Emerging autonomous vehicles (AV) can either supplement the public transportation (PT) system or compete with it. This study examines the competitive perspective where both AV and PT operators are profit-oriented with dynamic adjustable supply strategies under five regulatory structures regarding whether the AV operator is allowed to change the fleet size and whether the PT operator is allowed to adjust headway. Four out of the five scenarios are constrained competition while the other one focuses on unconstrained competition to find the Nash Equilibrium. We evaluate the competition process as well as the system performance from the standpoints of four stakeholders -- the AV operator, the PT operator, passengers, and the transport authority. We also examine the impact of PT subsidies on the competition results including both demand-based and supply-based subsidies. A heuristic algorithm is proposed to update supply strategies for AV and PT based on the operators' historical actions and profits. An agent-based simulation model is implemented in the first-mile scenario in Tampines, Singapore. We find that the competition can result in higher profits and higher system efficiency for both operators compared to the status quo. After the supply updates, the PT services are spatially concentrated to shorter routes feeding directly to the subway station and temporally concentrated to peak hours. On average, the competition reduces the travel time of passengers but increases their travel costs. Nonetheless, the generalized travel cost is reduced when incorporating the value of time. With respect to the system efficiency, the bus supply adjustment increases the average vehicle load and reduces the total vehicle kilometer traveled measured by the passenger car equivalent (PCE), while the AV supply adjustment does the opposite.