid:10) δ (∆) (cid:11) ∼ ∆ . ), which confirms our scale-specificassumption (see additional evidence below). Moreover, theinset of Fig. 1(b) shows a wide distribution of TASD aroundthe average ARS, characteristic of ergodicity breaking, and asharply peaked distribution for the commuting phase.In Fig. 2 we use simulations to show the markedly differentbehavior between subdiffusive CTRW and simple RW, in thecase of a confined movement within predefined domain walls.In Fig. 2(a-b) we show that for exponentially distributed WTs,the TASD saturates upon interacting with the boundaries andthat there is no dependence on the measurement time. Incontrast, in Fig. 2(c-d), we show that the averaged TASDdoes not saturate with ∆ and that the dependence on the timeof measurement t agrees with Eq. (4). ∗ The long commuting flights taken by the owl are strongly directed and relatively fast ( . – . m/s),and similar directional fast commuting also holds for kites and kestrels. Vilk et al.
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January 28, 2021 | vol. XXX | no. XX | R A F T Fig. 2.
CTRW simulations. (a-b) Exponentially-distributed WTs (a simple RW with τ = 10 s ). (c-d) Power law WTs, Eq. (2) , with α = 0 . . In (a,c) each of the redlines are the TASD calculated for a single simulation versus the time difference ∆ ,and the blue line is the average TASD, Eq. (3) , over an ensemble of 200 simulations.In (b,d) we plot the averaged TASD versus the total time of the simulation. Simulationswere done in a bounded domain of × m . For subdiffusive CTRW, since very long WTs can occur,time-averaged measurable quantities are generally irrepro-ducible, such that realizations vary from one another even forvery long times (43), see Fig. 2(c). This variability of theTASD is quantified in terms of the dimensionless parameter ξ = δ (∆) / (cid:10) δ (∆) (cid:11) – the TASD distribution normalized byits average. At long measurement times the distribution of ξ satisfies a Mittag-Leffler distribution (49, 52, 53): φ ( ξ ) = Γ /α (1 + α ) αξ /α l α (cid:18) Γ /α (1 + α ) ξ /α (cid:19) , [5]where l α is the one-sided Lévy stable distribution with theLaplace transform L{ l α ( t ) } = exp( − u α ) and Γ( · ) is theGamma function. In the limit of Brownian diffusion, α → φ ( ξ ) ∼ δ ( ξ − α the distribution is wide and skewed; e.g.,for α = 0 . φ ( ξ ) tends to a half Gaussian with maximum at ξ = 0. Importantly, Eq. (5) also holds for subdiffusive CTRWin a bounded domain (49, 54) and accounts for the broaddistribution of TASD in ARS, see Fig. 1(b).Having segmented the data into commuting and ARSphases, we computed the average TASD, (cid:10) δ (∆) (cid:11) , for eachbird in the following way: for each ARS, we computed theTASD and then averaged over all TASDs with similar totalmeasurement times. Importantly, for all individuals, we aimedto average only TASDs within the same period, either withinor after the breeding season (see Methods). The TASD andaveraged TASD are plotted in Fig. 3(a) as a function of ∆ foran individual female owl, during the breeding season. For thisfemale, for ∆ (cid:28) t , the slope on a log-log plot approaches avalue of 0 . ± . † . In Fig. 3(b), we plot the averaged TASDas a function of t for different values of ∆. Here we see anexplicit dependence on the time of measurement, which is char-acteristic of ageing as longer measurement times infer smallermeasured diffusivity (43). Thus, Fig. 3(a-b) strongly indicatesthat the process is subdiffusive. Moreover, the dependence † The fit was performed on the range of < ∆ < min to ensure the validity of the theory andthat we were above the noise limit. We computed the slope for different measurement times, andthe error reflects a 95% confidence interval in the slope variations for different measurement times. Fig. 3. (a-b) TASD and averaged TASD of ARS of a female owl (tag 4789) during thebreeding season of 2018. (a) TASD for single trajectories (red lines) versus ∆ , andthe average of all lines ( Eq. (3) ) (blue solid line). The total measurement time is t = 50 min. The lower dashed line scales as ∆ . . (b) The averaged TASD versusthe total measurement time t for specific values of ∆ = 10 , , s (bottom to toplines), yielding powers of − . , − . and − . respectively. (c) The distributionof ξ of ARS for the same individual in the same time period (plotted using kernelestimation with a bandwidth of 0.1). Each of the + markers represents