Noisy pursuit and pattern formation of self-steering active particles

We consider a moving target and an active pursing agent, modeled as an intelligent active Brownian particle capable of sensing the instantaneous target location and adjusting its direction of motion accordingly. An analytical and simulation study in two spatial dimensions reveals that pursuit performance depends on the interplay between self-propulsion, active reorientation, limited maneuverability, and random noise. Noise is found to have two opposing effects: (i) it is necessary to disturb regular, quasi-elliptical orbits around the target, and (ii) slows down pursuit by increasing the traveled distance of the pursuer. For a stationary target, we predict a universal scaling behavior of the mean pursuer-target distance and of the mean first-passage time as a function of Pe²/Ω, where the Péclet number Pe characterizes the activity and Ω the maneuverability. Importantly, the scaling variable Pe²/Ω depends implicitly on the level of thermal or active noise. A similar behavior is found for a moving target, but modified by the velocity ratio α = u ₀/v ₀ of target and pursuer velocities u ₀ and v ₀, respectively. We also propose a strategy to sort active pursuers according to their motility by circular target trajectories.