Skip to main content
SpringerLink
Log in
Book cover

Dispersal, Individual Movement and Spatial Ecology pp 33–52Cite as

Lévy or Not? Analysing Positional Data from Animal Movement Paths

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2071))

Abstract

The Lévy walk hypothesis asserts that the optimal search strategy for a forager under specific conditions is to make successive movement steps that have uniformly random directions and lengths drawn from a probability distribution that is heavy-tailed. This idea has generated a huge amount of interest, with numerous studies providing empirical evidence in support of the hypothesis and others criticising some of the methods employed in these. The most common method for identifying Lévy walk behaviour in movement data is to fit a set of candidate distributions to the observed step lengths using maximum likelihood methods. Commonly used candidate distributions are the exponential distribution and the power-law (Pareto) distribution, both on an infinite and a finite (truncated) range. Data sets for which the relative fit of a power-law distribution is better than that of an exponential are typically classified as Lévy walks. However, the movement pattern of the Lévy walk is similar to that of an animal that switches between two behavioural modes in a composite correlated random walk (CCRW) movement process. Recent studies have shown that standard approaches can misidentify the CCRW process as a Levy walk. This misidentification can be due to the methods used to sample and process the data, a failure to assess the absolute fit of the candidate distributions, or the lack of a strong alternative model. In this chapter, we simulate a CCRW process and show that including a composite exponential distribution in the set of candidate distributions can alleviate the problem of misidentification. However, in some cases sampling and processing of the CCRW data can cause a power-law distribution to have a better fit than a composite exponential. In such cases, the absolute goodness-of-fit of the power-law distribution is typically poor, indicating that none of the candidate distributions are a good model for the data. We discuss the relevance of these results for the analysis of empirical movement data.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. R.P.D. Atkinson, C.J. Rhodes, D.W. Macdonald, R.M. Anderson, Scale-free dynamics in the movement patterns of jackals. OIKOS 98, 134–140 (2002)

    Article  Google Scholar 

  2. M. Auger-Méthé, C.C. St. Clair, M.A. Lewis, A.E. Derocher, Sampling rate and misidentification of Lévy and non-Lévy movement paths: comment. Ecology 92, 1699–1701 (2011)

    Google Scholar 

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

    Article  Google Scholar 

  4. F. Bartumeus, Behavioral intermittence, Lévy patterns, and randomness in animal movement. Oikos 118, 488–494 (2009)

    Google Scholar 

  5. F. Bartumeus, S.A. Levin, Fractal reorientation clocks: linking animal behavior to statistical patterns of search. Proc. Natl. Acad. Sci. 105, 19072–19077 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. S. Benhamou, Spatial memory and searching efficiency. Anim. Behav. 47, 1423–1433 (1994)

    Article  Google Scholar 

  8. S. Benhamou, How to reliably estimate the tortuosity of an animal’s path: straightness, sinuosity, or fractal dimension? J. Theor. Biol. 229, 209–220 (2004)

    Article  MathSciNet  Google Scholar 

  9. S. Benhamou, Detecting an orientation component in animal paths when the preferred direction is individual-dependent. Ecology 87, 518–528 (2006)

    Article  Google Scholar 

  10. S. Benhamou, How many animals really do the Lévy walk? Ecology 88, 1962–1969 (2007)

    Article  Google Scholar 

  11. S. Benhamou, P. Bovet, Distinguishing between elementary orientation mechanisms by means of path analysis. Anim. Behav. 43, 371–377 (1992)

    Article  Google Scholar 

  12. O. Bénichou, M. Coppey, P.-H. Suet, R. Voituriez, Optimal search strategies for hidden targets. Phys. Rev. Lett. 94, 198101 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. P. Bovet, S. Benhamou, Spatial analysis of animals’ movements using a correlated random walk model. J. Theor. Biol. 131, 419–433 (1988)