a distributionfor a different ∆ ( ∆ = 100 , , , seconds). The total measurementtime is seconds. The dashed line is the theoretical distribution ( Eq. (5) ) for α = 0 . . Inset shows the same plot on a linear scale. (d) The autocorrelationfunction defined as R XX ( τ ) = E [ X ( t ) X ( t + τ )] , versus the time difference τ ,such that R XX ( τ = 0) = 1 . Here X is the location of the bird normalized by themean location. The blue dots are R XX averaged over 100 owl ARS segments oflength s . The orange dashed and the green dash-dotted lines are averages over1000 simulations of subdiffusive CTRW (with α = 0 . ) and simple RW, respectively. (cid:10) δ (∆) (cid:11) ∼ ∆ . indicates that for this individual, α ’ . ‡ .We have measured the subdiffusivity for all individuals in thesame manner and recorded the value of α . The mean valueof α for different subgroups within each species is given inTable 1. Notably, within each species, the values of α did notstatistically differ between males and females, fledglings andadults, and during or after the breeding season (see Methods).To evaluate the reproducibility of individual ARS, we cal-culated the distribution of ξ for different ∆ for each individual.Using Eq. (5), we compare in Fig. 3(c), between theoretical(for α = 0 .
6) and experimental results for one female owl.Importantly, this broad distribution [see also Fig. 1(b)], whichdoes not seem to depend on ∆, as predicted by Eq. (5), isobserved for all individuals and serves as further evidenceof weak ergodicity breaking. The fact that the experimentalTASD distribution is more sharply peaked than the theoreticalprediction, can be explained by the presence of noise in ourdata, which skews the ξ distribution and yields lower thanexpected values close to ξ = 0 (55). Additionally, in Fig. 3(d),we plot the autocorrelation function computed for owl ARSaveraged over many ARSs, revealing that simulations basedon subdiffusive CTRW fit the data much better than thosebased on simple RW. This is also confirmed independently bya p-variation test (56) (see SI).Direct evidence for long WTs is shown in Fig. 4, by plottingthe distribution of WTs within a radius of 15 m (about thesize of a local cluster within ARS) for all three study species.The WTs are calculated using a spatiotemporal criterion forthe segmentation procedure with a threshold R th ; see theinset of Fig. 1. We chose the threshold to be above the noise ‡ We do not extract the value of α from the dependence of the averaged TASD on t , as here, ageingeffects can play a significant role, see Ref. (43) et al. R A F T Table 1.
Mean α values for species subgroups, including the individualsnumber in each subgroup and a confidence interval around the mean. species subgroup group size h α i ± owls breeding adults 14 0.69 0.05post-breeding adults 14 0.65 0.05fledglings 30 0.66 0.03kites adults 6 0.56 0.05fledglings 10 0.51 0.05kestrels adults 10 0.51 0.09 limit of 5–10 m and much smaller than the typical size of anARS patch; the results do not vary significantly when R th is between 10 and 25 m. We fitted the data to a power-lawdistribution [Eq. (2)] using the method of maximum likelihood(57) § . In Fig. 4, we plot the distribution of all individualswithin each species; similar results were separately obtainedfor each individual (see SI). For most individual birds, theresults of this fit were within the error of the value of α ,independently found by fitting the averaged TASD to Eq. (4),see Table 1. Importantly, the distribution of stops within a15 m radius anywhere within the foraging range (excludingtheir nest and its vicinity) shows a tendency of power-lawdependence to break down at an average time of 40 min, aswas corroborated by fitting a truncated power law to thedata (Fig. 4(a-c)). Focusing on ARS, the distribution of stopdurations only within ARS reveals a clear phase transition at30–60 min for all three species (Fig. 4(d-f)). For short times( <
40 min), the distribution is best fitted by a power-law with α = 1 .
15, whereas for longer times ( >
40 min), the data isbest fitted by a power-law with α > ¶ . We thus suggest that the long stopswithin ARS, driven by the motivation to hunt from a perch,to rest, or by other reasons, are not only spatially but alsotemporally confined, such that beyond ∼
40 min the functionalgain from a long stop is diminished, driving the bird to moveto another location within or outside the local patch.
