Considering such a fact that travellers dynamically adjust their routes and the resultant link traffic flows in a network evolve over time,this paper proposes a dynamical evolutionary model of the traffic assignment problem with endogenous origin-destination (OD) demands.The model’s stability is analysed and the resultant user equilibrium (UE) state is shown to be stable under certain conditions.Numerical results in a grid network indicate that the model can generate convergent flow patts and finally terminates at the UE state.Impacts by the parameters associated with OD demand function and link cost function are also investigated.