Modeling Growth Process of \b{eta} index of Transport Network Based on Nonlinear Spatial Dynamics

The \b{eta} index is one of important measurements reflecting development level of traffic networks. However, how to explain and predict the \b{eta} index growth for a geographical region is a pending problem. With the help of mathematical reasoning and empirical analysis, this paper is devoted to modeling the growing curve of \b{eta} index. A new measurement termed {\delta} index is introduced and a set of new models are constructed. Suppose there is a nonlinear relation between human settlements and roads. A pair differential equations are built for describing the nonlinear dynamics of traffic networks. A logistic function of \b{eta} index growth is derived from the two spatial dynamic equations. On the other hand, based on verified empirical models about urbanization level, economic development level, and \b{eta} index of traffic network, a Boltzmann equation of \b{eta} index growth can be derived. Normalizing the \b{eta} index, Boltzmann equation will become logistic function. This lends an indirect support to the theoretical model of \b{eta} index from the prospective of positive studies. The models proposed in this work provided new approaches for explanation and prediction of spatio-temporal evolution of traffic networks.