Assessing Lévy walks as models of animal foraging

Journal of the Royal Society Interface

Volumn 8, Issue 62, 2011, Pages 1233-1247

  James, Alex ^a   Plank, Michael J ^a   Edwards, Andrew M ^b  

^a UNIVERSITY OF CANTERBURY   (New Zealand)

^b PACIFIC BIOLOGICAL STATION   (Canada)

Author keywords

Animal movement; Efficiency; Heavy tailed; Power law; Random search; Random walk

Indexed keywords

ANIMAL MOVEMENT; HEAVY-TAILED; POWER LAW; RANDOM SEARCH; RANDOM WALK;

OPTIMIZATION;

ANIMALS;

ANIMAL BEHAVIOR; FORAGING BEHAVIOR; HYPOTHESIS; RANDOMIZATION; REVIEW; STATISTICAL ANALYSIS; STATISTICAL MODEL; TECHNIQUE; THEORETICAL MODEL;

ANIMALS; BEHAVIOR, ANIMAL; COMPUTER SIMULATION; FEEDING BEHAVIOR; MODELS, BIOLOGICAL; STOCHASTIC PROCESSES;

ANIMALIA;

EID: 79961124638     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2011.0200     Document Type: Review

References (90)

1. Bovet, P. & Benhamou, S. 1988 Spatial analysis of animals' movements using a correlated randomwalk model. J. Theor. Biol. 131, 419-433. (doi:10.1016/S0022-5193(88)80038-9)
2. Benhamou, S. & Bovet, P. 1989 How animals use their environment: a new look at kinesis. Anim. Behav. 38, 375-383. (doi:10.1016/S0003-3472(89)80030- 2)
3. Benhamou, S. & Bovet, P. 1992 Distinguishing between elementary orientation mechanisms by means of path analysis. Anim. Behav. 43, 371-377. (doi:10.1016/S0003-3472(05)80097-1)
4. Hill, N. A. & Häder, D. P. 1997 A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186, 503-526. (doi:10.1006/jtbi.1997.0421)
5. Codling, E. A., Hill, N. A., Pitchford, J. W. & Simpson, S. 2004 Random walk models for the movement and recruitment of reef fish larvae. Mar. Ecol. Prog. Ser. 279, 215-224. (doi:10.3354/meps279215)
6. Codling, E. A., Plank, M. J. & Benhamou, S. 2008 Random walks in biology. J. R. Soc. Interface 5, 813-834. (doi:10.1098/rsif.2008.0014)
7. Shlesinger, M. F., West, B. J. & Klafter, J. 1987 Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100-1103. (doi:10.1103/PhysRev Lett.58.1100)
8. Edwards, A. M. 2008 Using likelihood to test for Lévy flight search patterns and for general power-law distributions in nature. J. Anim. Ecol. 77, 1212-1222. (doi:10.1111/j.1365-2656.2008.01428.x)
9. Viswanathan, G. M., Afanasyev, V., Buldyrev, S. V., Murphy, E. J., Prince, P. A. & Stanley, H. E. 1996 Lévy flight search patterns of wandering albatrosses. Nature 381, 413-415. (doi:10.1038/381413a0)
10. Okubo, A. & Levin, S. A. 2001 Diffusion and ecological problems: modern perspectives. Berlin, Germany: Springer.
11. Pyke, G. H. 1984 Optimal foraging theory: a critical review. Annu. Rev. Ecol. Syst. 15, 523-575.
12. Stephens, D. W. & Krebs, J. R. 1986 Foraging theory. Princeton, NJ: Princeton University Press.
13. Pitchford, J. W., James, A. & Brindley, J. 2005 Quantifying the effects of individual and environmental variability in fish recruitment. Fish. Oceanogr. 14, 156-160. (doi:10.1111/j.1365-2419.2004.00299.x)
14. Stephens, D. W., Brown, J. S. & Ydenberg, R. C. 2007 Foraging: behaviour and ecology. Chicago, IL: University of Chicago Press.
15. Cole, B. J. 1995 Fractal time in animal behaviour: the movement activity of Drosophila. Anim. Behav. 50, 1317-1324. (doi:10.1016/0003-3472(95)80047-6)
16. Viswanathan, G. M., Buldyrev, S. V., Havlin, S., da Luz, M. G. E., Raposo, E. P. & Stanley, H. E. 1999 Optimising the success of random searches. Nature 401, 911-914. (doi:10.1038/44831)
