We present a dynamic model describing the time evolution of travelers number, fleet units effort and fleet fare price. The fare price is fixed by the gap between demand and supply and is assumed to evolve on a fast time scale. A system of two equations is derived from a system of three equations through aggregation of variable method. Long term behavior of the model is then studied by determination of equilibrium points, stability and bifurcation analysis. Three equilibrium points are obtained as; unstable origin, stable carrying capacity and a stable free equilibrium. In the later case, a stable equilibrium corresponds to high cost of running fleet units hence negative effort, whereas unstable equilibrium corresponds to high demand for public transport which allow positive effort hence a saddle.
Wilberforce et al. (2025) studied this question.