Universal Phenomenology, Divergent Mechanism: A Behavioural Gradient of Mobility from Foraging to Commute

📄 Manuscript Status

This single-author manuscript has been submitted to Nature as an Article and is currently under editorial consideration. While the paper is in peer review, the manuscript text, the cover letter and the preregistration are not redistributed here; this repository hosts only the figures and the figure-level numerical data (CSV) needed to inspect, verify or re-plot the published results. The full text and supporting documents will be linked here upon acceptance. In the meantime, the corresponding author can supply them on reasonable request.

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👤 Author

Author	Affiliation	Contact
Prof. David L. Azevedo	Instituto de Física, Universidade de Brasília (UnB), Campus Darcy Ribeiro, 70910-900 Brasília-DF, Brazil	📧 david888azv@unb.br

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📋 Abstract (short form)

Three influential scaling laws of human mobility — the truncated power law of step lengths and the bounded radius of gyration [1], the universal visitation law ρ ∝ (rf)⁻² [2], and the hierarchical-container architecture [3] — were established on datasets of 10⁵–10⁷ humans and have since been treated as a human-specific phenomenology. Animal-movement ecology has developed in parallel around a different paradigm based on Lévy-flight foraging [4–6], optimal-foraging theory [7] and home-range allometry. Here I apply a single estimator battery to GPS records of eleven vertebrate species spanning a behavioural-energetic gradient — nomadic ungulates (Mongolian gazelle, Burchell's zebra), home-ranging and central-place mammals (African elephant, olive baboon), seabirds and migrants (Northern gannet, Galápagos albatross, white stork), bats (Eidolon helvum, two studies), and marine reptiles (loggerhead turtle, Mediterranean and North-Pacific populations) — and compare the results with the published human values.

The phenomenological architecture is universal across all eleven species; the underlying microscopic mechanism is not. The Song preferential-return relations [8] hold closely only when the visitation Zipf exponent ζ approaches unity, defining a continuous gradient: from low-CP nomadism (ζ ≈ 0.3 for nomadic gazelles, oceanic turtles, and elephants), through migratory and seabird regimes (ζ ≈ 0.7–0.9), to strict central-place colonial life (ζ ≈ 1.6 for olive baboons and Galápagos albatrosses), bracketing the human value ζ ≈ 1.0 from both sides. The Schläpfer η = 2 [2] is a regime rather than a universal: all eleven non-human species sit at η ∈ [0.35, 1.13] (linear-cost regime), whereas humans on motorised infrastructure return η ≈ 2 (kinetic-cost regime). The Mongolian gazelle, a fully nomadic ungulate, exhibits the heaviest tail among non-human animals (α = 2.29 ± 0.11) — the closest non-human approach to the human Lévy reference α = 1.75 ± 0.15. Restricting the stork record to the obligate-CP breeding window yields α_breed = 1.70 ± 0.05, statistically indistinguishable from the human value. Humans are therefore not a separate class of mobile organism but the extremum of a continuous behavioural-energetic axis.

Key findings

• ✅ Phenomenology is universal across 11 species. Truncated power-law step lengths, logarithmic saturation of ⟨r_g(t)⟩, four to five hierarchically nested lognormal containers and circadian peaks of return are observed in every species, from nomadic gazelles to colonial baboons.
• 📈 A continuous gradient in central-place strength. ζ spans an unbroken interval ζ ∈ [0.26, 1.64] across the eleven species, bracketing the human value ζ ≈ 1.0 from both sides.
• 🐺 Mongolian gazelle is the heaviest-tailed non-human. α = 2.29 ± 0.11 is the closest non-human approach to the human Lévy reference α = 1.75 ± 0.15.
• 🚗 Transport-regime split. All eleven animal species sit at η ∈ [0.35, 1.13] (linear-cost regime); only humans on motorised infrastructure return η ≈ 2 (kinetic-cost regime). The split is governed by velocity heterogeneity, not phylogeny.
• 🐣 Falsifiable test passes. Restricting the stork record to the obligate-CP breeding window yields α = 1.70 ± 0.05, coinciding with the published human value 1.75 ± 0.15.
• 📐 Two-parameter unification. A minimal generative model with central-place strength γ_CP and velocity heterogeneity η_v places all twelve points (eleven species + human reference) on a continuous (γ_CP, η_v) diagonal.

What this reshapes

The result reconciles Christaller's central-place theory with Charnov's marginal-value theorem under a single conserved-effort principle, offers a testable prediction for how "15-minute city" planning should shift urban η towards 1, and reinterprets the long-running Lévy-flight controversy as an artefact of fitting a single power law to a hierarchical-container mixture.

