Uniqueness of the swarm steady-state from arbitrary initial conditions

Establish whether the swarm density produced by the stochastic sensing-and-motion model with collision ejections—defined by the agent-level Markov reactions in Eqs.(1)–(2) and the mean-field dynamical system in Eq.(MFpatternFormation)—converges to a unique steady-state for arbitrary initial conditions across the parameter ranges considered.

Background

The paper introduces a stochastic swarming model in which agents make uncertain local density measurements and move to reduce error with a known target density, while collisions can eject agents to neighboring patches. From these agent-level processes, the authors derive a mean-field system (Eq.(MFpatternFormation)) and analyze its behavior under small fluctuation approximations, showing monotonic convergence toward a steady-state in that regime.

Although the small-fluctuation analysis indicates uniqueness of the steady-state within that approximation, the global uniqueness of the swarm’s steady-state for arbitrary initial conditions under the full model and parameter ranges is not established. This motivates an explicit open question about whether the steady-state is unique beyond the approximations used.