scholarly journals Intermittent inverse-square Lévy walks are optimal for finding targets of all sizes

2021 ◽  
Vol 7 (15) ◽  
pp. eabe8211
Author(s):  
Brieuc Guinard ◽  
Amos Korman
Keyword(s):  
Random Walk ◽  
Mathematical Proof ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Power Law Distribution ◽  
Arbitrary Size ◽  
Lévy Walk ◽  
Lévy Walks ◽  
Levy Walk ◽  
Biological Organisms

Lévy walks are random walk processes whose step lengths follow a long-tailed power-law distribution. Because of their abundance as movement patterns of biological organisms, substantial theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here, we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the Lévy foraging hypothesis and are thus expected to affect future experiments on animals performing Lévy walks.

scholarly journals Simulations of Lévy walk

2020 ◽  
Author(s):  
Venkat Abhignan ◽  
Sinduja Rajadurai
Keyword(s):  
Movement Pattern ◽  
Stable Distributions ◽  
Lévy Distribution ◽  
Lévy Walk ◽  
Lévy Walks ◽  
Levy Distribution ◽  
Levy Walk ◽  
The Ideal

AbstractWe simulate stable distributions to study the ideal movement pattern for the spread of a virus using autonomous carrier. We observe Lévy walks to be the most ideal way to spread and further study how the parameters in Lévy distribution affects the spread.

Noisy Lévy walk analog of two-dimensional DNA walks for chromosomes of S. cerevisiae

1998 ◽  
Vol 58 (1) ◽  
pp. 914-918 ◽  
Author(s):  
Guillermo Abramson ◽  
Pablo A. Alemany ◽  
Hilda A. Cerdeira
Keyword(s):  
Two Dimensional ◽  
Lévy Walk ◽  
Levy Walk

scholarly journals Differentiating the Lévy walk from a composite correlated random walk

2015 ◽  
Vol 6 (10) ◽  
pp. 1179-1189 ◽  
Author(s):  
Marie Auger‐Méthé ◽  
Andrew E. Derocher ◽  
Michael J. Plank ◽  
Edward A. Codling ◽  
Mark A. Lewis
Keyword(s):  
Random Walk ◽  
Correlated Random Walk ◽  
Lévy Walk ◽  
Levy Walk

scholarly journals Signatures of active and passive optimized Lévy searching in jellyfish

2014 ◽  
Vol 11 (99) ◽  
pp. 20140665 ◽  
Author(s):  
Andy M. Reynolds
Keyword(s):  
Simulated Annealing ◽  
Water Column ◽  
Movement Patterns ◽  
Search Theory ◽  
Search Pattern ◽  
Global Maximum ◽  
Distributed Resources ◽  
Lévy Walk ◽  
Lévy Walks ◽  
Levy Walk

Some of the strongest empirical support for Lévy search theory has come from telemetry data for the dive patterns of marine predators (sharks, bony fishes, sea turtles and penguins). The dive patterns of the unusually large jellyfish Rhizostoma octopus do, however, sit outside of current Lévy search theory which predicts that a single search strategy is optimal. When searching the water column, the movement patterns of these jellyfish change over time. Movement bouts can be approximated by a variety of Lévy and Brownian (exponential) walks. The adaptive value of this variation is not known. On some occasions movement pattern data are consistent with the jellyfish prospecting away from a preferred depth, not finding an improvement in conditions elsewhere and so returning to their original depth. This ‘bounce’ behaviour also sits outside of current Lévy walk search theory. Here, it is shown that the jellyfish movement patterns are consistent with their using optimized ‘fast simulated annealing’—a novel kind of Lévy walk search pattern—to locate the maximum prey concentration in the water column and/or to locate the strongest of many olfactory trails emanating from more distant prey. Fast simulated annealing is a powerful stochastic search algorithm for locating a global maximum that is hidden among many poorer local maxima in a large search space. This new finding shows that the notion of active optimized Lévy walk searching is not limited to the search for randomly and sparsely distributed resources, as previously thought, but can be extended to embrace other scenarios, including that of the jellyfish R. octopus . In the presence of convective currents, it could become energetically favourable to search the water column by riding the convective currents. Here, it is shown that these passive movements can be represented accurately by Lévy walks of the type occasionally seen in R. octopus . This result vividly illustrates that Lévy walks are not necessarily the result of selection pressures for advantageous searching behaviour but can instead arise freely and naturally from simple processes. It also shows that the family of Lévy walkers is vastly larger than previously thought and includes spores, pollens, seeds and minute wingless arthropods that on warm days disperse passively within the atmospheric boundary layer.