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🗂️ Repository structure

universal-mobility-gradient/
├── pipeline/                    # Python analysis pipeline (16 scripts, A/B/C/D blocks)
├── requirements.txt             # numpy, pandas, scipy, matplotlib, requests
├── data_raw/                    # CC0 raw GPS records — see README for download steps
├── figures/
│   ├── main/                    # Figures 1–4 of the main text (PDF + PNG, 300 dpi)
│   ├── extended_data/           # Extended Data Figures 1–4
│   └── supplementary/           # Per-species intermediate-block outputs
├── data/                        # Figure-level CSV exports (tidy)
│   ├── elephant/                #   – one CSV per figure, named fig<N>_<quantity>_<species>.csv
│   ├── gannet/
│   ├── stork/
│   ├── albatross/               #   ── new species (2026 sweep) ──
│   ├── bat_scharf/
│   ├── bat_abedi/
│   ├── turtle_med/
│   ├── turtle_pac/
│   ├── zebra/
│   ├── baboon/
│   ├── gazelle/
│   └── cross_species/           # master table + unified-model sweep
├── assets/logos/                # Institutional logos used in acknowledgements
├── LICENSE                      # CC BY 4.0
└── README.md                    # You are here

📝 Note. The full manuscript PDF, the cover letter, the preregistration and the BibTeX file are intentionally not included while the paper is in peer review. They will be added (or linked to the published Nature DOI) once the editorial process is complete.

Data folder — naming convention

Every CSV lives in data/<species>/ and follows the pattern fig<figure_number>_<quantity>_<species>.csv, so a reviewer can locate the numerical content of any panel from its filename alone. For example:

File	Content
data/gazelle/fig1_step_length_gazelle.csv	Log-binned P(Δr) for 36 Mongolian gazelles
data/baboon/fig2_zipf_rank_frequency_baboon.csv	Top-50 pooled rank–frequency (Mpala troop)
data/albatross/fig3_bootstrap_exponents_albatross.csv	α, ζ, μ, γ, β, η with bootstrap SE and 95% CI
data/stork/fig4a_seasonal_alpha_stork.csv	Stork tail exponent per seasonal phase (Fig. 4a)
data/cross_species/master_table_all_exponents.csv	Master table — all 11 species + human reference
data/cross_species/fig4bc_unified_model_sweep.csv	9 × 7 (γ_CP, η_v) sweep producing Fig. 4b,c & Ext. Fig. 4
data/<species>/extdata1_containers_<species>.csv	Container effective sizes per grid level (Ext. Fig. 1)

Figures folder

• figures/main/ — publication versions of Figures 1–4 (vector PDF and 300 dpi PNG on Nature's double-column width of 183 mm). All four main figures now incorporate the eleven-species sweep.
• figures/extended_data/ — four Extended Data figures:
  • Ext. Fig. 1 — Alessandretti hierarchical containers (three benchmark species).
  • Ext. Fig. 2 — Per-species P(Δr) overlay, raw and rescaled by the median step length.
  • Ext. Fig. 3 — Forest plot of all six exponents across the eleven species + human.
  • Ext. Fig. 4 — Unified-model parameter-sweep heatmaps with all twelve points overlaid.
• figures/supplementary/<species>/ — intermediate results from each of the four estimator blocks (A — González; B — Song; C — Alessandretti; D — Schläpfer).

Manuscript and supporting documents

The submitted manuscript, the cover letter, the OSF-style preregistration (four falsifiable hypotheses H1–H4 with explicit refutation criteria) and the BibTeX database are withheld from public release while peer review is in progress at Nature. They are available on reasonable request from the corresponding author and will be linked from this repository once the manuscript is accepted or made publicly available as a preprint.

Scripts (pipeline/)

The full analysis pipeline (Python 3.13 — NumPy 2.4, pandas 3.0, SciPy 1.17, Matplotlib 3.10) is included in this repository under pipeline/. It is structured as four modular blocks plus auxiliary modules:

• Block A (blocoA.py, González 2008): P(Δr) tail fitting, r_g distribution, return probability, Zipf.
• Block B (blocoB.py, Song 2010): S(t), P_new(S), waiting times, three scaling relations.
• Block C (blocoC.py, Alessandretti 2020): multi-scale container grid + lognormal KS test.
• Block D (blocoD.py, Schläpfer 2021): ρ(r,f) visitation regression on 4-month windows.
• Bootstrap (bootstrap.py): block-bootstrap by individual; deterministic seed numpy.random.default_rng(42).
• Stork seasonal split (stork_seasonal.py): yields the breeding-window α = 1.70 of Fig. 4a.
• Unified-model simulator (unified_model.py): two-parameter sweep producing Fig. 4b,c and Ext. Fig. 4.
• Multi-species driver (run_all.py): loops every block over the eleven species automatically.
• Helpers (_helpers.py, ingest_new_species.py, download_movebank.py, cross_species_table.py, nature_figures.py, export_csv.py).

