A Locust-Inspired Model of Collective Marching on Rings

We study the collective motion of autonomous mobile agents in a ringlike environment. The agents' dynamics are inspired by known laboratory experiments on the dynamics of locust swarms. In these experiments, locusts placed at arbitrary locations and initial orientations on a ring-shaped arena are observed to eventually all march in the same direction. In this work we ask whether, and how fast, a similar phenomenon occurs in a stochastic swarm of simple locust-inspired agents. The agents are randomly initiated as marching either clockwise or counterclockwise on a discretized, wide ring-shaped region, which we subdivide into k concentric tracks of length n. Collisions cause agents to change their direction of motion. To avoid this, agents may decide to switch tracks to merge with platoons of agents marching in their direction. We prove that such agents must eventually converge to a local consensus about their direction of motion, meaning that all agents on each narrow track must eventually march in the same direction. We give asymptotic bounds for the expected time it takes for such convergence or "stabilization" to occur, which depends on the number of agents, the length of the tracks, and the number of tracks. We show that when agents also have a small probability of "erratic", random track-jumping behavior, a global consensus on the direction of motion across all tracks will eventually be reached. Finally, we verify our theoretical findings in numerical simulations.