scholarly journals Random Walks of a Cell With Correlated Speed and Persistence Influenced by the Extracellular Topography

2021 ◽  
Vol 9 ◽  
Author(s):  
Kejie Chen ◽  
Kai-Rong Qin
Keyword(s):  
Cell Migration ◽  
Random Walk ◽  
Cell Types ◽  
Coarse Grained ◽  
Search Efficiency ◽  
Complex Environments ◽  
Unit Distance ◽  
Lévy Walk ◽  
Levy Walk ◽  
A Cell

Cell migration through extracellular matrices is critical to many physiological processes, such as tissue development, immunological response and cancer metastasis. Previous models including persistent random walk (PRW) and Lévy walk only explain the migratory dynamics of some cell types in a homogeneous environment. Recently, it was discovered that the intracellular actin flow can robustly ensure a universal coupling between cell migratory speed and persistence for a variety of cell types migrating in the in vitro assays and live tissues. However, effects of the correlation between speed and persistence on the macroscopic cell migration dynamics and patterns in complex environments are largely unknown. In this study, we developed a Monte Carlo random walk simulation to investigate the motility, the search ability and the search efficiency of a cell moving in both homogeneous and porous environments. The cell is simplified as a dimensionless particle, moving according to PRW, Lévy walk, random walk with linear speed-persistence correlation (linear RWSP) and random walk with nonlinear speed-persistence correlation (nonlinear RWSP). The coarse-grained analysis showed that the nonlinear RWSP achieved the largest motility in both homogeneous and porous environments. When a particle searches for targets, the nonlinear coupling of speed and persistence improves the search ability (i.e. find more targets in a fixed time period), but sacrifices the search efficiency (i.e. find less targets per unit distance). Moreover, both the convex and concave pores restrict particle motion, especially for the nonlinear RWSP and Lévy walk. Overall, our results demonstrate that the nonlinear correlation of speed and persistence has the potential to enhance the motility and searching properties in complex environments, and could serve as a starting point for more detailed studies of active particles in biological, engineering and social science fields.

scholarly journals Swimming Escherichia coli explore the environment by Lévy walk

2021 ◽  
Author(s):  
Haiyan Huo ◽  
Rui He ◽  
Rongjing Zhang ◽  
Junhua Yuan
Keyword(s):  
Random Walk ◽  
Response Regulator ◽  
Bacterial Chemotaxis ◽  
Random Fluctuation ◽  
Aqueous Environment ◽  
E Coli ◽  
Lévy Walk ◽  
Diffusion Characteristics ◽  
Levy Walk ◽  
Run Lengths

E. coli cells swim in aqueous environment in a random walk of alternating runs and tumbles. The diffusion characteristics of this random walk remains unclear. Here, by tracking the swimming of wildtype cells in a 3d homogeneous environment, we found that their trajectories are super diffusive, consistent with Lévy walk behavior. For comparison, we tracked the swimming of mutant cells that lack the chemotaxis signaling noise (the steady-state fluctuation of the concentration of the chemotaxis response regulator CheY-P), and found that their trajectories are normal diffusive. Therefore, wildtype E. coli cells explore the environment by Lévy walk, which originates from the chemotaxis signaling noise. This Lévy walk pattern enhances their efficiency in environmental exploration. Importance E. coli cells explore the environment in a random walk of alternating runs and tumbles. By tracking the 3d trajectories of E. coli cells in aqueous environment, we find that their trajectories are super diffusive, with a power-law shape for the distribution of run lengths, which is characteristics of Lévy walk. We further show that this Lévy walk behavior is due to the random fluctuation of the output level of the bacterial chemotaxis pathway, and it enhances the efficiency of the bacteria in exploring the environment.