Install dependencies with pip install -r requirements.txt. The eleven raw datasets are not redistributed here (they are CC0 at Movebank/Dryad); see data_raw/README.md for one-shot download instructions using the bundled download_movebank.py helper. The bootstrap is deterministic, so any reviewer can reproduce the exponents reported in the manuscript bit for bit. A Code Ocean capsule of this same pipeline will be made available to editors and peer reviewers during evaluation and will receive a public DOI on acceptance.

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📊 Master summary

Mean ± standard error from bootstrap by individual (N_boot = 500 for every species except white stork at N_boot = 50). Full confidence intervals are listed in data/cross_species/master_table_all_exponents.csv.

Species (n indiv)	α	ζ	μ	γ	β	η
Mongolian gazelle (n=36)	2.29 ± 0.11	0.38 ± 0.06	0.92 ± 0.02	−0.11 ± 0.04	1.56 ± 1.19	0.53 ± 0.09
Loggerhead turtle, N. Pacific (n=12)	2.72 ± 0.20	0.29 ± 0.06	0.94 ± 0.05	−0.01 ± 0.04	2.33 ± 0.42	1.13 ± 0.34
Northern gannet (n=25)	2.83 ± 0.32	0.89 ± 0.05	0.93 ± 0.04	−0.13 ± 0.05	1.20 ± 0.89	0.65 ± 0.07
Eidolon helvum, Burkina (n=63)	3.24 ± 0.38	0.71 ± 0.07	0.76 ± 0.08	−0.42 ± 0.07	0.71 ± 0.03	0.77 ± 0.08
African elephant (n=14)	3.40 ± 0.18	0.28 ± 0.07	0.82 ± 0.03	0.09 ± 0.03	1.89 ± 0.70	1.03 ± 0.04
White stork (n=92)	3.48 ± 0.12	0.86 ± 0.04	1.98 ± 0.11	−0.56 ± 0.02	1.06 ± 0.60	1.09 ± 0.05
↳ stork, breeding subset only	1.70 ± 0.05	—	—	—	—	—
Loggerhead turtle, Mediterranean (n=11)	3.56 ± 0.27	0.26 ± 0.06	0.91 ± 0.04	0.02 ± 0.05	—	0.44 ± 0.20
Galápagos albatross (n=28)	3.90 ± 0.36	1.64 ± 0.09	1.35 ± 0.15	−0.15 ± 0.13	1.14 ± 0.23	0.96 ± 0.12
Eidolon helvum, Ghana (n=28)	4.29 ± 1.11	0.97 ± 0.10	0.84 ± 0.15	−0.57 ± 0.16	0.78 ± 0.05	0.67 ± 0.09
Olive baboon, Mpala (n=26)	4.35 ± 0.08	1.58 ± 0.02	0.50 ± 0.03	—	3.15 ± 4.69	0.35 ± 0.09
Burchell's zebra (n=7)	5.74 ± 1.59	0.56 ± 0.10	0.97 ± 0.06	−0.09 ± 0.08	3.37 ± 0.65	0.75 ± 0.11
Human (literature)	1.75 ± 0.15 [1]	≈ 1.0 [1]	0.60 [8]	0.21 [8]	0.80 [8]	2.05 ± 0.02 [2]

Bold entries mark the values that lie closest to the published human reference: the Mongolian gazelle has the heaviest non-human tail (closest to the human α), and the stork's breeding-window subset coincides with the human α to within 0.05.

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🔗 Primary data sources

All eleven GPS tracking datasets are hosted in public repositories under permissive Creative Commons licences (CC0 1.0 in every case). I gratefully acknowledge the original data owners for making the records publicly available:

