Emergent Strategies for Shepherding a Flock

We investigate how a shepherd should move to effectively herd a flock towards a target. Using an agent-based (ABM) and a coarse-grained (ODE) model for the flock, we pose and solve for the optimal strategy of a shepherd that must keep the flock cohesive and coerce it towards a target. Three distinct strategies emerge naturally as a function of the scaled herd size {and} the scaled shepherd speed: (i) mustering, where the shepherd circles the herd to ensure compactness, (ii) droving, where the shepherd chases the herd in a desired direction while sweeping back and forth, and (iii) driving, where the flock surrounds a shepherd that drives it from within. A minimal dynamical model for the size, shape, and position of the herd captures the effective behavior of the ABM and further allows us to characterize the different herding strategies in terms of the behavior of the shepherd that librates (mustering), oscillates (droving), or moves steadily (driving).