# Lévy flight foraging hypothesis

The Lévy flight foraging hypothesis is a hypothesis in the field of biology that may be stated as follows:

Since Lévy flights and walks can optimize search efficiencies, therefore natural selection should have led to adaptations for Lévy flight foraging.[1]

## Background

The movement of animals closely resembles in many ways the random walks of dust particles in a fluid.[2] This similarity led to interest in trying to understand how animals move via the analogy to Brownian motion. This conventional wisdom held until the early 1990s. However, starting in the late 1980s, evidence began to accumulate that did not fit the theoretical predictions.[2]

In 1999, a theoretical investigation of the properties of Lévy flights showed that an inverse square distribution of flight times or distances could optimize the search efficiency under certain circumstances.[3] Specifically, a search based on an inverse-square Lévy walk, consisting of a constant velocity search following a path whose length is distributed over an inverse square Levy stable distribution, is optimal for searching sparsely and randomly distributed revisitable targets in the absence of memory. The team of researchers, consisting of Gandhimohan M. Viswanathan, Sergey V. Buldyrev, Marcos Gomes E. da Luz, Shlomo Havlin, Ernesto P. Raposo and H. Eugene Stanley, published these results in 1999 in the journal Nature.

There has been some controversy about the reality of Lévy flight foraging. Early studies were limited to a small range of movement, and thus the type of motion could not be unequivocally determined; and in 2007 flaws were found in a study of wandering albatrosses which was the first empirical example of such a strategy.[4] There are however many new studies backing the Lévy flight foraging hypothesis.[5][6][7][8]

Recent studies use newer statistical methods[9] and larger data sets showing longer movement paths.[10] Studies published in 2012 and 2013 re-analysed wandering albatross foraging paths and concluded strong support for truncated Lévy flights and Brownian walks consistent with predictions of the Lévy flight foraging hypothesis.[11][12]

On the theoretical point of view, a recent study[13] disputes the validity of the optimality result published in 1999, by concluding that for bi- or tri-dimensional random walks, this result is only valid for very specific conditions: (i) once a target has been foraged, it has to reappear infinitely fast, (ii) the typical scale of the animal displacement has to be very small compared to the typical size of the targets, (iii) after a target is found, the animal has to start a new random walk infinitely close to the border of this target. If any of this condition is not valid, the optimality result does not hold: inverse-square Levy walks are not optimal, and the gain of any optimal Levy walk over others is necessarily marginal (in the sense that it does not diverge when the density of targets is low).

