Swarm dynamics may give rise to Lévy flights

Abstract

"Continuous-Time correlated random walks" are now gaining traction as models of scale-finite animal movement patterns because they overcome inherent shortcomings with the prevailing paradigm-discrete random walk models. Continuous-Time correlated random walk models are founded on the classic Langevin equation that is driven by purely additive noise. The Langevin equation is, however, changed fundamentally by the smallest of multiplicative noises. The inclusion of such noises gives rise to Lévy flights, a popular but controversial model of scale-free movement patterns. Multiplicative noises have not featured prominently in the literature on biological Lévy flights, being seen, perhaps, as no more than a mathematical contrivance. Here we show how Langevin equations driven by multiplicative noises and incumbent Lévy flights arise naturally in the modelling of swarms. Model predictions find some support in three-dimensional, time-resolved measurements of the positions of individual insects in laboratory swarms of the midge Chironomus riparius. We hereby provide a new window on Lévy flights as models of movement pattern data, linking patterns to generative processes.