Skip to main content
SpringerLink
Log in

Weierstrassian Lévy walks are a by-product of crawling

  • Regular Article - Living Systems
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract

Weierstrassian Lévy walks are one of the simplest random walks which do not satisfy the central limit theorem and have come to epitomize scale invariance even though they were initially regarded as being a mathematical abstraction. Here, I show how these Lévy walks can be generated intrinsically as a by-product of crawling, a common but ancient form of locomotion. This may explain why Weierstrassian Lévy walks provide accurate representations of the movement patterns of a diverse group of molluscs—certain mussels, mud snails and limpets. I show that such movements are not specific to molluscs as they are also evident in Drosophila larvae. The findings add to the growing realization that there are many idiosyncratic, seemingly accidental pathways to Lévy walking. And that the occurrence of Lévy walks need not be attributed to the execution of an advantageous searching strategy.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: The data that support the findings of this study were obtained by Sims et al. [15].The data are freely available at https://elifesciences.org/articles/50316.]

References

  1. A.M. Reynolds, Liberating Lévy walk research from the shackles of optimal foraging. Phys. Life Rev. 14, 59–83 (2015)

    Article  ADS  Google Scholar 

  2. A.M. Reynolds, Current status and future directions of Lévy walk research. Biol. Open 7, bio030106 (2017)

    Article  Google Scholar 

  3. A.M. Reynolds, Mussels realize Weierstrassian Lévy walks as composite correlated random walks. Sci. Rep. 4, 4409 (2014)

    Article  ADS  Google Scholar 

  4. A.M. Reynolds, F. Bartumeus, A. Kölzsch, J. van de Koppel, Signatures of chaos in animal search patterns. Sci. Rep. 6, 23492 (2016)

    Article  ADS  Google Scholar 

  5. A.M. Reynolds, G. Santini, G. Chelazzi, S. Focardi, The Weierstrassian movement patterns of snails. Roy. Soc. Open Sci. 4, 160941 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  6. G. Ariel, A. Be’er, A.M. Reynolds, Chaotic model of Lévy walks in swarming bacteria. Phys. Rev. Lett. 118, 228102 (2017)

    Article  ADS  Google Scholar 

  7. J. Klafter, M.F. Schlesinger, G. Zumofen, Beyond Brownian motion. Phys. Today 49, 33–39 (1996)

    Article  Google Scholar 

  8. B.D. Hughes, M.F. Schlesinger, E.W. Montroll, Random walks with self-similar clusters. Proc. Natl. Acad. Sci. USA 78, 3287–3291 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  9. G.M. Zaslavsky, D. Stevens, H. Weitzner, Self-similar transport in incomplete chaos. Phys. Rev. E 48, 1683–1694 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  10. G.M. Zaslavsky, From Hamiltonian chaos to Maxwell’s Demon. Chaos 5, 653–661 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  11. J. Loveless, K. Lagogiannis, B. Webb, Modelling the mechanics of exploration in larval Drosophila. PLOS Comp. Biol. 5, e1006635 (2019)

    Article  Google Scholar 

  12. G.M. Viswanathan, E.P. Raposo, M.G.E. da Luz, Lévy flights and super diffusion in the context of biological encounters and random searches. Phys. Life Rev. 5, 133–150 (2008)

    Article  ADS  Google Scholar 

  13. N. Levernier, J. Textor, O. Bénichou, R. Voituriez, Inverse square Lévy walks are not optimal search strategies for \(d\geqslant 2\). Phys. Rev. Lett. 124, 080601 (2020). See Reynolds A.M. Extending Lévy search theory from one to higher dimensions: Lévy walking favours the blind. Proc. Roy. Soc. A 471, 20150123 (2015)

  14. J.H. Lai, J.C. del Alamo, J. Rodriguez-Rodriguez, J.C. Lasheras, The mechanics of the adhesive locomotion of terrestrial gastropods. J. Exp. Biol. 213, 3920–3933 (2010)

    Article  Google Scholar 

  15. D.W. Sims, N.E. Humphries, N. Hu, V. Meda, J. Berni, Optimal searching behaviour generated intrinsically by the central pattern generator for locomotion. eLife 8, e50316 (2019)

  16. N.E. Humphries, H. Weimerskirch, D.W. Sims, A new approach for objective identification of turns and steps in organism movement data relevant to random walk modelling. Meth. Ecol. Evol. 4, 930–938 (2013)

    Google Scholar 

  17. A. Clauset, C.R. Shalizi, M.E.J. Newman, Power-law distributions in empirical data. SIAM Rev. 51, 661–703 (2009)

  18. M.S. Abe, Functional advantages of Lévy walks emerging near a critical point. Proc. Nat. Acad. Sci. 117, 24336–24344 (2020)

    Article  ADS  Google Scholar 

  19. H. Hayashi, S. Ishizuka, M. Ohta, K. Hirakawa, Chaotic behavior in the onchidium giant neuron under sinusoidal stimulation. Phys. Lett. A. 88, 435–438 (1982)

  20. A.O. Komendantov, N.I. Kononenko, Deterministic chaos in mathematical model of pacemaker activity in bursting neurons of snail. Helix Pomatia. J. Theor. Biol. 183, 219–230 (1996)

    Article  ADS  Google Scholar 

  21. A.M. Reynolds, Passive particles Lévy walk through turbulence mirroring the diving patterns of marine predators. J. Phys. Comm. 2, 085003 (2018)

    Article  ADS  Google Scholar 

  22. A.M. Reynolds, J.G. Cecere, V.H. Paiva, J.A. Ramos, S. Focardi, Pelagic seabird flight patterns are consistent with a reliance on olfactory maps for oceanic navigation. Roy. Soc. Proc. B 282, 20150468 (2015)

    Article  Google Scholar 

  23. M.L. Rabinovich, H.D.I. Abarbanel, The role of chaos in neural systems. Neurosci. 87, 5–14 (1998)

    Article  Google Scholar 

  24. S. Kuroda, I. Kunita, Y. Tanaka, A. Ishiguro, R. Kobayashi, T. Nakagaki, Common mechanics of mode switching in locomotion of limbless and legged animals. J. Roy. Soc. Int. 11, 20140205 (2014)

    Article  Google Scholar 

  25. M.F. Shlesinger, Weierstrassian Lévy flights and self-avoiding random walks. J. Chem. Phys. 78, 416–420 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  26. D.W. Sims, A.M. Reynolds, V.J. Wearmouth, N.E. Humphries, E.J. Southall, B. Metcalfe et al., Hierarchical random walks in trace fossils and the origin of optimal search behaviour. Proc. Nat. Acad. of Sci. USA 111, 11073–11078 (2014)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The work at Rothamsted forms part of the Smart Crop Protection (SCP) strategic programme (BBS/OS/CP/000001) funded through Biotechnology and Biological Sciences Research Council’s Industrial Strategy Challenge Fund.

Author information

Authors and Affiliations

  1. Rothamsted Research, Harpenden, AL5 2JQ, UK

    Andy M. Reynolds

Authors
  1. Andy M. Reynolds
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.M.R. undertook the research and wrote the paper.

Corresponding author

Correspondence to Andy M. Reynolds.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reynolds, A.M. Weierstrassian Lévy walks are a by-product of crawling. Eur. Phys. J. E 44, 96 (2021). https://doi.org/10.1140/epje/s10189-021-00100-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/s10189-021-00100-2

Search

Navigation