17. Viswanathan, G. M., Afanasyev, V., Buldyrev, S. V., Havlin, S., da Luz, M. G. E., Raposo, E. P. & Stanley, H. E. 2000 Lévy flights in random searches. Physica A 282, 1-12. (doi:10.1016/S0378-4371(00)00071-6)
18. Mårell, A., Ball, J. P. & Hofgaard, A. 2002 Foraging and movement paths of female reindeer: insights from fractal analysis, correlated random walks, and Lévy flights. Can. J. Zool. 80, 854-865. (doi:10.1139/z02-061)
19. Ramos-Fernández, G., Mateos, J. L., Miramontes, O., Cocho, G., Larralde, H. & Ayala-Orozco, B. 2004 Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi). Behav. Ecol. Sociobiol. 55, 223-230. (doi:10.1007/s00265-003-0700-6)
20. Austin, D., Bowen, W. D. & McMillan, J. I. 2004 Intraspecific variation in movement patterns: modelling individual behaviour in a large marine predator. Oikos 105, 15-30. (doi:10.1111/j.0030-1299.1999.12730.x)
21. Reynolds, A. M. & Frye, M. A. 2007 Free-flight odor tracking in Drosophila is consistent with an optimal intermittent scale-free search. PLoS ONE 2, e354. (doi:10.1371/journal.pone.0000354)
22. Reynolds, A. M., Smith, A. D., Menzel, R., Greggers, U., Reynolds, D. R. & Riley, J. R. 2007 Displaced honey bees perform optimal scale-free search flights. Ecology 88, 1955-1961. (doi:10.1890/06-1916.1)
23. Reynolds, A. M., Reynolds, D. R., Smith, A. D., Svensson, G. P. & Lofstedt, C. 2007 Adaptive flight patterns of male Agrotis segetum moths over landscape scales. J. Theor. Biol. 245, 141-149. (doi:10.1016/j.jtbi.2006.10.007)
24. Sims, D. W. et al. 2008 Scaling laws of marine predator search behaviour. Nature 451, 1098-1103. (doi:10.1038/nature06518)
25. Humphries, N. E. et al. 2010 Environmental context explains Lévy and Brownianmovement patterns ofmarine predators. Nature 465, 1066-1069. (doi:10.1038/nature09116)
26. Bertrand, S., Burgos, J. M., Gerlotto, F. & Atiquipa, J. 2005 Lévy trajectories of Peruvian purse-seiners as an indicator of the spatial distribution of anchovy (Engraulis ringens). ICES J. Mar. Sci. 62, 477-482. (doi:10.1016/j.icesjms.2004.12.002)
27. Brown, C. T., Liebovitch, L. S. & Glendon, R. 2007 Lévy flights in Dobe Ju/'hoansi foraging patterns. Hum. Ecol. 35, 129-138. (doi:10.1007/s10745-006-9083-4)
28. Bartumeus, F., Catalan, J., Fulco, U. L., Lyra, M. L. & Viswanathan, G. M. 2002 Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies. Phys. Rev. Lett. 88, 097901. (doi:10.1103/ PhysRevLett.88.097901)
29. Raposo, E. P., Buldyrev, S. V., da Luz, M. G. E., Santos, M. C., Stanley, H. E. & Viswanathan, G. M. 2003 Dynamical robustness of Lévy search strategies. Phys. Rev. Lett. 91, 240601. (doi:10.1103/PhysRevLett.91.240601)
30. Santos, M. C., Raposo, E. P., Viswanathan, G.M. & da Luz, M. G. E. 2004 Optimal random searches of revisitable targets: crossover from super diffusive to ballistic random walks. Europhys. Lett. 67, 734-740. (doi:10.1209/epl/i2004-10114-9)
31. Reynolds, A. M. 2006 On the intermittent behaviour of foraging animals. Europhys. Lett. 75, 517-520. (doi:10.1209/epl/i2006-10157-x)
32. Edwards, A. M. et al. 2007 Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449, 1044-1048. (doi:10.1038/nature06199)
33. Sims, D. W., Righton, D. & Pitchford, J. W. 2007 Minimizing errors in identifying Lévy flight behaviour of organisms. J. Anim. Ecol. 76, 222-229. (doi:10.1111/j.1365-2656.2006.01208.x)
34. White, E. P., Enquist, B. J. & Green, J. L. 2008 On estimating the exponent of power-law frequency distributions. Ecology 89, 905-912. (doi:10.1890/07-1288.1)