A two-dimensional modified Lévy-walk model for the DNA sequences

2006 ◽  
Vol 369 (2) ◽  
pp. 688-698 ◽  
Author(s):  
Yuan-Yen Tai ◽  
Ping-Cheng Li ◽  
Hsen-Che Tseng
Keyword(s):  
Dna Sequences ◽  
Two Dimensional ◽  
Lévy Walk ◽  
Levy Walk

scholarly journals Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns

2010 ◽  
Vol 7 (53) ◽  
pp. 1753-1758 ◽  
Author(s):  
Andy M. Reynolds
Keyword(s):  
Animal Movement ◽  
Probability Distributions ◽  
Movement Patterns ◽  
Lévy Walk ◽  
Lévy Walks ◽  
Correlated Random Walks ◽  
Levy Walk ◽  
Simple Scaling ◽  
Experimental And Theoretical Studies ◽  
Intense Debate

For many years, the dominant conceptual framework for describing non-oriented animal movement patterns has been the correlated random walk (CRW) model in which an individual's trajectory through space is represented by a sequence of distinct, independent randomly oriented ‘moves’. It has long been recognized that the transformation of an animal's continuous movement path into a broken line is necessarily arbitrary and that probability distributions of move lengths and turning angles are model artefacts. Continuous-time analogues of CRWs that overcome this inherent shortcoming have appeared in the literature and are gaining prominence. In these models, velocities evolve as a Markovian process and have exponential autocorrelation. Integration of the velocity process gives the position process. Here, through a simple scaling argument and through an exact analytical analysis, it is shown that autocorrelation inevitably leads to Lévy walk (LW) movement patterns on timescales less than the autocorrelation timescale. This is significant because over recent years there has been an accumulation of evidence from a variety of experimental and theoretical studies that many organisms have movement patterns that can be approximated by LWs, and there is now intense debate about the relative merits of CRWs and LWs as representations of non-orientated animal movement patterns.

scholarly journals Lévy walk dynamics explain gamma burst patterns in primate cerebral cortex

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Yuxi Liu ◽  
Xian Long ◽  
Paul R. Martin ◽  
Samuel G. Solomon ◽  
Pulin Gong
Keyword(s):  
Cerebral Cortex ◽  
Oscillatory Activity ◽  
Motion Processing ◽  
Empirical Observation ◽  
Lévy Walk ◽  
Visual Motion Processing ◽  
Lévy Walks ◽  
Levy Walk ◽  
Primate Cerebral Cortex ◽  
Burst Patterns

AbstractLévy walks describe patterns of intermittent motion with variable step sizes. In complex biological systems, Lévy walks (non-Brownian, superdiffusive random walks) are associated with behaviors such as search patterns of animals foraging for food. Here we show that Lévy walks also describe patterns of oscillatory activity in primate cerebral cortex. We used a combination of empirical observation and modeling to investigate high-frequency (gamma band) local field potential activity in visual motion-processing cortical area MT of marmoset monkeys. We found that gamma activity is organized as localized burst patterns that propagate across the cortical surface with Lévy walk dynamics. Lévy walks are fundamentally different from either global synchronization, or regular propagating waves, because they include large steps that enable activity patterns to move rapidly over cortical modules. The presence of Lévy walk dynamics therefore represents a previously undiscovered mode of brain activity, and implies a novel way for the cortex to compute. We apply a biophysically realistic circuit model to explain that the Lévy walk dynamics arise from critical-state transitions between asynchronous and localized propagating wave states, and that these dynamics yield optimal spatial sampling of the cortical sheet. We hypothesise that Lévy walk dynamics could help the cortex to efficiently process variable inputs, and to find links in patterns of activity among sparsely spiking populations of neurons.

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