Single integrodifferential wave equation for a Lévy walk

Fedotov, Sergei

doi:10.1103/physreve.93.020101

Cited by 29 publications

(58 citation statements)

References 46 publications

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“…This result finds analytical support in Ref. [42] in the strong ballistic case (α < 2). When the transport process is superdiffusive (2 < α < 3), it should similarly hold, due to the large statistical weight of events characterized by particles keeping their velocity for a very long time.…”

Section: Extension To Levy Walkssupporting

confidence: 86%

Reaction fronts in persistent random walks with demographic stochasticity

et al. 2019

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Standard Reaction-Diffusion (RD) systems are characterized by infinite velocities and no persistence in the movement of individuals, two conditions that are violated when considering living organisms. Here we consider a discrete particle model in which individuals move following a persistent random walk with finite speed and grow with logistic dynamics. We show that when the number of individuals is very large, the individual-based model is well described by the continuous Reactive Cattaneo Equation (RCE), but for smaller values of the carrying capacity important finitepopulation effects arise. The effects of fluctuations on the propagation speed are investigated both considering the RCE with a cutoff in the reaction term and by means of numerical simulations of the individual-based model. Finally, a more general Lévy walk process for the transport of individuals is examined and an expression for the front speed of the resulting traveling wave is proposed.

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Section: Extension To Levy Walkssupporting

confidence: 86%

Reaction fronts in persistent random walks with demographic stochasticity

et al. 2019

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“…While for subdiffusion fractional diffusion equations have been derived based on subordination or continuous time random walk theory [3,11], this problem turned out to be much more nontrivial for Lévy walkers due to the spatiotemporal coupling imposing finite velocities [10]. Only very recently progress was made by deriving an integrodifferential wave equation for a onedimensional LW [13]. More generally, in position and velocity space a fractional Klein-Kramers equation containing an n-dimensional correlated LW as a special case was given in Refs.…”

Section: Introductionmentioning

confidence: 99%

Fractional diffusion equation for ann-dimensional correlated Lévy walk

et al. 2016

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Lévy walks define a fundamental concept in random walk theory that allows one to model diffusive spreading faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n-dimensional correlated Lévy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short-range auto-correlated Lévy walks in the large time limit, and we solve it. Our derivation discloses different dynamical mechanisms leading to correlated Lévy walk diffusion in terms of quantities that can be measured experimentally.

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“…To implement nonlinear effects we use the structural density approach together with a population density-dependent turning rate. This method has been used by the authors for the analysis of subdiffusive random walks [40,41] and Lévy walks [42,43].…”

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confidence: 99%

Emergence of Lévy walks in systems of interacting individuals

Fedotov¹,

Korabel²

2017

Phys. Rev. E

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We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the collision and repulsion interactions lead to the density-dependent exponential tempering of power-law distributions. This qualitatively explains the experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases [M. [15][16][17][18]. Recently an intriguing behavior of swarming bacteria was found: they fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics [19]. The extraordinary feature of this movement is that the emergence of a superdiffusive motility is a result of the interactions between bacteria rather than the standard mechanism of controlling the individual frequency of tumbling. However, it is still an open question how Lévy walks emerge in systems of interacting self-propelled particles. The current theory of Lévy walk [8] assumes noninteracting particles and power-law distribution of traveled distances from the inception.The collective behavior of large groups of interacting individuals such as bird flocks, fish schools, and the collective migration of cells or bacteria is another rapidly growing area of active matter research [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30]. In this Rapid Communication we are only concerned with the nonlinear kinetic and macroscopic equations. They have been used to describe interactions between individuals and investigate the formation of a large variety of spatiotemporal self-organized aggregations. Most of these models converge to the density-dependent diffusive transport and do not lead to non-Markovian Lévy walks.In this Rapid Communication we propose a kinetic nonlinear Lévy walk model for interacting individuals in which...

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Single integrodifferential wave equation for a Lévy walk

Cited by 29 publications

References 46 publications

Reaction fronts in persistent random walks with demographic stochasticity

Reaction fronts in persistent random walks with demographic stochasticity

Fractional diffusion equation for ann-dimensional correlated Lévy walk

Emergence of Lévy walks in systems of interacting individuals

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