35. Edwards, A. M. 2011 Overturning conclusions of Lévy flight movement patterns by fishing boats and foraging animals. Ecology 92, 1247-1257. (doi:10.1890/10-1182.1)
36. Plank, M. J. & James, A. 2008 Optimal foraging: Lévy pattern or process? J. R. Soc. Interface 5, 1077-1086. (doi:10.1098/rsif.2008.0006)
37. Plank, M. J. & Codling, E. A. 2009 Sampling scale and misidentification of Lévy and non-Lévy movement paths. Ecology 90, 3546-3553. (doi:10.1890/09-0079.1)
38. Benhamou, S. 2007 How many animals really do the Lévy walk? Ecology 88, 1962-1969. (doi:10.1890/06-1769.1)
39. James, A., Plank, M. J. & Brown, R. 2008 Optimizing the encounter rate in biological interactions: ballistic versus Lévy versus Brownian strategies. Phys. Rev. E 78, 051128. (doi:10.1103/PhysRevE.78.051128)
40. James, A., Pitchford, J. W. & Plank, M. J. 2010 Efficient or inaccurate? Analytical and numerical modelling of random search strategies. Bull. Math. Biol. 72, 896-913. (doi:10.1007/s11538-009-9473-z)
41. Reynolds, A. M. 2008 How many animals really do the Lévy walk? Comment. Ecology 89, 2347-2351. (doi:10.1890/07-1688.1)
42. Benhamou, S. 2008 How many animals really do the Lévy walk? Reply. Ecology 89, 2351-2352. (doi:10.1890/08-0313.1)
43. Bartumeus, F., Fernández, P., da Luz, M. G. E., Catalan, J., Solé, R. V. & Levin, S. A. 2008 Superdiffusion and encounter rates in diluted, low dimensional worlds. Eur.Phys. J. Special Topics 157, 157-166. (doi:10.1140/epjst/e2008-00638-6)
44. da Luz, M. G. E., Buldyrev, S. V., Havlin, S., Raposo, E. P., Stanley, H. E. & Viswanathan, G. M. 2001 Improvements in the statistical approach to random Lévy flight searches. Physica A 295, 89-92. (doi:10.1016/S0378- 4371(01)00058-9)
45. Buldyrev, S. V., Gitterman, M., Havlin, S., Kazakov, A. Y., da Luz, M. G. E., Raposo, E. P., Stanley, H. E. & Viswanathan, G. M. 2001 Properties of Lévy flights and walks on an interval with absorbing boundaries. Physica A 302, 148-161. (doi:10.1016/S0378-4371(01)00461-7)
46. Buldyrev, S. V., Havlin, S., Kazakov, A. Y., da Luz, M. G. E., Raposo, E. P., Stanley, H. E. & Viswanathan, G. M. 2001 Average time spent by Lévy flights and walks on an interval with absorbing boundaries. Phys. Rev. E 64, 041108. (doi:10.1103/PhysRevE.64.041108)
47. Santos, M. C., Viswanathan, G. M., Raposo, E. P. & da Luz, M. G. E. 2005 Optimization of random searches on regular lattices. Phys. Rev. E 72, 046143. (doi:10.1103/PhysRevE.72.046143)
48. Santos, M. C., Viswanathan, G. M., Raposo, E. P. & da Luz, M. G. E. 2008 Optimization of random searches on defective lattice networks. Phys. Rev. E 77, 041101. (doi:10.1103/PhysRevE.77.041101)
49. Raposo, E. P., Buldyrev, S. V., da Luz, M. G. E., Viswanathan, G. M. & Stanley, H. E. 2009 Lévy flights and random searches. J. Phys. A: Math. Theor. 42, 434003. (doi:10.1088/1751-8113/42/43/434003)
50. Bartumeus, F., da Luz, M. G. E., Viswanathan, G. M. & Catalan, J. 2005 Animal search strategies: a quantitative random walk analysis. Ecology 86, 3078-3087. (doi:10.1890/04-1806)
51. Patlak, C. S. 1953 Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311-338. (doi:10.1007/BF02476407)
52. Reynolds, A. M. 2008 Optimal random Lévy-loop searching: new insights into the searching behaviours of central-place foragers. Eur. Phys. Lett. 82, 20001. (doi:10.1209/0295-5075/82/20001)
53. Reynolds, A. M. & Bartumeus, F. 2009 Optimising the success of random destructive searches: Lévy walks can outperform ballistic motions. J. Theor. Biol. 260, 98-103. (doi:10.1016/j.jtbi.2009.05.033)