Discussion
Applying the CTRW framework to rich high-quality movementdatasets encompassing > localizations from 70 individualsof three avian predator species, we revealed that local ARSis uniquely characterized as nonergodic, irreproducible andsubdiffusive, whereas commuting is ergodic, reproducible andsuperdiffusive. That is, there is a typical commuting segment,but there is no typical ARS. These two distinct phases areseparated via a phase transition in movement occurring atcharacteristic spatial and temporal scales. Specifically, ARScombines short and very long WTs yielding subdiffusive non-ergodic motion, while the commuting mode is superdiffusivewith long directional flights, and ergodic as implied by theTASD distribution around the mean. Although the average § To test the quality of fit, we used a likelihood ratio test to compare between a power law andexponential fit for the distribution (57, 58). In all cases, a power law was a better fit. A similar testbetween a power law and a truncated power law showed that the latter was a better fit. ¶ As the mean WT and variance of the jump length distribution are finite at such spatiotemporalscales (see SI) the process corresponds to regular Brownian motion (43).
Fig. 4. (a-c) WT distributions (symbols) within a radius of m, for 14 owls (a), 6kites (b) and 10 kestrels (c), all adults. The solid line is a fit to a truncated powerlaw, P ( τ ) ∼ τ − α e − τ/τ , with α = 0 . , . , . , and τ = 40 , , in (a),(b) and (c), respectively. These values of α are the average fit values for the jointdistribution of all birds within each species, and we have checked that for all birds, thepower law truncates at τ = 20 – min. The dashed lines are power laws, Eq. (2) ,with the same α values. (d-f) Distributions of stop duration within ARS (symbols) forowls (d), kites (e) and kestrels (f), all adults. The dashed black and red lines are powerlaws, presenting a clear phase transition from α = 1 . for shorter τ in all threespecies to α = 2 . , . and . for longer τ , with typical transition times (whenthe two slopes intersect) of τ = 45 , and min in (d), (e) and (f), respectively. WT during ARS has been used to distinguish foraging tacticsamong species (59), our study shows that the distribution ofWTs during search is fat-tailed, with a cutoff around 40 min,indicating that ARS is composed of multiple foraging tactics.Moreover, while ARS is characterized by clear ageing, commut-ing birds maintain similar movement characteristics regardlessof commuting duration. Putting this in simple words, thereare many ways to hunt within a local patch but only a limitednumber of ways to commute between distant patches.Our findings contradict the scale-free Lévy foraging hy-pothesis that local within-ARS movement and long-distancecommuting between ARS stem from the same distribution. In-stead, movements of all three avian predators differ markedlybetween the two scales, though fat-tailed distributions of WTsand steps have been observed at both scales. Overall, our anal-ysis offers novel insights and a general formalism for resolvingthe long-standing conundrum of whether free-ranging animalsmove in a scale-specific or scale-free manner (5, 10).We hypothesized that local ARS in the three study specieswould tend towards the static side of the static-to-mobile for-aging continuum given their hunting behavior and despitetheir high mobility. Our findings support this hypothesis, asevident in the similar levels of subdiffusivity among speciesand individuals, indicating they operate within the same nar-row zone of the continuum. Within species, α values weresimilar for both adults and fledglings and were higher for adultowls during breeding, but neither statistically different fromthe subsequent non-breeding season, nor between males andfemales. This tendency of breeding owls to minimize WTswithin ARS is expected due to the urge to provision theirnesting mate and nestlings, and the lack of a significant differ-ence between the two periods can be attributed to the urgeof adult owls to provision their young also a few months afterthey fledge. Interestingly, the only key difference we found isamong species: while anomalously long in all three species,WTs of kestrels and kites were similar and longer (lower α )than those of owls, suggesting that nocturnally foraging owlstend to remain stationary for shorter periods compared to thetwo diurnal raptors. Overall, assuming that the power-law dis- Vilk et al.
