A PDE model for the dynamics of trail formation by ants

Armies of ants are known to move in trails. These trails are formed by a chemotactic force induced by pheromone secreted by the ants. In this paper we develop a mathematical model consisting of two partial differential equations, which explain when and how these trails are formed. The first equation, for the ants, includes the chemotaxis effect of pheromone and the dispersion caused by overcrowding. The second equation is a reaction–diffusion equation for the pheromone concentration. The strength of the chemotactic force, χ, plays a critical role in the analysis. We prove that trails cannot be formed if χ is small, while many trails exist if χ is large.