Day-to-day dynamics in a duopoly ride-sourcing market

We propose a dynamical system model to formulate the day-to-day dynamics of both traveler mode choice and ride-sourcing car supply in a duopoly ride-sourcing market. In the transport system, besides usual transport modes, e.g., private car and public transit, there are two platforms to provide ride-sourcing services, and travelers choose travel modes according to their perceived travel costs. Furthermore, the earnings and reservation wages of ride-sourcing car drivers are heterogeneous and follow continuous distributions. We mathematically prove the existence and stability of the stationary point of the model and the equivalence between the stationary point and the equilibrium point. Numerical results show that there may be multiple stationary points in the system, which show a positive correlation characteristic. The evolution trajectories of the system from different initial points may converge to different stationary points, and a lager (smaller) initial number of users using a service platform can result in more (fewer) users and higher (lower) profits of the service platform at the stationary state. When the two platforms provide the same service level to users, setting the same price level may not make their profits simultaneously maximum at the stationary state. The proposed model can be applied to capture the period-to-period dynamics of platform pricings.