Modeling and Cohesiveness Analysis of Midge Swarms

1. Introduction

Coordinated group motion has been studied extensively for a wide range of species (e.g., bacteria, insects, fish, and birds) via experiments, simulations, and mathematical analyses (Parrish & Hamner, 1997; Gueron et al., 1996; Mogilner & Edelstein-Keshet, 1999; Okubo et al., 2001; Couzin et al., 2002; Mogilner et al., 2003; Ballerini et al., 2008; Schultz et al., 2008). In a particularly early and influential study, Okubo and Chiang (1974) conducted a series of experiments where they filmed midge swarms flying over a white-colored “swarm marker,” an object over which midges are attracted to swarm for the purpose of mating (Chiang, 1968; Chiang et al., 1978, 1980). By using shadows from the sun on the white marker, they were able to distinguish between individual midge trajectories for all members of a swarm. This gave them position trajectories in the plane, from which they computed position variances and velocities for the midges at each sampling time. They computed the swarm position centroid as it varied over time. They showed that the speed distribution of the midges in a swarm obeyed a 2-dimensional Maxwell-Boltzmann distribution, and that the distributions of the velocities relative to the velocity centroid, in the and dimensions, were Gaussian. They showed that the velocity autocorrelation coefficient varied with lag, and that the density functions for midge position relative to the swarm position centroid are not Gaussian, but are peaked near zero. Moreover, they showed the mean midge velocity depends on the distance from the middle of the swarm, with a tendency for higher velocities near the edge of the swarm. Using the data sets from (Okubo & Chiang, 1974), Okubo et al. (1976) computed the acceleration field, which shows that accelerations are generally higher near the edges of the swarm. Mathematical analysis of midge swarms was initiated in Okubo and Chiang (1974), and later expanded upon in Okubo (1986). In both these works, the focus was on comparisons between diffusion processes and swarm dynamics in terms of autocorrelation functions.