It is widely acknowledged that, to create models for transportation planning that recognize the essential dynamic character of passenger network flows, one must consider two time scales: the so-called within-day time scale and the day-to-day time scale. Substantial progress has been made in modelling within-day dynamic flows for fixed trip matrices; one of the most widely acknowledged models for this purpose is the dynamic user equilibrium model proposed by Friesz et al. (1993) and studied by Xu et al. (1999), Wu et al. (1998), Friesz et al. (2001), Bliemer and Bovy (2003), and Friesz and Mookherjee (2006). In this chapter we propose two day-to-day models of demand growth compatible with a differential variational inequality formulation of the Friesz et al. (1993) model. The first of these employs dynamics inspired by evolutionary game theory, while the second uses the perspective of preferential attachment familiar from the network science and social network literature to create a model of demand growth. Additionally, numerical experiments to compare and contrast the two proposed theories of demand growth are described, along with hypotheses that one might address via such experiments.
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