754:
729:
834:, blue marlin and swordfish. The data showed that Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns. Birds and other animals (including humans) follow paths that have been modeled using Lévy flight (e.g. when searching for food). An example of an animal, specifically a beetle, that uses Lévy flight patterns is
826:, the random motion seen in swirling gas molecules, for Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids. Researchers analyzed over 12 million movements recorded over 5,700 days in 55 data-logger-tagged animals from 14 ocean predator species in the Atlantic and Pacific Oceans, including
425:
851:
Additionally, biological flight can also apparently be mimicked by other models such as composite correlated random walks, which grow across scales to converge on optimal Lévy walks. Composite
Brownian walks can be finely tuned to theoretically optimal Lévy walks but they are not as efficient as Lévy
1111:
Humphries, Nicolas E.; Queiroz, Nuno; Dyer, Jennifer R. M.; Pade, Nicolas G.; Musyl, Michael K.; Schaefer, Kurt M.; Fuller, Daniel W.; Brunnschweiler, Juerg M.; Doyle, Thomas K.; Houghton, Jonathan D. R.; Hays, Graeme C.; Jones, Catherine S.; Noble, Leslie R.; Wearmouth, Victoria J.; Southall, Emily
1057:
Sims, David W.; Southall, Emily J.; Humphries, Nicolas E.; Hays, Graeme C.; Bradshaw, Corey J. A.; Pitchford, Jonathan W.; James, Alex; Ahmed, Mohammed Z.; Brierley, Andrew S.; Hindell, Mark A.; Morritt, David; Musyl, Michael K.; Righton, David; Shepard, Emily L. C.; Wearmouth, Victoria J.; Wilson,
716:
256:
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Figure 2. An example of 1000 steps of an approximation to a
Brownian motion type of Lévy flight in two dimensions. The origin of the motion is at , the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e.
194:
2109:
732:
Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at , the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e.
45:
random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space.
635:
609:
420:{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=-{\frac {\partial }{\partial x}}f(x,t)\varphi (x,t)+\gamma {\frac {\partial ^{\alpha }\varphi (x,t)}{\partial |x|^{\alpha }}}}
818:
It has been found that jumping between climate states observed in the paleoclimatic record can be described as a Lévy flight or an alpha-stable process
Another application is the
223:. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a
1498:
89:
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search across most landscapes types, suggesting selection pressure for Lévy walk characteristics is more likely than multi-scaled normal diffusive patterns.
1047:
P. D. Ditlevsen, "Observation of alpha-stable noise and a bistable climate potential in an ice-core record", Geophys. Res. Lett 26, 1441-1444, 1999.
1251:; Reynolds, Andrew M.; Humphries, Nicholas E.; Southall, Emily J.; Wearmouth, Victoria J.; Metcalfe, Brett; Twitchett, Richard J. (July 29, 2014).
2127:
855:
Efficient routing in a network can be performed by links having a Levy flight length distribution with specific values of alpha.
2192:
1529:
1359:"Optimal foraging strategies: Lévy walks balance searching and patch exploitation under a very broad range of conditions"
791:
and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena. Examples include
711:{\displaystyle \left\langle |x|^{\theta }\right\rangle \propto t^{\theta /\alpha }\quad {\text{if }}\theta <\alpha .}
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848:
This behavior is optimal for widely dispersed prey that may not always be fully consumed at one time, such as slugs.
838:. When the beetles are hungry and food is scarce, they avoid searching for prey in locations that other individuals of
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819:
1606:
1426:
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246:. For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the
1199:
884:
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1320:"Avoidance of conspecific odour by carabid beetles: a mechanism for the emergence of scale-free searching patterns"
234:
The probability densities for particles undergoing a Levy flight can be modeled using a generalized version of the
1452:
2133:
1989:
1946:
1862:
1058:
Rory P.; Witt, Matthew J.; Metcalfe, Julian D. (2008). "Scaling laws of marine predator search behaviour".
621: = 2, i.e. Brownian motion. In general, the θ fractional moment of the distribution diverges if
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Another important property of the Lévy flight is that of diverging variances in all cases except that of
749:. Note the presence of large jumps in location compared to the Brownian motion illustrated in Figure 2.
