Stigmergic optimal transport

TL;DR

This work develops a stochastic optimal control framework for stigmergic transport, where agents lay and follow pheromone trails to minimize traversal time in heterogeneous domains. By employing arc-length parametrization and an adjoint-based trail-optimization procedure, local trail-following and corrective controls converge to time-minimizing geodesics of the refractive metric $ds_\nu=\nu ds$ without centralized coordination. Numerical experiments reproduce path straightening in homogeneous media and Snell-like refraction at interfaces, illustrating a fast–slow feedback loop that yields global optimality from local rules. The study connects stigmergy, geometric optics, and dynamic optimal transport, offering decentralized routing insights for swarm robotics and active-matter systems.

Abstract

Efficient navigation in swarms often relies on the emergence of decentralized approaches that minimize traversal time or energy. Stigmergy, where agents modify a shared environment that then modifies their behavior, is a classic mechanism that can encode this strategy. We develop a theoretical framework for stigmergic transport by casting it as a stochastic optimal control problem: agents (collectively) lay and (individually) follow trails while minimizing expected traversal time. Simulations and analysis reveal two emergent behaviors: path straightening in homogeneous environments and path refraction at material interfaces, both consistent with experimental observations of insect trails. While reminiscent of Fermat's principle, our results show how local, noisy agent+field interactions can give rise to geodesic trajectories in heterogeneous environments, without centralized coordination or global knowledge, relying instead on an embodied slow fast dynamical mechanism.

Figures (4)

• Figure 1: Stigmergic navigation and transport. Schematic of collective trail following, where an individual agent $i$, characterized by its position and orientation $(X_i(t), Y_i(t), \Theta_i(t))$, approximately follows a pheromone trail $\phi(t,x,y)$ laid by conspecifics. At a collective level, the density field $\rho(t,x,y)$ of all agents modulates and is in turn modulated by the pheromone field $\phi(t,x,y)$.
• Figure 2: Trail-following algorithm. (A) Representative trajectories of agents navigating from source (blue dot) to target (green dot) along a fixed pheromone trail (orange), governed by the Langevin dynamics \ref{['eq:langevin']} with trail-following control \ref{['eq:trailfollow']}, for increasing values of the dimensionless control-to-noise ratio $\beta / (\ell_0 D_\theta)$, with fixed trail sensitivity $\varepsilon = 0.1$. The shaded orange zone depicts the prescribed normalized pheromone concentration field $\phi$, ranging from white (low) to orange (high). As $\beta / (\ell_0 D_\theta)$ increases, agents transition from diffusive wandering to precise trail alignment. (B) Quantitative evaluation of trail-following accuracy as a function of the control-to-noise ratio $\beta / (\ell_0 D_\theta)$. The vertical axis reports the arc-length-averaged deviation between the agent and trail positions, $\int_0^1 \left\| \mathbf{x}_{\mathrm{agent}}(s) - \mathbf{x}_{\mathrm{trail}}(s) \right\| \, ds$, where $s \in [0, 1]$ denotes normalized arc-length. Shaded regions indicate 95% confidence intervals over 10 stochastic trials per parameter setting. Performance improves with increasing control strength, reaching an optimal regime of accurate trail tracking. Beyond this, excessively strong control leads to over-correction and instability, degrading trail-following performance.
• Figure 3: Stigmergic trail laying and following. Schematic of the path integral control procedure. To compute the trail optimizing control, we simulate $n$ uncontrolled forward trajectories using the Langevin dynamics reparametrized by arc-length (Eq. \ref{['eq:langevin_reparam']}), each with independently sampled angular noise. The adjoint dynamics governed by Eq. \ref{['eq:adjoint_backprop']} is integrated along the forward trajectories and the trail optimizing control $\omega_i^{\mathrm{ctrl}}$ is computed using \ref{['eq:ctrl']}. For the pseudocode, see End Matter.
• Figure 4: Convergence to Snell-optimal trails and exploration–exploitation tradeoff. Pheromone-guided refinement across stigmergic cycles. Red curves: agent trajectories; orange colormap: pheromone field. Agents move from source (blue) to target (green), depositing pheromone en route. (A) Homogeneous medium ($\nu=1.0$): trajectories converge to the straight-line optimal path. (B) Two-medium environment with interface (dashed line; $\nu=1.0$ below, $\nu=10.0$ above). With $\beta/(\ell_0 D_\theta)=1.0$, agents retain stochasticity to explore and converge to a refracted trail consistent with Snell’s law.