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January 28, 2021 | vol. XXX | no. XX | R A F T tributed WTs encapsulate a wide spectrum of possible tactics,our framework offers a way to quantify fairly subtle variationalong the static-to-mobile foraging continuum, based directlyon movement patterns within ARS.What is the origin of these slow, subdiffusive, dynamicswithin ARS? Although slow dynamics are counter-intuitive,they make sense if we recall that the predators primarily forageto find their target rather than to cover the most ground. Wepostulate that hunting efficiency can increase via highly vari-able, irreproducible foraging tactics compared to less variable,reproducible ones. Such variability may hold an evolutionaryadvantage through higher individual fitness of avian preda-tors that combine different foraging behaviors (11, 12). Wealso suggest that long WTs and ergodicity breaking are evolu-tionarily coupled to prey behavior. Barn owls, for example,are well known for their acute night vision and their highauditory sensitivity enabling high spatial resolution in soundlocalization of their prey even in complete darkness (60, 61).Their prey, however, have evolved to avoid owl predation usingvarious strategies such as minimizing exposure in risky timesand habitats (62) and adopting escape strategies during anactive owl attack (63, 64). Controlled experiments in a closedarena revealed that owls had higher success in catching sta-tionary rather than moving prey, and they tended to postponetheir attack until their prey became motionless (64). Anotherexperiment in the same settings revealed high variation incapture duration (from first attack to a successful capture) ofspiny mice ( Acomys cahirinus ) and Günther’s voles (
Microtusguentheri ), ranging from 0.5 sec to 43 min (63). The longestprey captures occurred while hunting voles, the preferred preyof owls in Israel (65, 66) and in the Hula Valley in particular(67). Thus, as it is hard to catch a moving target, adoptingirreproducible (and thus unpredictable) movement tactics mayprove beneficial for a predator. In other words, it is moreefficient for a predator to adopt a wide spectrum of huntingtactics and behavioral modes among which it can alternate,rather than committing to a single tactic or behavior.In summary, movement patterns of the three avian preda-tors examined here are scale specific, with a unique scalinglaw for local scales that does not carry on to larger scales. De-spite the urge to provision their offspring, our results provideevidence that free-ranging birds in the wild tend to remainstationary in their hunting sites for long periods, presumablydue to co-evolutionary constraints imposed by the availabilityand behavior of their prey.
Materials and Methods
Tracking method.
Sixty owls were tracked in the Hula Valley, Israel(33.10N, 35.61E) between May and December 2018. These include18 adults tracked both during breeding and subsequent post breedingand 42 fledglings tracked for the first few months after fledging.For the adults, we used the hatching date to define the breedingseason by assuming that the 90 days following hatching are withinthe season, as nestlings are still dependent on their parents (41).Twenty-one kites were tracked in the Hula Valley between July 2019and September 2020. Six were adults that actively bred duringthe tracking period, and the other 15 were fledglings. As kites canhave multiple broods in a year, we defined their breeding season bydirectly observing their nests. Lastly, fifteen kestrels were trackedin the Judean Plains, Israel (31.74N, 34.84E) between March andAugust 2019. Among these, 11 were actively nesting, and thebreeding season was defined by direct observations.All individuals were tracked using ATLAS (37, 50), which re-ports localization errors in the form of a 2 × > m , defined in terms of the trace over the covariance matrix.Furthermore, we filtered out days or nights in which many local-izations are missing ( > > × localizations for44 owls, 1,619 days and > localizations for 16 kites, and 508days and > × localizations for 10 kestrels, mostly during theirrespective breeding seasons. Segmentation.
ATLAS tracks were first segmented to exclude stopsin or around the (known) nests of breeding adults. The remainingtracks were considered foraging excursions and segmented by detect-ing distinct switching points separating ARS and commuting modes.We used the Penalized Contrast Method suggested by Barraquandand Benhamou (51), a non-parametric method in which the initialnumber of segments is unknown and estimated by minimizing apenalized contrast function. First passage time (FPT) was used asthe focal metric (51, 68). Each point was assigned an FPT outside aradius of R s and data was segmented such that points with similarFPT that were close in time were clustered together (69). Dataare then split into commuting and ARS according to a thresholdon the mean FPT chosen in accordance with the animal’s velocity,which during commuting is 7 −
10 m/s for all three species. In oursegmentation we choose R s = 100 m and a threshold of 50 s; yet ourresults are insensitive to small changes in these parameters: R s wastested between 70 −
150 m and the threshold on the FPT between40–120 s (see SI). Note that the choice of threshold reflects thetime it takes the bird to cross the area defined by R s . For instance,taking R s = 100 m, we define the threshold for a commuting flightssuch that crossing a diameter of 200 m takes <
50 s, as if the birdflies in a velocity of 4 m/s across in a straight line.