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189:{\displaystyle \Pr(U>u)={\begin{cases}1&:\ u<1,\\u^{-D}&:\ u\geq 1.\end{cases}}}
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41:. When defined as a walk in a space of dimension greater than one, the steps made are in
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1114:"Environmental context explains Lévy and Brownian movement patterns of marine predators"
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2016:
2011:
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1226:"Navigating Our World Like Birds and some authors have claimed that the motion of bees"
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The particular case for which
Mandelbrot used the term "Lévy flight" is defined by the
1253:"Hierarchical random walks in trace fossils and the origin of optimal search behavior"
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Guy, Adam G.; Bohan, David A.; Powers, Stephen J.; Reynolds, Andrew M. (2008-09-01).
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Chechkin, Alexei V.; Metzler, Ralf; Klafter, Joseph; Gonchar, Vsevolod Yu. (2008).
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Shlesinger, Michael F.; Klafter, Joseph; Zumofen, Gert (December 1999).
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A comparison of the paintings of
Jackson Pollock to a Lévy flight model
822:. When sharks and other ocean predators cannot find food, they abandon
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The definition of a Lévy flight stems from the mathematics related to
72:(which is not an example of a heavy-tailed probability distribution).
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property, and they are used to model data that exhibits clustering.
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The exponential scaling of the step lengths gives Lévy flights a
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1421:(Updated and augm. ed.). New York: W. H. Freeman.
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for the case where the distribution of step sizes is a
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This can be easily extended to multiple dimensions.
803:, signals analysis as well as many applications in
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1453:"Fractal and nonfractal behavior in Levy flights"
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207:and the distribution is a particular case of the
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93:
1257:Proceedings of the National Academy of Sciences
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1238:
604:{\displaystyle F_{k}\left=-|k|^{\alpha }F_{k}}
434:is a constant akin to the diffusion constant,
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910:"Introduction to the Theory of Lévy Flights"
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941:
250:. In one dimension, the equation reads as
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16:Random walk with heavy-tailed step lengths
1499:"Above, below and beyond Brownian motion"
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727:
48:The term "Lévy flight" was coined after
2128:List of fractals by Hausdorff dimension
2175:
1224:Reynolds, Gretchen (January 1, 2014).
1555:
1197:
1357:Humphries, N.E.; Sims, D.W. (2014).
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993:
242:. The equation requires the use of
219:Lévy flights are, by construction,
79:of the distribution of step-sizes,
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2110:How Long Is the Coast of Britain?
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238:, which is usually used to model
29:in which the step-lengths have a
1460:Journal of Mathematical Physics
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1311:
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890:Lévy flight foraging hypothesis
820:Lévy flight foraging hypothesis
782:
690:
438:is the stability parameter and
68:for when the distribution is a
2134:The Fractal Geometry of Nature
1417:The Fractal Geometry of Nature
1366:Journal of Theoretical Biology
1200:"Sharks hunt via Lévy flights"
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885:Lévy alpha-stable distribution
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1451:Cheng, Z.; Savit, R. (1987).
1403:
1336:10.1016/j.anbehav.2008.04.004
1198:Dacey, James (11 June 2010).
950:"Navigation in a small world"
214:
1581:
7:
2150:Chaos: Making a New Science
1506:American Journal of Physics
1112:J.; Sims, David W. (2010).
858:
248:Riesz fractional derivative
10:
2209:
1386:10.1016/j.jtbi.2014.05.032
745: = 0 which is a
2193:Paul Lévy (mathematician)
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2025:
1974:
1945:
1861:
1831:
1813:
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1174:"Sharks Have Math Skills"
922:10.1002/9783527622979.ch5
875:Heavy-tailed distribution
948:J. M. Kleinberg (2000).
895:
35:probability distribution
1278:10.1073/pnas.1405966111
870:Fat-tailed distribution
836:Pterostichus melanarius
227:due to the generalized
2142:The Beauty of Fractals
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712:
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244:fractional derivatives
236:Fokker–Planck equation
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1411:Mandelbrot, Benoit B.
797:financial mathematics
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713:
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229:central limit theorem
191:
2088:Lewis Fry Richardson
2083:Hamid Naderi Yeganeh
1873:Burning Ship fractal
1805:Weierstrass function
916:. pp. 129–162.