54. Viswanathan, G. M., Bartumeus, F., Buldyrev, S. V., Catalan, J., Fulco, U. L., Havlin, S., da Luz, M. G. E., Lyra, M. L., Raposo, E. P. & Stanley, H. E. 2002 Lévy flight random searches in biological phenomena. Physica A 314, 208-213. (doi:10.1016/S0378-4371(02)01157-3)
55. Benhamou, S. 1994 Spatial memory and searching efficiency. Anim. Behav. 47, 1423-1433. (doi:10.1006/anbe.1994.1189)
56. Kramer, D. L. & McLaughlin, R. L. 2001 The behavioural ecology of intermittent locomotion. Am. Zool. 41, 137-153. (doi:10.1668/0003-1569(2001) 041[0137:TBEOIL]2.0.CO;2)
57. Morales, J. M., Haydon, D. T., Frair, J., Holsinger, K. E. & Fryxell, J. M. 2004 Extracting more out of relocation data: building movement models as mixtures of random walks. Ecology 85, 2436-2445. (doi:10.1890/03-0269)
58. Klassen, R. H. G., Nolet, B. A. & Bankert, D. 2006 Movement of foraging tundra swans explained by spatial pattern in cryptic food densities. Ecology 87, 2244-2254. (doi:10.1890/0012-9658(2006)87[2244:MOFTSE]2.0.CO;2)
59. Jonsen, I. D., Myers, R. A. & James, M. C. 2007 Identifying leatherback turtle foraging behaviour from satellite telemetry using a switching state-space model. Mar. Ecol. Prog. Ser. 337, 255-264. (doi:10.3354/meps337255)
60. Reynolds, A. M. 2009 Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios. Physica A 388, 561-564. (doi:10.1016/j.physa.2008.11.007)
61. Bartumeus, F. & Levin, S. A. 2008 Fractal reorientation clocks: linking animal behavior to statistical patterns of search. Proc. Natl Acad. Sci. USA 105, 19 072-19 077. (doi:10.1073/pnas.0801926105)
62. Bénichou, O., Coppey, M., Suet, P.-H. & Voituriez, R. 2005 Optimal search strategies for hidden targets. Phys. Rev. Lett. 94, 198101. (doi:10.1103/PhysRevLett.94.198101)
63. Bénichou, O., Loverdo, C., Moreau, M. & Voituriez, R. 2006 Two-dimensional intermittent search processes: an alternative to Lévy flight strategies. Phys. Rev. E 74, 020102. (doi:10.1103/PhysRevE.74.020102)
64. Lomholt, M. A., Koren, T., Metzler, R. & Klafter, J. 2008 Lévy strategies in intermittent search processes are advantageous. Proc. Natl Acad. Sci. USA 105, 11 055-11 059. (doi:10.1073/pnas.0803117105)
65. Bénichou, O., Coppey, M., Moreau, M. & Voituriez, R. 2006 Intermittent search strategies: when losing tine becomes efficient. Europhys. Lett. 75, 349-354. (doi:10.1209/epl/i2006-10100-3)
66. Newman, M. E. J. 2006 Power laws, Pareto distributions and Zipf's law. Cont. Phys. 46, 323-351. (doi:10.1080/00107510500052444)
67. Burnham, K. P. & Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. New York, NY: Springer.
68. Elliott, K. H., Bull, R. D., Gaston, A. J. & Davoren, G. K. 2009 Underwater and above-water search patterns of an Arctic seabird: reduced searching at small spatiotemporal scales. Behav. Ecol. Sociobiol. 63, 1773-1785. (doi:10. 1007/s00265-009-0801-y)
69. Grimmett, G. R. & Stirzaker, D. R. 2001 Probability and random processes. Oxford, UK: Oxford University Press.
70. Reynolds, A. M. 2010 Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns. J. R. Soc. Interface 7, 1753-1758. (doi:10.1098/rsif.2010.0292)
71. Petrovskii, S. & Morozov, A. 2009 Dispersal in a statistically structured population. Am. Nat. 173, 278-289. (doi:10.1086/595755)
72. Hapca, S., Crawford, J. W. & Young, I. M. 2009 Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level. J. R. Soc. Interface 6, 111-122. (doi:10.1098/rsif.2008.0261)
