Emergence of L\'{e}vy Walks in Systems of Interacting Individuals

TL;DR

This paper presents a nonlinear, non-Markovian model explaining how collective interactions among individuals can lead to emergent superdiffusive Lévy walk behavior, transitioning from simple diffusion to complex movement patterns observed in biological systems.

Contribution

It introduces a novel nonlinear non-Markovian persistent random walk model that explains the emergence of Lévy walks as a collective phenomenon, not just from individual run length distributions.

Findings

Alignment interactions induce superdiffusive growth.

Power law run length distribution with infinite variance emerges.

Density-dependent exponential tempering explains transition from superdiffusion to diffusion.

Abstract

Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed an intriguing behavior of swarming bacteria: they fundamentally change their collective motion from simple diffusion into a superdiffusive L\'{e}vy walk dynamics. We introduce a nonlinear non-Markovian persistent random walk model that explains the emergence of superdiffusive L\'{e}vy walks. We show that the alignment interaction between individuals can lead 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 nonlinear collective phenomenon, rather than due to the standard assumption of the power law distribution of run distances from the inception. At the same time, we find that the repulsion/collision effects lead to the density dependent exponential tempering of power law distributions. This qualitatively explains 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)).