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In many countries, public transport (PT) services are offered by multiple operators with different modes, such as trains, metros, and buses, which are interchangeable across a single journey. This paper presents a model with one origin (O)-destination(D) path operated by two operators, each of which is responsible for different parts of the OD journey. One operator competes with potential third-party transport companies by offering discount incentives. Such an abstract settings has not been discussed in the literature. We explore how prices, demand, profits, and social welfare change with discounts through a theoretical analysis and numerical simulations under five scenarios. The results indicate that in all the scenarios the operator offering a discount incentive can always attract more passengers and increase its profits. Moreover, reducing the service time of operators offering discounts contributes to an increase in social welfare. Notably, this paper deduces for the first time that the demand scenario aimed at maximizing social welfare is twice as high as that aimed at maximizing total profit. However, in the scenario of maximum social welfare, the profitability of operators becomes challenging. [ABSTRACT FROM AUTHOR]

With the rise of the sharing economy, shared bicycles have become an important component of urban transportation. This paper explores the nonlinear dual oligopoly system for the Cournot model in the bike-sharing market; both operators have maximized profits as their competitive goals. The analysis of pivotal factors influencing passenger preferences, including pricing discounts and comfort levels, is meticulously depicted by a bifurcation diagram. A new chaotic attractor—the shared bicycle attractor—is discovered. The research results indicate that larger discounts and adjustment speeds can cause the system to be in a chaotic state, which is not conducive to the long-term development of operators, although discounts can indeed attract more passengers to a certain extent. On the other hand, the increase in the marginal cost of comfort loss can also make it difficult for enterprises to operate, which requires continuous technological innovation to improve the comfort of cycling. [ABSTRACT FROM AUTHOR]

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