Ergodicity Breaking in Area-Restricted Search of Avian Predators

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Ergodicity Breaking in Area-Restricted Search of Avian Predators

Ohad Vilk, Yotam Orchan, Motti Charter, Nadav Ganot, Sivan Toledo, Ran Nathan, and Michael Assaf

Phys. Rev. X 12, 031005 – Published 8 July 2022

See Focus story: Birds of a Feather Hunt Differently

Abstract

Quantifying and comparing patterns of dynamical ecological systems requires averaging over measurable quantities. For example, to infer variation in movement and behavior, metrics such as step length and velocity are averaged over large ensembles. Yet, in nonergodic systems, such averaging is inconsistent; thus, identifying ergodicity breaking is essential in ecology. Using rich, high-resolution, movement data sets (greater than $7 \times 10^{7}$ localizations) from 70 individuals and continuous-time random walk modeling, we find subdiffusive behavior and ergodicity breaking in the localized movement of three species of avian predators. Small-scale, within-patch movement was found to be qualitatively different, not inferrable and separated from large-scale interpatch movement. Local search is characterized by long, power-law-distributed waiting times with a diverging mean, giving rise to ergodicity breaking in the form of considerable variability uniquely observed at this scale. This implies that wild animal movement is scale specific, with no typical waiting time at the local scale.

Received 15 November 2021
Revised 13 February 2022
Accepted 20 May 2022

DOI:https://doi.org/10.1103/PhysRevX.12.031005

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anomalous diffusion Biological movement Continuous-time random walk Organismal, population, evolutionary & ecological systems Phase transitions

Data analysis Ergodic theory First passage problems

Statistical Physics & ThermodynamicsPhysics of Living SystemsNonlinear Dynamics

Focus

Birds of a Feather Hunt Differently

Published 8 July 2022

Tracking of bird movements shows that the animals don’t spread outward like molecules in a gas, as ecologists often assume.

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Authors & Affiliations

Ohad Vilk ^1,2,3,*, Yotam Orchan^2,3, Motti Charter^2,3,4, Nadav Ganot^2,3, Sivan Toledo^3,5, Ran Nathan ^2,3,†, and Michael Assaf ^1,‡

¹Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
²Movement Ecology Lab, Department of Ecology, Evolution and Behavior, Alexander Silberman Institute of Life Sciences, Faculty of Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
³Minerva Center for Movement Ecology, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
⁴The Shamir Research Institute and Department of Geography and Environmental Studies, University of Haifa, 199 Aba Hushi Boulevard, Mount Carmel, Haifa 3498838, Israel
⁵Blavatnik School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel

^*ohad.vilk@mail.huji.ac.il
^†ran.nathan@mail.huji.ac.il
^‡michael.assaf@mail.huji.ac.il

Popular Summary

Movements of animals typically vary across time and space: Real-life landscapes are typically heterogeneous, and animals commonly alternate between relatively long movements between resource-rich patches, and shorter searches for prey within patches. Understanding the drivers and mechanisms of these movement patterns is vital for resource management and conservation but requires detailed information on animal movement at a high spatiotemporal resolution. Here, using rich datasets of movements of three avian predators obtained from a novel reverse-GPS tracking system, we discover unique movement characteristics at previously unmeasured scales.

We employ the notion of ergodicity breaking in stochastic dynamical systems to reveal that local searches are irreproducible (i.e., nonergodic) while long-range commuting is reproducible (i.e., ergodic). The nonergodic nature of local searches implies, for example, that the average velocity across many different searches likely differs from the average velocity in one long search. Such discrepancies should be carefully considered in modeling of ecological systems.

Overall, our study reveals that there are many ways to hunt within a patch but only a limited number of ways to commute between distant patches. Using a continuous-time random-walk model, we identify a behavioral switch between local search and movement between patches. We also show that birds remain stationary during local searches for highly variable durations including exceedingly long times, suggesting multiple hunting tactics and behavioral modes.

We argue that hunting efficiency can increase via highly variable, irreproducible foraging tactics compared to less variable, reproducible ones. Such variability may hold an evolutionary advantage through the higher individual fitness of avian and possibly other predators combining different foraging behaviors. Finally, our results highlight the importance of considering ergodicity in movement-ecology research and other scientific disciplines dealing with stochastic dynamical systems.

