lévy walks
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2021 ◽  
Vol 104 (6) ◽  
Author(s):  
Yu. S. Bystrik ◽  
S. Denisov

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3219
Author(s):  
Viacheslav V. Saenko ◽  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Yuri E. Chamchiyan

The process of Levy random walks is considered in view of the constant velocity of a particle. A kinetic equation is obtained that describes the process of walks, and fractional differential equations are obtained that describe the asymptotic behavior of the process. It is shown that, in the case of finite and infinite mathematical expectation of paths, these equations have a completely different form. To solve the obtained equations, the method of local estimation of the Monte Carlo method is described. The solution algorithm is described and the advantages and disadvantages of the considered method are indicated.


Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2368
Author(s):  
Gaetano Zimbardo ◽  
Francesco Malara ◽  
Silvia Perri

Superdiffusive transport of energetic particles in the solar system and in other plasma environments is often inferred; while this can be described in terms of Lévy walks, a corresponding transport differential equation still calls for investigation. Here, we propose that superdiffusive transport can be described by means of a transport equation for pitch-angle scattering where the time derivative is fractional rather than integer. We show that this simply leads to superdiffusion in the direction parallel to the magnetic field, and we discuss some advantages with respect to approaches based on transport equations with symmetric spatial fractional derivates.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Carlos Garcia-Saura ◽  
Eduardo Serrano ◽  
Francisco B. Rodriguez ◽  
Pablo Varona

AbstractAutonomous robotic search problems deal with different levels of uncertainty. When uncertainty is low, deterministic strategies employing available knowledge result in most effective searches. However, there are domains where uncertainty is always high since information about robot location, environment boundaries or precise reference points is unattainable, e.g., in cave, deep ocean, planetary exploration, or upon sensor or communications impairment. Furthermore, latency regarding when search targets move, appear or disappear add to uncertainty sources. Here we study intrinsic and environmental factors that affect low-informed robotic search based on diffusive Brownian, naive ballistic, and superdiffusive strategies (Lévy walks), and in particular, the effectiveness of their random exploration. Representative strategies were evaluated considering both intrinsic (motion drift, energy or memory limitations) and extrinsic factors (obstacles and search boundaries). Our results point towards minimum-knowledge based modulation approaches that can adjust distinct spatial and temporal aspects of random exploration to lead to effective autonomous search under uncertainty.


2021 ◽  
Author(s):  
Ketika Garg ◽  
Christopher T. Kello ◽  
Paul E Smaldino

Search requires balancing exploring for more options and exploiting the ones previously found. Individuals foraging in a group face another trade-off: whether to engage in social learning to exploit the solutions found by others or to solitarily search for unexplored solutions. Social learning can decrease the costs of finding new resources, but excessive social learning can decrease the exploration for new solutions. We study how these two trade-offs interact to influence search efficiency in a model of collective foraging under conditions of varying resource abundance, resource density, and group size. We modeled individual search strategies as Lévy walks, where a power-law exponent (μ) controlled the trade-off between exploitative and explorative movements in individual search. We modulated the trade-off between individual search and social learning using a selectivity parameter that determined how agents responded to social cues in terms of distance and likely opportunity costs. Our results show that social learning is favored in rich and clustered environments, but also that the benefits of exploiting social information are maximized by engaging in high levels of individual exploration. We show that selective use of social information can modulate the disadvantages of excessive social learning, especially in larger groups and with limited individual exploration. Finally, we found that the optimal combination of individual exploration and social learning gave rise to trajectories with μ ≈ 2 and provide support for the general optimality such patterns in search. Our work sheds light on the interplay between individual search and social learning, and has broader implications for collective search and problem-solving.


Author(s):  
Seongyu Park ◽  
Samudrajit Thapa ◽  
Yeong-Jin Kim ◽  
Michael A Lomholt ◽  
Jae-Hyung Jeon

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2573
Author(s):  
Davide Cocco ◽  
Massimiliano Giona

This paper addresses the generalization of counting processes through the age formalism of Lévy Walks. Simple counting processes are introduced and their properties are analyzed: Poisson processes or fractional Poisson processes can be recovered as particular cases. The stationarity assumption in the renewal mechanism characterizing simple counting processes can be modified in different ways, leading to the definition of generalized counting processes. In the case that the transition mechanism of a counting process depends on the environmental conditions—i.e., the parameters describing the occurrence of new events are themselves stochastic processes—the counting processes is said to be influenced by environmental stochasticity. The properties of this class of processes are analyzed, providing several examples and applications and showing the occurrence of new phenomena related to the modulation of the long-term scaling exponent by environmental noise.


2021 ◽  
Author(s):  
Winston John Neil Armstrong Campeau ◽  
Andrew M Simons ◽  
Brett Stevens

Lévy flight is a type of random walk that models the behaviour of many phenomena across a multiplicity of academic disciplines; within biology specifically, the behaviour of fish, birds, insects, mollusks, bacteria, plants, slime molds, t-cells, and human populations. The Lévy flight foraging hypothesis states that because Lévy flights can maximize an organism’s search efficiency, natural selection should result in Lévy-like behaviour. Empirical and theoretical research has provided ample evidence of Lévy walks in both extinct and extant species, and its efficiency across models with a diversity of resource distributions. However, no model has addressed the maintenance of Lévy flight foraging through evolutionary processes, and existing models lack ecological breadth. We use numerical simulations, including lineage-based models of evolution with a distribution of move lengths as a variable and heritable trait, to test the Lévy flight foraging hypothesis. We include biological and ecological contexts such as population size, searching costs, lifespan, resource distribution, speed, and consider both energy accumulated at the end of a lifespan and averaged over a lifespan. We demonstrate that selection often results in Lévy-like behaviour, although conditional; smaller populations, longer searches, and low searching costs increase the fitness of Lévy-like behaviour relative to Brownian behaviour. Interestingly, our results also evidence a bet-hedging strategy; Lévy-like behaviour reduces fitness variance, thus maximizing geometric mean fitness over multiple generations.


2021 ◽  
Vol 127 (11) ◽  
Author(s):  
Siddhartha Mukherjee ◽  
Rahul K. Singh ◽  
Martin James ◽  
Samriddhi Sankar Ray

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