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

TL;DR

This paper develops nonlinear spatial dynamic models to explain and predict the growth of the ta index in traffic networks, integrating empirical data and mathematical reasoning to enhance understanding of urban development.

Contribution

It introduces a new elta index, constructs nonlinear differential equation models, and derives logistic and Boltzmann equations to describe ta index growth.

Findings

Derived a logistic growth model for ta index.

Established a Boltzmann equation linking urbanization and ta index.

Provided new methods for modeling traffic network evolution.

Abstract

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.