A model of the vicious cycle of a bus line
  Several authors have noted that in a non-regulated environment the development of public transport service is self-adjusting: Faced with a decreasing demand, operators will tend to reduce service to cut costs, resulting in a decrease in the level-of-service, which then triggers a further drop in demand. The opposite may also occur: high demand will induce the operator to increase supply, e.g. through an increase in frequency, which results in a higher level-of-service and a subsequent increase in passenger numbers, triggering another round of service improvements. This paper adds to the literature by presenting an analytic model for analyzing these phenomena of vicious or virtuous cycles. The model formalizes passengers decisions to use a public transport service depending on waiting time and employs field data regarding passengers variation in willingness-to-wait for a public transport service. The paper investigates the dynamics of the line service and shows how the emergence of a vicious or virtuous cycle depends on total number of potential passengers and share of captive riders. It ends with a discussion of the implications of the findings for the planning of public transport services. 
Asaf Bar-Yosef, Karel Martens, Itzhak Benenson
physics.soc-ph