73. Gurarie, E., Anderson, J. J. & Zabel, R. W. 2009 Continuous models of population-level heterogeneity inform analysis of animal dispersal and migration. Ecology 90, 2233-2242. (doi:10.1890/08-0359.1)
74. Fritz, H., Said, S. & Weimerskirch, H. 2003 Scale-dependent hierarchical adjustments of movement patterns in a long-range foraging seabird. Proc. R. Soc. Lond. B 270, 1143-1148. (doi:10.1098/rspb.2003.2350)
75. Turchin, P. 1991 Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology 72, 1253-1266. (doi:10.2307/1941099)
76. Viswanathan, G. M., Raposo, E. P., Bartumeus, F., Catalan, J. & da Luz, M. G. E. 2005 Necessary criterion for distinguishing true superdiffusion from correlated random walk processes. Phys. Rev. E 72, 011111. (doi:10.1103/PhysRevE.72.011111)
77. Reynolds, A. M., Swain, J. L., Smith, A. D., Martin, A. P. & Osborne, J. L. 2009 Honeybees use a Lévy flight search strategy and odour-mediated anemotaxis to relocate food sources. Behav. Ecol. Sociobiol. 64, 115-123. (doi:10.1007/s00265-009-0826-2)
78. Codling, E. A. & Plank, M. J. In press. Turn designation, sampling rate and the misidentification of power laws in movement path data using maximum likelihood estimates. Theor. Ecol. (doi:10.1007/s12080-010-0086-9)
79. Reynolds, A. M. 2005 Scale-free movement patterns arising from olfactory-driven foraging. Phys. Rev. E 72, 041928. (doi:10.1103/PhysRevE.72. 041928)
80. Reynolds, A. M. 2007 Avoidance of conspecific odour trails results in scale-free movement patterns and the execution of an optimal searching strategy. Europhys. Lett. 79, 30006. (doi:10.1209/0295-5075/79/30006)
81. Guy, A. G., Bohan, D. A., Powers, S. J. & Reynolds, A. M. 2008 Avoidance of conspecific odour by carabid beetles: a mechanism for the emergence of scale-free searching patterns. Anim. Behav. 76, 585-591. (doi:10.1016/j.anbehav.2008.04.004)
82. Reynolds, A. M. 2008 Deterministic walks with inverse-square power-law scaling are an emergent property of predators that use chemotaxis to locate randomly distributed prey. Phys. Rev. E 78, 011906. (doi:10.1103/PhysRevE.78. 011906)
83. Reynolds, A. M. 2009 Lévy flight patterns are predicted to be an emergent property of a bumblebees' foraging strategy. Behav. Ecol. Sociobiol. 64, 19-23. (doi:10.1007/s00265-009-0813-7)
84. Reynolds, A. M. & Rhodes, C. J. 2009 The Lévy flight paradigm: random search patterns and mechanisms. Ecology 90, 877-887. (doi:10.1890/08-0153.1)
85. Reynolds, A. M. 2009 Scale-free animal movement patterns: Lévy walks outperform fractional Brownian motions and fractional Lévy motions in random search scenarios. J. Phys. A: Math. Theor. 42, 434006. (doi:10. 1088/1751-8113/42/43/434006)
86. Mandelbrot, B. B. & Ness, J. W. V. 1968 Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422-437. (doi:10.1137/1010093)
87. Benhamou, S. 2006 Detecting an orientation component in animal paths when the preferred direction is individualdependent. Ecology 87, 518-528. (doi:10.1890/05-0495)
88. Hammer, M. & Menzel, R. 1995 Learning and memory in the honeybee. J. Neurosci. 15, 1617-1630.
89. Grant, V. & Tembles, E. J. 1992 Foraging ability of rufous hummingbirds on hummingbird flowers and hawkmoth flowers. Proc. Natl Acad. Sci. USA 89, 9400-9404. (doi:10.1073/pnas.89.20.9400)
90. Viswanathan, G. M., Raposo, E. P. & da Luz, M. G. E. 2008 Lévy flights and superdiffusion in the context of biological encounters and random searches. Phys. Life Rev. 5, 133-150. (doi:10.1016/j.plrev.2008.03.002)