Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer

Abstract

The study of animal foraging behaviour is of practical ecological importance1, and exemplifies the wider scientific problem of optimizing search strategies2. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails3,4, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical3,4,5 and chemical6 systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface7. This well known finding2,4,8,9 was followed by similar inferences about the search strategies of deer10 and bumblebees10. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer13, microzooplankton14, grey seals15, spider monkeys16 and fishing boats17. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data7 using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood18 and Akaike weights19,20. We apply this to the four original deer and bumblebee data sets10, finding that none exhibits evidence of Lévy flights, and that the original graphical approach10 is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms13,14,15,16,17, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool17. Our results question the strength of the empirical evidence for biological Lévy flights.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Rank/frequency plot 23 of 2004 wandering albatross data, showing no evidence for Lévy flight behaviour.
Figure 2: Data for the six wandering albatross trips in 1992 that have known departure and return times.
Figure 3: When corrected, the 1992 wandering albatross flight durations no longer follow a power law.
Figure 4: Foraging times of deer previously concluded to demonstrate Lévy flight behaviour10.

References

  1. Turchin, P. Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants (Sinauer, Sunderland, Massachusetts, 1998)

    Google Scholar 

  2. Shlesinger, M. F. Search research. Nature 443, 281–282 (2006)

    Article  ADS  CAS  Google Scholar 

  3. Shlesinger, M. F., Zaslavsky, G. M. & Frisch, U. Lévy Flights and Related Topics in Physics (Springer, Berlin, 1995)

    Book  Google Scholar 

  4. ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, Cambridge, 2000)

    Book  Google Scholar 

  5. Bardou, F., Bouchaud, J.-P., Aspect, A. & Cohen-Tannoudji, C. Lévy Statistics and Laser Cooling: How Rare Events Bring Atoms to Rest (Cambridge Univ. Press, Cambridge, 2002)

    MATH  Google Scholar 

  6. Ott, A., Bouchaud, J. P., Langevin, D. & Urbach, W. Anomalous diffusion in “living polymers”: a genuine Lévy flight? Phys. Rev. Lett. 65, 2201–2204 (1990)

    Article  ADS  CAS  Google Scholar 

  7. Viswanathan, G. M. et al. Lévy flight search patterns of wandering albatrosses. Nature 381, 413–415 (1996)

    Article  ADS  CAS  Google Scholar 

  8. Metzler, R. & Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–R208 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  9. Klafter, J. & Sokolov, I. M. Anomalous diffusion spreads its wings. Phys. World 18, 29–32 (2005)

    Article  CAS  Google Scholar 

  10. Viswanathan, G. M. et al. Optimizing the success of random searches. Nature 401, 911–914 (1999)

    Article  ADS  CAS  Google Scholar 

  11. Reynolds, A. M. Scale-free movement patterns arising from olfactory-driven foraging. Phys. Rev. E 72, 041928 (2005)

    Article  ADS  CAS  Google Scholar 

  12. Bénichou, O., Loverdo, C., Moreau, M. & Voituriez, R. Two-dimensional intermittent search processes: An alternative to Lévy flight strategies. Phys. Rev. E 74, 020102(R) (2006)

    Article  ADS  Google Scholar 

  13. Mårell, A., Ball, J. P. & Hofgaard, A. Foraging and movement paths of female reindeer: insights from fractal analysis, correlated random walks, and Lévy flights. Can. J. Zool. 80, 854–865 (2002)

    Article  Google Scholar 

  14. Bartumeus, F., Peters, F., Pueyo, S., Marrasé, C. & Catalan, J. Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton. Proc. Natl Acad. Sci. USA 100, 12771–12775 (2003)

    Article  ADS  CAS  Google Scholar 

  15. Austin, D., Bowen, W. D. & McMillan, J. I. Intraspecific variation in movement patterns: modeling individual behaviour in a large marine predator. Oikos 105, 15–30 (2004)

    Article  Google Scholar 

  16. Ramos-Fernández, G. et al. Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi). Behav. Ecol. Sociobiol. 55, 223–230 (2004)

    Article  Google Scholar 

  17. Bertrand, S., Burgos, J. M., Gerlotto, F. & Atiquipa, J. Lévy trajectories of Peruvian purse-seiners as an indicator of the spatial distribution of anchovy (Engraulis ringens). ICES J. Mar. Sci. 62, 477–482 (2005)