Species (common name)	Location and period	Repository / DOI	Depositors
Loxodonta africana (African elephant)	Kruger NP, South Africa, 2007–2009	10.5441/001/1.403h24q5	R. Slotow, M. Thaker & A. T. Vanak
Morus bassanus (Northern gannet)	Cape St Mary's, Newfoundland, 2019–2022	Movebank handle 10255/move.1592	K. J. N. d'Entremont, G. K. Davoren & W. A. Montevecchi
Ciconia ciconia (white stork)	SW Germany breeding grounds, 2013–2023	Movebank handle 10255/move.1685	Y. Cheng, W. Fiedler, M. Wikelski & A. Flack
Phoebastria irrorata (Galápagos albatross)	Española Island, Galápagos, 2008	10.5441/001/1.3hp3s250	F. H. Cruz, P. D. Proaño, D. J. Anderson, K. P. Huyvaert, M. Wikelski
Eidolon helvum (straw-coloured fruit bat) — Scharf et al. 2019	Burkina Faso / Ghana, 2009–2014	10.5441/001/1.k8n02jn8	A. Scharf, J. Fahr, M. Abedi-Lartey, K. Safi, D. K. N. Dechmann, M. Wikelski, M. T. O'Mara
Eidolon helvum (straw-coloured fruit bat) — Abedi-Lartey 2016	Ghana, 2016 (seed-dispersal study)	10.5441/001/1.44183438	M. Abedi-Lartey, D. K. N. Dechmann, M. Wikelski, A. Scharf, J. Fahr
Caretta caretta (loggerhead sea turtle) — Mediterranean	Western Mediterranean Sea, 2017+	10.5441/001/1.1f1h87r8	S. Hochscheid
Caretta caretta (loggerhead sea turtle) — N. Pacific	Japan / North Pacific, 2016+	10.5441/001/1.m3c90703	J. Okuyama et al.
Equus quagga burchellii (Burchell's zebra)	Okavango / Makgadikgadi, Botswana, 2007–2009	10.5441/001/1.f3550b4f	H. L. A. Bartlam-Brooks & S. Harris
Papio anubis (olive baboon)	Mpala Research Centre, Kenya, August 2012	10.5441/001/1.kn0816jn	M. C. Crofoot, R. Kays, M. Wikelski (Strandburg-Peshkin et al. 2015)
Procapra gutturosa (Mongolian gazelle)	Eastern Mongolian Steppe, 2007–2010	Dryad 10.5061/dryad.45157	C. H. Fleming, J. M. Calabrese, T. Mueller, K. A. Olson, P. Leimgruber, W. F. Fagan

Human reference values are taken directly from refs. [1, 2, 3, 8]; no proprietary human data were analysed.

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🔑 Keywords

human mobility · animal movement ecology · scaling laws · Lévy flight · central-place foraging · Zipf law · visitation law · hierarchical containers · preferential return · Movebank · Dryad · cross-species synthesis · statistical physics

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🙏 Acknowledgements

       

D. L. Azevedo acknowledges financial and infrastructural support from the Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF) under calls 04/2017 and 09/2022, and from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) through research-productivity fellowship 306456/2025-7 (PQ-C). Computational resources were provided by the Centro Nacional de Processamento de Alto Desempenho em São Paulo (CENAPAD-SP).

This work would not have been possible without the eleven public datasets that the original investigators chose to release under CC0 1.0; the author thanks them and the curators of the Movebank Data Repository (Max Planck Institute of Animal Behavior) and the Dryad Digital Repository for sustaining the long-term infrastructure that makes cross-species synthesis tractable. Specific dataset depositors are listed in the table above.

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📖 Citation

If you use this repository in your research, please cite both the manuscript (currently under review) and the underlying public datasets.

@article{Azevedo2026Mobility,
  author  = {Azevedo, David L.},
  title   = {Universal phenomenology, divergent mechanism:
             a behavioural gradient of mobility from foraging to commute},
  journal = {Nature},
  year    = {2026},
  note    = {Submitted}
}

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📜 License

The analysis outputs and the manuscript are released under the Creative Commons Attribution 4.0 International License. The underlying GPS tracking data remain under their original CC0 1.0 Universal terms as deposited on Movebank and Dryad.

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📬 Contact

For questions, pipeline requests, or collaboration enquiries:

Prof. David L. Azevedo Instituto de Física, Universidade de Brasília (UnB) 📧 david888azv@unb.br

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🔖 Key references

1. González, Hidalgo & Barabási. Nature 453, 779 (2008).
2. Schläpfer et al. Nature 593, 522 (2021).
3. Alessandretti, Aslak & Lehmann. Nature 587, 402 (2020).
4. Viswanathan et al. Nature 381, 413 (1996).
5. Sims et al. Nature 451, 1098 (2008).
6. Humphries et al. Nature 465, 1066 (2010).
7. Charnov. Theor. Popul. Biol. 9, 129 (1976).
8. Song, Koren, Wang & Barabási. Nat. Phys. 6, 818 (2010).
9. Cheng, Fiedler, Wikelski & Flack. Ecol. Evol. 9, 8945 (2019).

The full bibliography is included in the submitted manuscript and will be linked here upon publication.