We present a model of self-propelled agents navigating on a landscape guided by a pheromone signal. The model is motivated by ant navigation. We begin by deducing the model from biological considerations, where each individual orients itself according to the presence of the pheromone in a small circular sector around it. We present some stability properties of trail-like solutions when the trail is given, showing that trail-following behavior is a stable feature of the model. Next, we show the model's consistency with some experimental results. We proceed to the analysis of the model in the case where the pheromone field is produced by a large number of interacting agents. This gives rise to a system of N ODEs coupled by a diffusion equation, which we analyze. Finally, we provide some numerics to illustrate the collective, self-organizing behavior.
This work is a collaboration with T.Goudon and F.Peruani (INRIA, U. Côte d'Azur).