Extending Lévy search theory from one to higher dimensions: Lévy walking favours the blind

A diverse range of organisms, including T cells, E. coli , honeybees, sharks, turtles, bony fish, jellyfish, wandering albatrosses and even human hunter–gatherers have movement patterns that can be approximated by Lévy walks (LW; sometimes called Lévy flights in the biological and ecological literature). These observations lend support to the ‘Lévy flight foraging hypothesis’ which asserts that natural selection should have led to adaptations for Lévy flight foraging, because Lévy flights can optimize search efficiencies. The hypothesis stems from a rigorous theory of one-dimensional searching and from simulation data for two-dimensional searching. The potential effectiveness of three-dimensional Lévy searches has not been examined but is central to a proper understanding of marine predators and T cells which have provided the most compelling empirical evidence for LW. Here I extend Lévy search theory from one to three dimensions. The new theory predicts that three-dimensional Lévy searching can be advantageous but only when targets are large compared with the perceptual range of the searchers, i.e. only when foragers are effectively blind and need to come into contact with a target to establish its presence. This may explain why effective blindness is a common factor among three-dimensional Lévy walkers.

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Lévy flight movement patterns in marine predators may derive from turbulence cues

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0408 ◽

2014 ◽

Vol 470 (2171) ◽

pp. 20140408 ◽

Cited By ~ 8

Author(s):

A. M. Reynolds

Keyword(s):

Three Dimensional ◽

Movement Patterns ◽

Lévy Flight ◽

Telemetry Data ◽

Lévy Flights ◽

Levy Flight ◽

Marine Predators ◽

Levy Flights ◽

Natural Response ◽

Kinematic Simulations

The Lévy-flight foraging hypothesis states that because Lévy flights can optimize search efficiencies, natural selection should have led to adaptations for Lévy flight foraging. Some of the strongest evidence for this hypothesis has come from telemetry data for sharks, bony fish, sea turtles and penguins. Here, I show that the programming for these Lévy movement patterns does not need to be very sophisticated or clever on the predator's part, as these movement patterns would arise naturally if the predators change their direction of travel only after encountering patches of relatively strong turbulence (a seemingly natural response to buffeting). This is established with the aid of kinematic simulations of three-dimensional turbulence. Lévy flights movement patterns are predicted to arise in all but the most quiescent of oceanic waters.

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LÉVY FLIGHTS: CHAOTIC, TURBULENT, AND RELATIVISTIC

Fractals ◽

10.1142/s0218348x95000412 ◽

1995 ◽

Vol 03 (03) ◽

pp. 491-497 ◽

Cited By ~ 5

Author(s):

MICHAEL F. SHLESINGER ◽

JOSEPH KLAFTER ◽

GERT ZUMOFEN

Keyword(s):

Random Walk ◽

Turbulent Diffusion ◽

Random Variables ◽

Lévy Flight ◽

Scale Invariant ◽

Lévy Flights ◽

Levy Flight ◽

Levy Flights ◽

Mathematical Investigation ◽

Infinite Moments

Lévy flights were introduced through the mathematical investigation of the algebra of random variables with infinite moments. For many years Lévy flights remained an abstract topic in mathematics. Mandelbrot recognized that the Lévy flight prescription had a deep connection to scale-invariant fractal random walk trajectories. We review the utility of Lévy flights in several physics topics involving chaotic and turbulent diffusion and introduce the scaling to describe trajectories in relativistic turbulence.

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Lévy flights in binary optimization

Archives of Control Sciences ◽

10.2478/acsc-2013-0027 ◽

2013 ◽

Vol 23 (4) ◽

pp. 447-454 ◽

Cited By ~ 1

Author(s):

Martin Klimt ◽

Jaromír Kukal ◽

Matej Mojzeš

Keyword(s):

Statistical Analysis ◽

Local Search ◽

Quantum Tunneling ◽

Mutation Operator ◽

Lévy Flight ◽

Lévy Flights ◽

Levy Flight ◽

Binary Optimization ◽

Levy Flights ◽

Optimization Heuristics

Abstract There are many optimization heuristics which involves mutation operator. Reducing them to binary optimization allows to study properties of binary mutation operator. Modern heuristics yield from Lévy flights behavior, which is a bridge between local search and random shooting in binary space. The paper is oriented to statistical analysis of binary mutation with Lévy flight inside and Quantum Tunneling heuristics.

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LÉVY FLIGHT SUPERDIFFUSION: AN INTRODUCTION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021877 ◽

2008 ◽

Vol 18 (09) ◽

pp. 2649-2672 ◽

Cited By ~ 164

Author(s):

A. A. DUBKOV ◽

B. SPAGNOLO ◽

V. V. UCHAIKIN

Keyword(s):

Brownian Motion ◽

Characteristic Function ◽

Lévy Process ◽

Levy Process ◽

Lévy Flight ◽

Lévy Flights ◽

Levy Flight ◽

Barrier Crossing ◽

Lévy Motion ◽

Levy Flights

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

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Chaos and Lévy flights in the three-body problem

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa1722 ◽

2020 ◽

Vol 497 (3) ◽

pp. 3694-3712

Author(s):

Viraj Manwadkar ◽

Alessandro A Trani ◽

Nathan W C Leigh

Keyword(s):

Power Law ◽

Body Problem ◽

Lifetime Distribution ◽

Lévy Flight ◽

Three Body Problem ◽

Power Law Distribution ◽

Lévy Flights ◽

Levy Flight ◽

Levy Flights ◽

Three Body

ABSTRACT We study chaos and Lévy flights in the general gravitational three-body problem. We introduce new metrics to characterize the time evolution and final lifetime distributions, namely Scramble Density $\mathcal {S}$ and the Lévy flight (LF) index $\mathcal {L}$, that are derived from the Agekyan–Anosova maps and homology radius $R_{\mathcal {H}}$. Based on these metrics, we develop detailed procedures to isolate the ergodic interactions and Lévy flight interactions. This enables us to study the three-body lifetime distribution in more detail by decomposing it into the individual distributions from the different kinds of interactions. We observe that ergodic interactions follow an exponential decay distribution similar to that of radioactive decay. Meanwhile, Lévy flight interactions follow a power-law distribution. Lévy flights in fact dominate the tail of the general three-body lifetime distribution, providing conclusive evidence for the speculated connection between power-law tails and Lévy flight interactions. We propose a new physically motivated model for the lifetime distribution of three-body systems and discuss how it can be used to extract information about the underlying ergodic and Lévy flight interactions. We discuss ejection probabilities in three-body systems in the ergodic limit and compare it to previous ergodic formalisms. We introduce a novel mechanism for a three-body relaxation process and discuss its relevance in general three-body systems.

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