762:) distribution with
737:) distribution with
636:
461:
257:
90:
1846:Space-filling curve
1823:Multifractal system
1706:Space-filling curve
1691:Sierpinski triangle
1518:1999AmJPh..67.1253S
1472:1987JMP....28..592C
1378:2014JThBi.358..179H
1269:2014PNAS..11111073S
1263:(30): 11073–11078.
1143:10.1038/nature09116
1135:2010Natur.465.1066H
1129:(7301): 1066–1069.
1082:10.1038/nature06518
1074:2008Natur.451.1098S
1068:(7182): 1098–1102.
966:2000Natur.406..845K
914:Anomalous Transport
865:Anomalous diffusion
776:normal distribution
766: = 2 and
747:Cauchy distribution
741: = 1 and
225:stable distribution
209:Pareto distribution
70:normal distribution
62:Cauchy distribution
31:stable distribution
2073:Aleksandr Lyapunov
2053:Desmond Paul Henry
2017:Self-avoiding walk
2012:Percolation theory
1656:Iterated function
1597:Fractal dimensions
1231:The New York Times
1172:Witze, Alexandra.
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186:
181:
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2116:Coastline paradox
2093:Wacław Sierpiński
2078:Benoit Mandelbrot
2002:Fractal landscape
1910:Misiurewicz point
1815:Strange attractor
1696:Apollonian gasket
1686:Sierpinski carpet
1512:(12): 1253–1259.
694:
535:
452:Fourier Transform
415:
314:
293:
205:fractal dimension
169:
132:
77:survival function
54:Benoît Mandelbrot
2200:
2188:Markov processes
2033:Michael Barnsley
1900:Lyapunov fractal
1758:Sierpiński curve
1711:Blancmange curve
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1528:. Archived from
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1000:Mandelbrot (1982
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2048:Felix Hausdorff
2021:
1985:Brownian motion
1970:
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1827:
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1800:Pythagoras tree
1657:
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1646:Self-similarity
1590:Characteristics
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1526:10.1119/1.19112
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1445:Further reading
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1541:External links
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1535:on 2012-03-28.
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1330:(3): 585–591.
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2030:
2028:
2024:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1996:
1993:
1991:
1990:Brownian tree
1988:
1987:
1986:
1983:
1982:
1980:
1977:
1973:
1967:
1964:
1962:
1959:
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1954:
1953:
1951:
1948:
1944:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1917:Multibrot set
1915:
1911:
1908:
1907:
1906:
1903:
1901:
1898:
1894:
1893:Douady rabbit
1891:
1889:
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1783:
1781:
1778:
1776:
1773:
1771:
1768:
1764:
1763:Z-order curve
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1733:Hilbert curve
1731:
1729:
1726:
1722:
1719:
1718:
1717:
1716:De Rham curve
1714:
1712:
1709:
1708:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1681:Menger sponge
1679:
1677:
1674:
1672:
1669:
1667:
1666:Barnsley fern
1664:
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1531:
1527:
1523:
1519:
1515:
1511:
1507:
1500:
1495:
1490:
1489:2027.42/70735
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1454:
1449:
1448:
1438:
1434:
1430:
1428:0-7167-1186-9
1424:
1419:
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1412:
1408:
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1379:
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1367:
1360:
1353:
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1333:
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1325:
1321:
1314:
1306:
1302:
1297:
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1194:
1179:
1178:discovery.com
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1115:
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1079:
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1063:
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1020:
1013:
1008:
1001:
996:
994:
985:
981:
976:
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963:
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959:
955:
951:
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923:
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915:
911:
904:
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876:
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862:
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853:
849:
847:
846:
841:
840:P. melanarius
837:
833:
829:
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821:
816:
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806:
802:
798:
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790:
777:
773:
769:
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755:
748:
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736:
730:
726:
724:
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702:
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696:
685:
681:
677:
673:
669:
665:
659:
649:
640:
632:
631:
630:
628:
625: ≤
624:
620:
615:
592:
589:
586:
580:
572:
568:
562:
552:
544:
541:
537:
529:
519:
503:
500:
497:
491:
486:
475:
469:
465:
457:
456:
455:
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449:
445:
441:
437:
433:
409:
399:
383:
380:
377:
371:
366:
355:
352:
346:
343:
340:
334:
328:
325:
322:
316:
310:
298:
295:
289:
278:
275:
272:
266:
253:
252:
251:
249:
245:
241:
237:
232:
230:
226:
222:
212:
210:
206:
202:
176:
173:
170:
164:
157:
154:
150:
142:
139:
136:
133:
127:
122:
116:
111:
105:
102:
99:
86:
85:
84:
82:
78:
73:
71:
67:
63:
59:
58:Cauchy flight
55:
51:
46:
44:
40:
36:
32:
28:
24:
19:
2162:Chaos theory
2157:Kaleidoscope
2148:
2140:
2132:
2058:Gaston Julia
2038:Georg Cantor
2006:
1863:Escape-time
1795:Gosper curve
1743:Lévy C curve
1728:Dragon curve
1607:Box-counting
1530:the original
1509:
1505:
1463:
1459:
1416:
1369:
1365:
1352:
1327:
1323:
1313:
1260:
1256:
1229:
1219:
1207:. Retrieved
1203:
1193:
1181:. Retrieved
1177:
1167:
1126:
1120:
1106:
1065:
1059:
1052:
1043:
1031:
1019:
1007:
957:
953:
913:
903:
880:Lévy process
854:
850:
844:
843:
842:have visited
839:
828:silky sharks
817:
801:cryptography
789:chaos theory
786:
783:Applications
771:
767:
763:
742:
738:
720:
626:
622:
618:
616:
613:
447:
443:
439:
435:
431:
429:
233:
218:
200:
198:
80:
74:
65:
57:
47:
39:heavy-tailed
22:
20:
18:
2153:(1987 book)
2145:(1986 book)
2137:(1982 book)
2123:Fractal art
2043:Bill Gosper
2007:Lévy flight
1753:Peano curve
1748:Moore curve
1634:Topological
1619:Correlation
1372:: 179–193.
1209:22 February
1183:22 February
27:random walk
23:Lévy flight
2177:Categories
1961:Orbit trap
1956:Buddhabrot
1949:techniques
1937:Mandelbulb
1738:Koch curve
1671:Cantor set
1466:(3): 592.
1404:References
793:earthquake
215:Properties
2068:Paul Lévy
1947:Rendering
1932:Mandelbox
1878:Julia set
1790:Hexaflake
1721:Minkowski
1641:Recursion
1624:Hausdorff
1344:0003-3472
1287:0027-8424
805:astronomy
703:α
697:θ
686:α
678:θ
670:∝
660:θ
581:φ
563:α
545:−
530:α
512:∂
492:φ
487:α
483:∂
410:α
392:∂
372:φ
367:α
363:∂
356:γ
335:φ
308:∂
304:∂
299:−
287:∂
267:φ
264:∂
174:≥
155:−
50:Paul Lévy
43:isotropic
2183:Fractals
1978:fractals
1865:fractals
1833:L-system
1775:T-square
1583:Fractals
1413:(1982).
1394:24882791
1305:25024221
1151:20531470
1090:18305542
984:10972276
859:See also
693:if
666:⟩
641:⟨
629:. Also,
83:, being
37:that is
1927:Tricorn
1780:n-flake
1629:Packing
1612:Higuchi
1602:Assouad
1514:Bibcode
1468:Bibcode
1437:7876824
1374:Bibcode
1296:4121825
1265:Bibcode
1159:4316766
1131:Bibcode
1098:4412923
1070:Bibcode
962:Bibcode
813:physics
809:biology
2026:People
1976:Random
1883:Filled
1851:H tree
1770:String
1658:system
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811:, and
760:stable
735:stable
430:where
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131:
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2102:Other
1533:(PDF)
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1362:(PDF)
1155:S2CID
1117:(PDF)
1094:S2CID
896:Notes
772:i.e.,
199:Here
25:is a
1433:OCLC
1423:ISBN
1390:PMID
1340:ISSN
1301:PMID
1283:ISSN
1211:2013
1185:2013
1147:PMID
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980:PMID
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