Comparative analysis.
We used an unpaired t-test to compare thevalue of α between species. After verifying variance homogeneity(Levene test) and normal distribution (Shapiro-Wilk test) we per-formed for each two species an unpaired t-test for the means oftwo independent samples, with the null hypothesis being that themeans are equal. P-values were corrected for multiple comparisonsusing Tukey’s honest significance test. Based on these tests, wefound no difference between kestrels and kites ( p > . α than either kites ( p = 0 . p = 0 . α during breeding for all 14individual owls, but the null hypothesis of identical averages wasnot rejected ( p = 0 . ACKNOWLEDGMENTS.
For fieldwork and technical assistancewe thank Y. Bartan, A. Levi, S. Margalit, R. Shaish, G. Rozmanand other members of the Movement Ecology Lab and the MinervaCenter for Movement Ecology. We also thank R. Metzler for usefulcomments. O.V. and M.A. acknowledge support from the ISFgrant 531/20. ATLAS development, maintenance, and studies havebeen supported by the Minerva Center for Movement Ecology, theMinerva Foundation, and ISF grants 965/15 and 1919/19 to R.Nand S.T.; research on black-winged kites was supported also byJNF/KKL grant 60-01-221-18. R.N. also acknowledges supportfrom Adelina and Massimo Della Pergola Chair of Life Sciences. et al. R A F T Appendix
Here we provide technical details and additional results to supportthe derivations presented above.
Mathematical background.
Here we give further details oncontinuous-time random walks (CTRWs) and ergodicity breaking,and provide a detailed derivation of Eq. (4). We also discuss theuniversality of this result with respect to the confining potentialform and jump-length distribution.CTRWs are defined in terms of the waiting time (WT) τ be-tween successive jumps, which is a random variable drawn from aprobability distribution ψ ( τ ). Assuming ψ ( τ ) ∼ τ − (1+ α ) , [A1]the (one-dimensional) fractional Fokker-Planck equation governingthe probability density W ( x, t ) of being at position x at time t is(20): ∂∂t W ( x, t ) = K α D − αt (cid:26) − ∂∂x h F ( x ) k B T i + ∂ ∂x (cid:27) W ( x, t ) , [A2]where V ( x ) = − R x F ( x ) dx is the confining potential of the ran-dom walker, while for a free (unconfined) walker, F ( x ) = 0. Further-more, K α is a generalized diffusion parameter, and the RiemannLiouville operator D − αt ≡ ( ∂/∂t ) D − αt is defined for 0 < α < D − αt φ ( x, t ) ≡ α ) ∂∂t Z t dt φ ( x, t )( t − t ) − α . [A3]In the limit of α →
1, Eq. Eq. (A2) reduces to the Fokker-Planckequation as the Riemann Liouville operator reduces to the unityoperator. For an unconfined random walker the mean square dis-placement (MSD) is given by (20) (cid:10) x (cid:11) = 2 K α t α Γ( a + 1) , [A4]while the averaged time-averaged square displacement (TASD),defined by Eq. (3), satisfies (53): (cid:10) δ (∆) (cid:11) ∼ K α Γ(1 + α ) ∆ t − α . [A5]As the dependence of the averaged TASD on ∆ is different from thedependence of the MSD on the measurement time t , the process issaid to display weak ergodicity breaking, as the time average differsfrom the ensemble average. Moreover, since the averaged TASDdepends on the total measurement time t , the process displaysageing (43).For a bounded random walker, F ( x ) = 0, one obtains an a-priorisurprising result for the averaged TASD. For Brownian motion andmost ergodic processes the TASD saturates upon interacting withthe confinement. Yet, for subdiffusive CTRW in a bounded domaingoverned by Eq. Eq. (A2), the averaged TASD does not saturateand satisfies (49): (cid:10) δ (∆) (cid:11) ∼ (cid:0)(cid:10) x (cid:11) B − h x i B (cid:1) πα )(1 − α ) απ (cid:16) ∆ t (cid:17) − α . [A6]Here h x n i B = Z − R ∞∞ x n exp ( − V ( x ) /k B T ) is the n th moment ofthe Boltzmann distribution, and Z = R ∞∞ exp ( − V ( x ) /k B T ) is itsnormalizing factor. Eq. Eq. (A6) is valid for 1 (cid:28) ∆ (cid:28) t , i.e. longtime lags that are much smaller than the total measurement time.This analysis indicates that, in contrast to the unbounded CTRWresult [Eq. Eq. (A6)] in which (cid:10) δ (∆) (cid:11) ∼ ∆ /t − α , for boundedCTRW we have (cid:10) δ (∆) (cid:11) ∼ (∆ /t ) − α , see Eq. (4). We stress thatthe result in Eq. Eq. (A6), which was verified both via simulationsand in experiments (see Ref. (43) and references therein) is universal,and in the leading order, does not depend on the confining potential.Indeed, the potential only enters in the prefactor, and via the firsttwo moments of the distribution (49). Importantly, in light of thisanalysis (which was also confirmed by our simulations), knowledge Fig. A1.
