Stochastic dynamics of order and disorder in milling whirligig beetles

The dynamics of polar order have received much attention in the literature on collective motion. However, milling dynamics, where organisms exhibit rotational motion around the group center, is understudied. Here, we investigate the stochastic dynamics of milling swarms of whirligig beetles [ (Coleoptera: Gyrinidae)] in small to medium groups (50 to 200) where noise arising due to the finite sizes of the swarms cannot be ignored. We quantify the milling order parameter of beetle swarms and show that they exhibit a “transition” from a disordered state to a milling state as a function of group size. We employ a method of data-driven discovery of dynamical equations to obtain stochastic differential equations (SDEs) that disentangle deterministic and stochastic factors driving the milling dynamics. Crucially, we find that the milling order at large group sizes can be explained largely by the deterministic terms of the SDE, while intrinsic stochasticity switches the system between clockwise and counterclockwise milling states. Furthermore, we find that an agent-based model in which each agent aligns with the average direction of its local neighbors and attracts toward neighbors on a larger spatial scale produces stochastic dynamics of milling (and corresponding SDEs) that are consistent with the data-derived SDEs. However, contrary to expectations from mean field theory, both data and explicit agent-based models show that the dynamical equations can change their qualitative features with the group size even when individual interaction rules do not change. In summary, we show evidence for the milling order transition and characterize the role of intrinsic stochastic dynamics in swarming beetles.