Ants use indirect communication via pheromone deposits (stigmergy) to collectively find shortest paths between nest and food sources.
Stigmergy (Grassé 1959): ants deposit pheromone on paths to food; shorter paths accumulate more pheromone (traveled more frequently), creating positive feedback. ACO (Dorigo 1992) uses this to solve NP-hard optimization. Pheromone obeys diffusion-evaporation PDE: ∂φ/∂t = D∇²φ − κφ + S.