WTs distributions within a radius of m (a-c) and within a radius of m (d-f), for 14 adult owls (a,d), 6 adult kites (b,e) and 12 adult kestrels (c,f). In eachpanel, different shapes and colors represent different individuals. In (a-c) the dashedblack lines indicate a power law, Eq. (2) , with α = 0 . , . and . for (a),(b) and (c) respectively, and are plotted to guide the eye. The solid black line is theaverage fit for a truncated power law P ( τ ) ∼ τ − α e − τ/τ . For all birds the powerlaw truncates between τ = 20 and τ = 80 min. The average fit values, for the jointdistribution of all individuals within each species, were α = 0 . , . , . and τ = 40 , , min for the owls, kites, and kestrels respectively. In (d-f) the dashedblack line indicates a power law with α = 1 . and the dashed red line indicates apower law with α = 2 . , . and . , for the owls, kites and kestrels respectively.At short times the birds are almost stationary indicating motion within ARS (arearestricted search), while at long times the WTs display a different movement phase. Fig. A2. (a-c) Distribution of local jump length within
ARS for adult owls (a), adultkites (b) and adult kestrels (c). The dashed lines indicate a power law, φ ( δr ) ∼ δr β ,with β = 2 . , . and . for (a), (b) and (c), respectively, and represent the bestfit parameters. (d-f) Distribution of commuting flight distances between ARS for adultowls (d), adult kites (e) and adult kestrels (f). The dashed lines indicate an exponentialdistribution φ ( δr ) = e δr/λ /λ with λ = 437 , and m for (d), (e) and (f),respectively. of the exact form of the confining potential of individual birds is not crucial for any of our conclusions.Notably, in CTRW the jump length δr can be also taken asa random variable following distribution φ ( δr ). For instance,Pearson’s random walk (45) in discrete space is retrieved when ψ ( τ ) = τ − exp( − τ/τ ) and φ ( δr ) = δ δr, , where τ is the meanWT and δ i,j is the Kronecker delta. In our simulations [see Fig. 2],the jump lengths are such that at each simulation step the randomwalker can perform a step to any point within the predefined do-main walls with equal probability, but can not exit the walls. It wasverified by additional simulations that our conclusions, and in par-ticular the form given in Eq. (4), do not depend on the jump-lengthdistribution φ ( δr ). Waiting-time and jump-length distributions.
Here we give additionalresults supporting the discussion above. Specifically, we plot the WTdistribution for individual birds, and the jump-length distributionfor each species.In Fig. 4(a-c) we plotted the WT distribution within a radiusof 15 m for each species. Here, in Fig. A1(a-c), we plot the WTdistribution within the same radius for each individual. A powerlaw was fitted to the distribution of each individual, and to testthe quality of fit, we performed a likelihood ratio test to comparebetween a power-law and exponential fit for the distribution (57, 58).
Vilk et al.
