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Emergence of Lévy walks in systems of interacting individuals

Sergei Fedotov and Nickolay Korabel

  • School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom

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Phys. Rev. E 95, 030107(R) – Published 28 March, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.030107

Abstract

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. de Jager et al., Proc. R. Soc. B 281, 20132605 (2014)].

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Phys. Rev. E 95, 030107(R)– Published 28 March, 2017
  • Received 30 June 2016

©2017 American Physical Society

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