    Article  Google Scholar 

  15. D. Boyer, G. Ramos-Fernández, O. Miramontes, J.L. Mateos, G. Cocho, H. Larralde, H. Ramos, F. Rojas, Scale-free foraging by primates emerges from their interaction with a complex environment. Proc. R. Soc. Lond. B 273, 1743–1750 (2006)

    Article  Google Scholar 

  16. E.A. Codling, N.A. Hill, Sampling rate effects on measurements of correlated and biased random walks. J. Theor. Biol. 233, 573–588 (2005)

    Article  MathSciNet  Google Scholar 

  17. E.A. Codling, M.J. Plank, Turn designation, sampling rate and the misidentification of power-laws in movement path data using maximum likelihood estimates. Theor. Ecol. 4, 397–406 (2011)

    Article  Google Scholar 

  18. E.A. Codling, M.J. Plank, S. Benhamou, Random walks in biology. J. R. Soc. Interface 5, 813–834 (2008)

    Article  Google Scholar 

  19. B.J. Cole, Fractal time in animal behaviour: the movement activity of Drosophila. Anim. Behav. 50, 1317–1324 (1995)

    Google Scholar 

  20. M. de Jager, F.J. Weissing, P.M.J. Herman, B.A. Nolet, J. van de Koppel1, Lévy walks evolve through interaction between movement and environmental complexity. Science 332, 1551–1553 (2011)

    Google Scholar 

  21. C. Dytham, Choosing and Using Statistics: A Biologist’s Guide, 3rd edn. (Wiley, New York, 2011)

    MATH  Google Scholar 

  22. A.M. Edwards, Using likelihood to test for Lévy flight search patterns and for general power-law distributions in nature. J. Anim. Ecol. 77, 1212–1222 (2008)

    Article  Google Scholar 

  23. A.M. Edwards, Overturning conclusions of Lévy flight movement patterns by fishing boats and foraging animals. Ecology 92(6), 1247–1257 (2011)

    Article  Google Scholar 

  24. A.M. Edwards, R.A. Phillips, N.W. Watkins, M.P. Freeman, E.J. Murphy, V. Afanasyev, S.V. Buldyrev, M.G.E. da Luz, E.P. Raposo, H.E. Stanley, G.M. Viswanathan, Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449, 1044–1048 (2007)

    Article  Google Scholar 

  25. M.R. Forster, Key concepts in model selection: performance and generalizability. J. Math. Psychol. 44, 205–231 (2000)

    Article  MATH  Google Scholar 

  26. L. Giuggioli, F. Bartumeus, Animal movement, search strategies, and behavioural ecology: a cross-disciplinary way forward. J. Anim. Ecol. 79, 906–909 (2009)

    Google Scholar 

  27. S. Hapca, J.W. Crawford, I.M. Young, Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level. J. R. Soc. Interface 6, 111–122 (2009)

    Article  Google Scholar 

  28. N.A. Hill, D.P. Häder, A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186, 503–526 (1997)

    Article  Google Scholar 

  29. H.-H. Hsu, S.-J. Joung, Y.-Y. Liao, K.-M. Liu, Satellite tracking of juvenile whale sharks, Rhincodon typus, in the northwestern pacific. Fish. Res. 84, 25–31 (2007)

    Article  Google Scholar 

  30. N.E. Humphries, N. Queiroz, J.R.M. Dyer, N.G. Pade1, M.K. Musy, K.M. Schaefer, D.W. Fuller, J.M. Brunnschweiler, T.K. Doyle, J.D.R. Houghton, G.C. Hays, C.S. Jones, L.R. Noble, V.J. Wearmouth, E.J. Southall, D.W. Sims, Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 465, 1066–1069 (2010)

    Google Scholar 

  31. A. James, M.J. Plank, A.M. Edwards, Assessing lvy walks as models of animal foraging. J. R. Soc. Interface 8, 1233–1247 (2011)

    Article  Google Scholar 

  32. I.D. Jonsen, R.A. Myers, M.C. James, Identifying leatherback turtle foraging behaviour from satellite telemetry using a switching state-space model. Mar. Ecol. Prog. Ser. 337, 255–264 (2007)