## References

1. ^ Viswanathan, G.M.; Raposo, E.P.; da Luz, M.G.E. (September 2008). "Lévy flights and superdiffusion in the context of biological encounters and random searches". Physics of Life Reviews. 5 (3): 133–150. Bibcode:2008PhLRv...5..133V. doi:10.1016/j.plrev.2008.03.002.
2. ^ a b Buchanan, Mark (5 June 2008). "Ecological modelling: The mathematical mirror to animal nature". Nature. Springer Nature. 453 (7196): 714–716. doi:10.1038/453714a. ISSN 0028-0836. PMID 18528368.
3. ^ Viswanathan, G. M.; Buldyrev, Sergey V.; Havlin, Shlomo; da Luz, M. G. E.; Raposo, E. P.; et al. (1999). "Optimizing the success of random searches". Nature. Springer Nature. 401 (6756): 911–914. Bibcode:1999Natur.401..911V. doi:10.1038/44831. ISSN 0028-0836. PMID 10553906. S2CID 4419834.
4. ^ Edwards, Andrew M.; Phillips, Richard A.; Watkins, Nicholas W.; Freeman, Mervyn P.; Murphy, Eugene J.; et al. (25 October 2007). "Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer" (PDF). Nature. Springer Nature. 449 (7165): 1044–1048. Bibcode:2007Natur.449.1044E. doi:10.1038/nature06199. ISSN 0028-0836. PMID 17960243. S2CID 4393885.
5. ^ Sims, David W.; Southall, Emily J.; Humphries, Nicolas E.; Hays, Graeme C.; Bradshaw, Corey J. A.; et al. (2008). "Scaling laws of marine predator search behaviour". Nature. Springer Nature. 451 (7182): 1098–1102. Bibcode:2008Natur.451.1098S. doi:10.1038/nature06518. ISSN 0028-0836. PMID 18305542. S2CID 4412923.
6. ^ Humphries, Nicolas E.; Queiroz, Nuno; Dyer, Jennifer R. M.; Pade, Nicolas G.; Musyl, Michael K.; et al. (2010). "Environmental context explains Lévy and Brownian movement patterns of marine predators" (PDF). Nature. Springer Nature. 465 (7301): 1066–1069. Bibcode:2010Natur.465.1066H. doi:10.1038/nature09116. ISSN 0028-0836. PMID 20531470. S2CID 4316766.
7. ^ Raichlen, D. A.; Wood, B. M.; Gordon, A. D.; Mabulla, A. Z. P.; Marlowe, F. W.; et al. (23 December 2013). "Evidence of Levy walk foraging patterns in human hunter-gatherers". Proceedings of the National Academy of Sciences. 111 (2): 728–733. Bibcode:2014PNAS..111..728R. doi:10.1073/pnas.1318616111. ISSN 0027-8424. PMC 3896191. PMID 24367098.
8. ^ Sims, D. W.; Reynolds, A. M.; Humphries, N. E.; Southall, E. J.; Wearmouth, V. J.; et al. (14 July 2014). "Hierarchical random walks in trace fossils and the origin of optimal search behavior". Proceedings of the National Academy of Sciences. 111 (30): 11073–11078. Bibcode:2014PNAS..11111073S. doi:10.1073/pnas.1405966111. ISSN 0027-8424. PMC 4121825. PMID 25024221.
9. ^ Clauset, Aaron; Shalizi, Cosma Rohilla; Newman, M. E. J. (4 November 2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51 (4): 661–703. arXiv:0706.1062. Bibcode:2009SIAMR..51..661C. doi:10.1137/070710111. ISSN 0036-1445. S2CID 9155618.
10. ^ Sims, David W.; Humphries, Nicolas E.; Bradford, Russell W.; Bruce, Barry D. (17 October 2011). "Lévy flight and Brownian search patterns of a free-ranging predator reflect different prey field characteristics". Journal of Animal Ecology. Wiley. 81 (2): 432–442. doi:10.1111/j.1365-2656.2011.01914.x. ISSN 0021-8790. PMID 22004140.
11. ^ Humphries, N. E.; Weimerskirch, H.; Queiroz, N.; Southall, E. J.; Sims, D. W. (23 April 2012). "Foraging success of biological Levy flights recorded in situ". Proceedings of the National Academy of Sciences. 109 (19): 7169–7174. Bibcode:2012PNAS..109.7169H. doi:10.1073/pnas.1121201109. ISSN 0027-8424. PMC 3358854. PMID 22529349.
12. ^ Humphries, Nicolas E.; Weimerskirch, Henri; Sims, David W. (2013). Freckleton, Robert (ed.). "A new approach for objective identification of turns and steps in organism movement data relevant to random walk modelling" (PDF). Methods in Ecology and Evolution. Wiley: 480–490. doi:10.1111/2041-210x.12096. ISSN 2041-210X.
13. ^ Levernier, Nicolas; Textor, Johannes; Bénichou, Olivier; Voituriez, Raphaël (2020-02-26). "Inverse Square L\'evy Walks are not Optimal Search Strategies for \$d\ensuremath{\ge}2\$". Physical Review Letters. 124 (8): 080601. doi:10.1103/PhysRevLett.124.080601. hdl:2066/217203. PMID 32167352. S2CID 211011297.