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January 28, 2021 | vol. XXX | no. XX | R A F T Fig. A3. (a) Value of α as a function of R s , and (b) value of α as a function of τ s , foran individual owl. The error bars reflect a confidence interval around the mean. For all individuals, a power law was a better fit. A similar testbetween a power-law and a truncated power-law fits showed thatthe latter is a better fit.In Fig. 4(d-f) we plotted the distribution of time spent withinARS to show that a qualitative phase transition occurs betweenARS and larger spatial modes. Here, in Fig. A1(d-f), we give furthervalidation of the phase transition by plotting the WT distributionwithin a larger radius of R th = 100 m for each individual (note theunits of hours). The WTs here were calculated using a spatiotem-poral segmentation procedure (71) and we verified that the resultsare not sensitive to small changes in R th , between 70 −
200 m. Thedistributions in A1(d-f) strongly resemble the distribution of timespent within ARS, as the power-law fits (black and red dotted lines)are similar to those shown in Fig. 4(d-f). At short times ( < α = 1 . >
40 min), the data is bestfitted by a power law with α > within
ARS [A2(a-c)], and large-scale jump lengths of commutingflights between
ARS [A2(d-f)] for owls, kites and kestrels. Both thelocal jumps and the commuting flights are obtained directly from oursegmentation procedure, and the distributions are fitted using themethod of maximum likelihood. To evaluate the distributions foreach flight mode we preformed a likelihood comparison test betweena power-law and an exponential distribution for each group (a-f).We find that in (a-c) a power law is a better fit while in (d-f) anexponential distribution is a better fit. Notably, for all distributionsin Fig. A2 it is plausible that other fat-tailed distributions will givea better fit; nonetheless, we view the qualitative difference betweenthe distributions at small and large length scales as further evidenceof a phase transition between the intensive ARS flight mode andthe extensive commuting flight mode.
Segmentation procedure.
Here we provide evidence that our mainresults do not depend on the segmentation parameters and segmen-tation method.In the Penalized Contrast Method we use R s = 100 m and athreshold of τ s = 50 s to classify commuting flights from ARS. InFig. A3(a) we vary R s between 70–150 m to show that the valueof α converges at around R s = 100 m to a constant value. In Fig.A3(b) we vary τ s between 40–200 s to show that the value of α does not depend on this parameter. Similar sensitivity analysis wasperformed for all individuals. Notably, due to the high frequencyand resolution of the data the segments are clearly visible to theeye, see Fig. 1, and are not sensitive to any specific segmentationprocedure. To further show this, switching points were also detectedusing spatiotemporal criteria segmentation, such that localizationsthat are in proximity to one another, both in space and time weresegmented together (71). Using this segmentation procedure didnot significantly alter any of the results reported in our study. p-variation test. Here we employ the p-variation test (see below)to further validate our claim that the movement within ARS is asubdiffusive CTRW.
Fig. A4.
A p-variation test on a randomly chosen movement segment of a femaleowl. In (a) shown is the test for p = 2 , and V ( p ) n ( t ) displays a monotonic step-likeincrease. In (b) shown is the test for p = 2 /α for α = 0 . (this was the value foundfor this female), and as expected V ( p ) n ( t ) tends to zero as n is increased. Note that n can only be increased up to n = N , N being the number of data points in thetrajectory (72). A p-variation test is performed in order to distinguish the CTRWfrom other types of subdiffusive behaviors such as fractal Brownianmotion (43, 56, 72). Notably, this test was applied in Ref. (55) toevaluate the effect of noise in subdiffusive CTRW, and it was foundin simulations that up to some noise level the p-variation test isvalid. The test is defined in terms of the sum of increments of atrajectory x ( t ) on the time interval [0 , T ]: V ( p ) n ( t ) = n − X j =0 (cid:12)(cid:12)(cid:12) x (cid:16) min n ( j + 1) T n , n o(cid:17) − x (cid:16) min n jT n , n o(cid:17)(cid:12)(cid:12)(cid:12) p . [A7]For subdiffusive CTRW V ( p ) ( t ) = lim n →∞ V ( p ) n ( t ) displays thefollowing properties: for p = 2 it shows a monotonic, step-likeincrease in time, while for p = 2 /α : V (2 /α ) ( t ) = 0 (56, 72). In Fig.A4 we show an example of this test on a randomly chosen ARSsegment of a female barn owl, and the results fit both theoreticalpredictions. This indicates that the motion at hand is a subdiffusiveCTRW. Notably, we have repeated the test on many randomlychosen trajectories of various individuals, and all gave similar results.
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