    Article  Google Scholar 

  18. Hilborn, R. & Mangel, M. The Ecological Detective: Confronting Models with Data (Vol. 28, Monographs in Population Biology, Princeton Univ. Press, New Jersey, 1997)

    Google Scholar 

  19. Burnham, K. P. & Anderson, D. R. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach 2nd edn (Springer, New York, 2002)

    MATH  Google Scholar 

  20. Johnson, J. B. & Omland, K. S. Model selection in ecology and evolution. Trends Ecol. Evol. 19, 101–108 (2004)

    Article  Google Scholar 

  21. Afanasyev, V. A miniature storing activity recorder for seabird species with 80 bytes of memory for data storage. NERC Tech. 1, 4–7 (1993)

    Google Scholar 

  22. Shlesinger, M. F. & Klafter, J. Lévy walks versus Lévy flights. In On Growth and Form: Fractal and Non-Fractal Patterns in Physics (eds Stanley, H. E. & Ostrowsky, N.) 279–283 (Martinus Nijhoff Publishers, Dordrecht, 1986)

    Chapter  Google Scholar 

  23. Newman, M. E. J. Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005)

    Article  ADS  Google Scholar 

  24. Sims, D. W., Righton, D. R. & Pitchford, J. W. Minimizing errors in identifying Lévy flight behaviour of organisms. J. Anim. Ecol. 76, 222–229 (2007)

    Article  Google Scholar 

  25. Pueyo, S. & Jovani, R. Comment on “A keystone mutualism drives pattern in a power function”. Science 313, 1739c (2006)

    Article  ADS  Google Scholar 

  26. Weimerskirch, H., Gault, A. & Cherel, Y. Prey distribution and patchiness: factors in foraging success and efficiency of wandering albatrosses. Ecology 86, 2611–2622 (2005)

    Article  Google Scholar 

  27. Focardi, S., Marcellini, P. & Montanaro, P. Do ungulates exhibit a food density threshold? A field study of optimal foraging and movement patterns. J. Anim. Ecol. 65, 606–620 (1996)

    Article  Google Scholar 

  28. Heinrich, B. Resource heterogeneity and patterns of movement in foraging bumblebees. Oecologia 40, 235–245 (1979)

    Article  ADS  Google Scholar 

  29. Sokal, R. R. & Rohlf, F. J. Biometry: The Principles and Practice of Statistics in Biological Research 3rd edn (W. H. Freeman and Company, New York, 1995)

    MATH  Google Scholar 

  30. Pueyo, S. Diversity: between neutrality and structure. Oikos 112, 392–405 (2006)

    Article  Google Scholar 

Download references

Acknowledgements

We thank P. Rothery, W. Blanchard and L. Thomas for statistical advice, and R. Myers, I. Jonsen, G. Breed, S.-J. Dunn, F. de Moura, J. Cressoni and M. Lyra for discussions. We acknowledge the work by all fieldworkers involved, in particular thanking B. Phalan and I. Forster for deploying devices at Bird Island. We thank M. Francis, A. Fukuda and H. Higuchi for providing instruments used in 2004, and J. Croxall for supporting albatross research at Bird Island. This work was funded by the UK Natural Environment Research Council and the Brazilian research agency CNPq. The work at the British Antarctic Survey represents a collaboration between the Discovery 2010 and Natural Complexity Programmes, and we appreciate P. Trathan’s efforts in facilitating it.

Author Contributions A.M.E. performed the analyses, computations and derivations presented in this paper, and led its preparation with input from all authors. The paper presents a synthesis of the work of two different teams of researchers who independently and concurrently conceived of re-examining the Lévy flight hypothesis with newer albatross data. One team (A) comprised N.W.W., M.P.F. and E.J.M. and was subsequently joined by A.M.E. The other team (B) comprised V.A., S.V.B., M.G.E.dL., E.P.R., H.E.S. and G.M.V.; R.A.P. provided albatross expertise to both teams. Team B were the first to show the implication of the long first and last dry sequences on the power-law distribution of albatross flight durations.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrew M. Edwards.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

The file contains extensive Supplementary Methods, Supplementary Tables 1-2 and Supplementary Figures 1-6 with Legends. (PDF 4331 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Edwards, A., Phillips, R., Watkins, N. et al. Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449, 1044–1048 (2007). https://doi.org/10.1038/nature06199

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature06199

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing