  We study the formation of trails in populations of self-propelled agents that
make oriented deposits of pheromones and also sense such deposits to which they
then respond with gradual changes of their direction of motion. Based on
extensive off-lattice computer simulations aiming at the scale of insects,
e.g., ants, we identify a number of emerging stationary patterns and obtain
qualitatively the non-equilibrium state diagram of the model, spanned by the
strength of the agent--pheromone interaction and the number density of the
population. In particular, we demonstrate the spontaneous formation of
persistent, macroscopic trails, and highlight some behaviour that is consistent
with a dynamic phase transition. This includes a characterisation of the mass
of system-spanning trails as a potential order parameter. We also propose a
dynamic model for a few macroscopic observables, including the sub-population
size of trail-following agents, which captures the early phase of trail
formation.