    Article  Google Scholar 

  33. P.M. Kareiva, N. Shigesada, Analyzing insect movement as a correlated random walk. Oecologia 56, 234–238 (1983)

    Article  Google Scholar 

  34. J. Klafter, B.S. White, M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk, in Biological Motion, ed. by G. Hoffmann, W. Alt (Springer, Berlin, 1989), pp. 281–296

    Google Scholar 

  35. R.H.G. Klassen, B.A. Nolet, D. Bankert, Movement of foraging tundra swans explained by spatial pattern in cryptic food densities. Ecology 87, 2244–2254 (2006)

    Article  Google Scholar 

  36. A.S. Knell, E.A. Codling, Classifying area-restricted search (ARS) using a partial sum approach. Theor. Ecol. 5, 325–339 (2012)

    Article  Google Scholar 

  37. D.L. Kramer, R.L. McLaughlin, The behavioural ecology of intermittent locomotion. Am. Zool. 41, 137–153 (2001)

    Article  Google Scholar 

  38. C.E. Kuhn, D.S. Johnson, R.R. Ream, T.S. Gelatt, Advances in the tracking of marine species: using GPS locations to evaluate satellite track data and a continuous-time movement model. Marine Ecol. Prog. Ser. 393, 97–109 (2009)

    Article  Google Scholar 

  39. K.V. Mardia, P.E. Jupp, Directional Statitics (Wiley, New York, 1999)

    Book  Google Scholar 

  40. A. Mårell, J.P. Ball, A. Hofgaard, 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 

  41. J.M. Morales, D.T. Haydon, J. Frair, K.E. Holsinger, J.M. Fryxell, Extracting more out of relocation data: building movement models as mixtures of random walks. Ecology 85, 2436–2445 (2004)

    Article  Google Scholar 

  42. C.S. Patlak, Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  43. T.A. Patterson, L. Thomas, C. Wilcox, O. Ovaskainen, J. Matthiopoulos, State-space models of individual animal movement. Trends Ecol. Evol. 23, 87–94 (2008)

    Article  Google Scholar 

  44. S. Petrovskii, A. Morozov, Dispersal in a statistically structured population. Am. Nat. 173, 278–289 (2009)

    Article  Google Scholar 

  45. M.J. Plank, E.A. Codling, Sampling scale and misidentification of Lévy and non-Lévy movement paths. Ecology 90, 3546–3553 (2009)

    Article  Google Scholar 

  46. M.J. Plank, E.A. Codling, Sampling rate and misidentification of Lévy and non-Lévy movement paths: reply. Ecology 92, 1701–1702 (2011)

    Article  Google Scholar 

  47. R Development Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2009). ISBN 3-900051-07-0

    Google Scholar 

  48. G. Ramos-Fernández, J.L. Mateos, O. Miramontes, G. Cocho, H. Larralde, B. Ayala-Orozco, Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi). Behav. Ecol. Sociobiol. 55, 223–230 (2004)

    Article  Google Scholar 

  49. A.M. Reynolds, How many animals really do the Lévy walk? Comment. Ecology 89, 2347–2351 (2008)

    Article  Google Scholar 

  50. A.M. Reynolds, Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios. Phys. A 388, 561–564 (2009)

    Article  Google Scholar 

  51. A.M. Reynolds, Balancing the competing demands of harvesting and safety from predation: Lévy walk searches outperform composite brownian walk searches but only when foraging under the risk of predation. Physica A 389, 4740–4746 (2011)

    Article  Google Scholar 

  52. A.M. Reynolds, M.A. Frye, Free-flight odor tracking in Drosophila is consistent with an optimal intermittent scale-free search. PLoS ONE 2, e354 (2007)

    Article  Google Scholar 

  53. A.M. Reynolds, A.D. Smith, R. Menzel, U. Greggers, D.R. Reynolds, J.R. Riley, Displaced honey bees perform optimal scale-free search flights. Ecology 88, 1955–1961 (2007)

