Emergence of Lévy walks in systems of interacting individuals

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)].

Figure 1

Emergence of the Lévy walks when alignment dominates repulsion. The log-log plot of the MSD [curve (2)] displays the superdiffusive behavior with exponent 1.7 (dashed-dotted line). We have used $a=1.5, \alpha =1, \beta =0$, and $r=0$. Other parameters are $\mu =3, {\tau }_0=0.1, v=1$. Without interactions $a=0, r=0$ [curve (1)], the MSD grows linearly in the long-time limit (dashed line has the slope 1).

Figure 2

(a) The run length PDF of the emerging Lévy walkers [curve (2)]. Parameters are the same as in Fig. 1. The run length PDF is a power law with the slope $-2.7$ (the dashed line), that is, the variance of the PDF is infinite. For noninteracting walkers the run length PDF is also a power law [curve (1)], but the slope is $-\mu -1=-4$ (the dotted line), so the variance is finite. (b) Typical trajectory of interacting Lévy walkers (curve $\left(2\right)$; for better clarity we use $a=2.5$) is very persistent, unlike the Brownian-like trajectory for a walker without interactions [curve $\left(1\right)]$.

Figure 3

(a) Alignment leads to the formation of two moving aggregates known as clumps. Density of the walkers $\rho$ (solid curve) develops two bumps corresponding to groups of walkers moving to the left ${\rho }_{-}$ (dashed curve) and to the right ${\rho }_{+}$ (dashed-dotted curve). Here $a=2.5$ and other parameters are the same as in Fig. 2. (b) For weak interactions, $a\to 0$, we find no Lévy walks and clumping behavior. The solid curve corresponds to $a=0.1$. Without interactions the density of walkers (dashed curve) is Gaussian apart from tails. All densities were calculated at $t=3$.

Figure 4

(a) The run length PDF of individuals interacting repulsively. Interactions lead to the transition to Brownian diffusion. Here we use $r={10}^3$ and $\mu =1.3$. Other parameters and the simulation procedure are the same. The run length PDF transitions from a power law with exponent $-\mu -1$ for the Lévy walk without interactions [curve (1)] to exponential distribution for the standard diffusion [curve (2)]. (b) Typical trajectories of walkers with interactions [curve (2)] and without interactions [curve (1)]. Repulsion interactions truncate the long runs of the Lévy walks.