A stimulus-response model of day-to-day network dynamics

A general structure of stimulus-response formula is presented to specify the interacted network dynamics under the assumption of a daily learning and adaptive travel behavior. By taking the time derivative of system variable as a response term, the evolution is formulated as a dynamic system. Issues of existence, uniqueness, and stability for the proposed differential equations are briefly discussed. Approximation of a time-varying route-choice model is derived from the addressed path-flow dynamics. Threshold effects on path-flow dynamics are encapsulated into the proposed general structure by incorporating a discontinuous stimulus term. Then, the quasi user equilibrium is achieved when all users feel indifferent between the experienced and predicted travel time provided by intelligent transportation systems, i.e., the whole system dynamics stay within a bounded range. The derived quasi user equilibrium is reduced to Wardrop's user equilibrium as the threshold effects of path-flow dynamics vanish.