    Article  Google Scholar 

  54. D. Rowat, M. Gore, Regional scale horizontal and local scale vertical movements of whale sharks in the Indian Ocean off Seychelles. Fish. Res. 84, 32–40 (2007)

    Article  Google Scholar 

  55. M.F. Shlesinger, B.J. West, J. Klafter, Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  57. D.W. Sims, E.J. Southall, N.E. Humphries, G.C. Hays, C.J.A. Bradshaw, J.W. Pitchford, A. James, M.Z. Ahmed, A.S. Brierley, M.A. Hindell, D. Morritt, M.K. Musyl, D. Righton, E.L.C. Shepard, V.J. Wearmouth, R.P. Wilson, M.J. Witt, J.D. Metcalfe, Scaling laws of marine predator search behaviour. Nature 451, 1098–1103 (2008)

    Article  Google Scholar 

  58. P. Turchin, Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology 72, 1253–1266 (1991)

    Article  Google Scholar 

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

    Google Scholar 

  60. P. Turchin, F.J. Odendaal, M.D. Rausher, Quantifying insect movement in the field. Environ. Entomol. 20, 955–963 (1991)

    Google Scholar 

  61. G.J.G. Upton, B. Fingleton, Spatial Data Analysis by Example. Volume 2: Categorical and Directional Data (Wiley, New York, 1989)

    Google Scholar 

  62. C. Varin, N. Reid, D. Firth, An overview of composite likelihood methods. Stat. Sinica 21, 5–42 (2011)

    MathSciNet  MATH  Google Scholar 

  63. A.W. Visser, T. Kiørboe, Plankton motility patterns and encounter rates. Oecologia 143, 538–546 (2006)

    Article  Google Scholar 

  64. G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. da Luz, E.P. Raposo, H.E. Stanley, Optimising the success of random searches. Nature 401, 911–914 (1999)

    Article  Google Scholar 

  65. G.M. Viswanathan, E.P. Raposo, F. Bartumeus, J. Catalan, M.G.E. da Luz, Necessary criterion for distinguishing true superdiffusion from correlated random walk processes. Phys. Rev. E 72, 011111 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  67. G.M. Viswanathan, M.G.E. da Luz, E.P. Raposo, H.E. Stanley, The Physics of Foraging: An Introduction to Random Searches and Biological Encounters (Cambridge University Press, London, 2011)

    Book  MATH  Google Scholar 

  68. W. Zucchini, I.L. MacDonald, Hidden Markov Models for Time Series: An Introduction Using R (Chapman and Hall/CRC, London, 2009)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

M.A.-M. thanks Drs. A. Derocher and M. A. Lewis and the Centre for Mathematical Biology for their support and Alberta Innovates-Technology Futures, Natural Sciences and Engineering Research Council of Canada, and the University of Alberta for graduate student scholarships.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

    Michael J. Plank

  2. Department of Biological Sciences, University of Alberta, Edmonton, Canada

    Marie Auger-Méthé

  3. Departments of Mathematical Sciences and Biological Sciences, University of Essex, Colchester, UK

    Edward A. Codling

Authors
  1. Michael J. Plank
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marie Auger-Méthé
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward A. Codling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edward A. Codling .

Editor information

Editors and Affiliations

  1. Mathematical & Statistical Sciences, University of Alberta, Central Academic Building 632, Edmonton, T6G 2G1, Alberta, Canada

    Mark A. Lewis

  2. Mathematical Institute, Centre for Mathematical Biology, University of Oxford, St. Giles Street 24-29, Oxford, OX1 3LB, United Kingdom

    Philip K. Maini

  3. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Sergei V. Petrovskii

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Plank, M.J., Auger-Méthé, M., Codling, E.A. (2013). Lévy or Not? Analysing Positional Data from Animal Movement Paths. In: Lewis, M., Maini, P., Petrovskii, S. (eds) Dispersal, Individual Movement and Spatial Ecology. Lecture Notes in Mathematics(), vol 2071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35497-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation