Nonlinear dynamics of a fuel-price-sensitive traffic flow model with economic and behavioural adaptations

This paper presents a new macroscopic traffic flow model formulated as a system of nonlinear partial differential equations that incorporates fuel-price sensitivity and behavioural adaptation of drivers. The model extends the classical second-order macroscopic framework by introducing a fuel price elasticity term that affects velocity in proportion to local density and price level, and a mode-switching sensitivity term that captures modal shift responses relative to a baseline price. These extensions enable the model to represent how rising fuel costs influence collective driver behaviour, including cautious driving and transitions to alternative transport modes. Linearization about the uniform equilibrium state yields a dispersion relation from which an explicit expression for the critical driver reaction time is obtained, identifying the threshold for the onset of stop-and-go waves as a function of behavioural and structural parameters. Near the stability boundary, a multiple-scale perturbation analysis leads to a Korteweg-de Vries (KdV)-type equation with fuel-sensitive coefficients, whose analytical soliton solutions describe self-sustained congestion waves generated by small disturbances. Numerical simulations based on a quasi-steady reduction of the model show that behavioural sensitivity governs whether fuel-price increases mitigate congestion through modal shift or paradoxically intensify it in the absence of viable transport alternatives. The results provide a mathematical link between fuel-price elasticity, adaptation delay, and nonlinear traffic dynamics, and demonstrate that pricing policies alone cannot guarantee congestion reduction without supportive modal options.