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Vol. 12, Iss. 3 — July - September 2022

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Figure 1
Foraging tracks are composed of superdiffusive commuting flights and subdiffusive local search. (a) Seventy-five minutes of tracking at 0.5 Hz (2280 localizations) of a randomly selected female owl (tag 4782), segmented into four ARS (red) of 4, 2, 4, and 56 min long, and three commuting segments (blue), each lasting approximately 1 min. The arrows show the direction of motion. The inset shows local search within a single ARS (size $200 \times 200 m^{2}$ , 56 min long), with five local clusters (WTs of 14, 36, 2, 3, and 1 min), each represented by a different color and highlighted by a red circle. The owl performs local jumps of 25 to 45 m between the local clusters.
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Figure 2
(a,b) Ensemble-averaged (red circles) and time-averaged (blue triangles) velocities of empirical movement tracks of a kite during commuting (a) and ARS (b). The dashed line in (a) denotes the long-time limit of the ensemble-averaged and time-averaged velocities. (c) Empirical averaged TASD [Eq. (3)] of ARS segments (red triangles) and commuting segments (blue dots). The dashed (blue) and dash-dotted (red) lines are power laws with $Δ^{1.80}$ and $Δ^{0.35}$ , respectively. The inset shows the probability density of TASD normalized by its average at time lag $Δ = 0.33 ≪ t$ (see text), for both cases.
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Figure 3
CTRW simulations. (a,b) Exponentially distributed WTs (a simple RW with $τ_{0} = 10 s$ ). (c,d) Power-law WTs, Eq. (2), with $α = 0.6$ . In panels (a) and (c), each of the red lines is the TASD calculated for a single simulation versus the time lag $Δ$ , and the blue line is the average TASD, Eq. (3), over an ensemble of 200 simulations. The blue dashed lines are the theoretical scaling. In panels (b) and (d), we plot the averaged TASD, Eq. (3), versus the total time of the simulation. Simulations were done in a bounded domain of $100 \times 100$ .
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Figure 4
(a) TASD (red solid lines) and averaged TASD (blue solid line) of 40-min ARSs of an owl, over time lag $Δ$ . The dashed line scales as $Δ^{0.4}$ . (b) Averaged TASD over time $t$ for $Δ = 10 s$ (blue), 40 s (orange), and 120 s (green), yielding powers of $- 0.26, - 0.27$ , and $- 0.28$ , respectively. (c) Probability density of the normalized TASD, $ϕ (ξ)$ [see Eq. (4)], during ARS for the same individual for $Δ = 100$ , 200, 400, and 700 s. The dashed line is the Mittag-Leffler distribution [Eq. (4)] for $α = 0.6$ . The inset shows the same plot on a linear scale. (d) Autocorrelation function (blue dots) averaged over 100 ARS segments of $2000 s$ . The orange dashed and green dash-dotted lines are averages over 1000 simulations of subdiffusive CTRW (with $α = 0.6$ ) and simple RW, respectively.
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Figure 5
WT probability densities (symbols) within a radius of 15 m, for 14 owls (a), 6 kites (b), and 10 kestrels (c), all adults. The solid line is a fit to a truncated power law, $P (τ) \sim τ^{- 1 - α} e^{- τ / τ_{0}}$ , with $α = 0.68$ , 0.51, 0.55, and $τ_{0} = 40$ , 35, 36 in panels (a), (b), and (c), respectively. These values of $α$ are the average fit values for the joint distribution of all birds within each species, and the power law truncates at $τ_{0} = 20 - 80 \min$ for all 70 individual birds. The dashed lines are power laws [Eq. (2)] with the same $α$ values.
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Figure 6
Distributions of stop durations $Δ T$ (symbols), defined as the total time spent within single ARSs (see text), for owls (a), kites (b), and kestrels (c), all adults. The dashed black and red lines are power laws, presenting a behavioral switch from $Δ T^{1.15}$ for shorter $Δ T$ for all three species to $Δ T^{2.97}, Δ T^{2.20}$ , and $Δ T^{2.61}$ for longer $Δ T$ , with typical transition times (when the two slopes intersect) of $Δ T = 45$ , 48, and 36 min in panels (a), (b), and (c), respectively.
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