Evolutionary equations of a sharing-bicycle system and their solutions

The development of sharing-bicycle systems is accompanied by serious phenomena such as indiscriminate parking of shared bikes. This study aims to explore causes of parking behaviour problems in a sharing-bicycle system. Based on user demand and the characteristics of the shared bicycle system, we establish the evolutionary equations for the two variables, the density of user groups on the road, and the density of parked shared bicycles on the road, and study conditions satisfied by a shock wave solution. The results of the model’s shock wave solution, time-invariant solution, and travelling wave solution allow us to reveal the phenomenon of aggregation of sharing-bicycles on the sidewalk due to the architecture of a sharing-bicycle system, as well as the fluctuation of the variables that still exists in a stable system. The results have a wide range of potential impacts on sharing-bicycle system operators. In terms of operation strategy, operators can predict the gathering situation of sharing-bicycle system at different times and places according to our model, so as to optimize vehicle scheduling and parking management and reduce the phenomenon of disorderly parking. In addition, in terms of cost control, operators can use resources more effectively and reduce operating costs through more accurate forecasting and scheduling. Our research findings, when combined the basic technologies of modern sharing-bicycle system (such as mobile payment and location technology), promise to significantly enhance system optimization, control, and management, thereby fostering the sustainable growth of the sharing-bicycle industry. • Investigated causes of indiscriminate parking in sharing-bicycle systems. • Established evolutionary equations to study user density and parked bike density. • Solutions to models reveal aggregation patterns and system stability. • Findings provide insights for optimizing bike-sharing systems.