## 3 The Case Study of Harbin
### Simulation Analysis of Multi-Modal Public Transportation Competitive Relationships  
The passenger volume data of urban mass transit, conventional buses, and cabs of the public transportation subsystem from 2012 to 2021 are brought into the previously constructed multi-modal travel competitiveness model to empirically analyze the competitiveness of multi-modal travel in Harbin. According to the model solving algorithm of Equations (4)-(11), the parameters of the model are solved, and the results of \(\hat{A}_{1}\), \(\hat{A}_{2}\), \(\hat{A}_{3}\) are obtained as  
\[\hat{A}_{1} =[0.747782,0.000015,0.000070,0.000096]\] \[\hat{A}_{2} =[0.289466,-0.000002,0.000020,-0.000031]\] \[\hat{A}_{3} =[0.656746,-0.000001,0.000055,0.000043]\]  
In turn, the L-V model of multi-modal public transportation in Harbin City is obtained as follows:  
\[\left\{ \begin{aligned} \frac{dx_{1}}{dt}&=0.747722x_{1}-0.0 00015x_{1}{}^{2}-0.000070x_{1}x_{2}-0.000096x_{1}x_{3}\\ \frac{dx_{2}}{dt}&=0.289466x_{2}+0.000002x_{1}x_{2}- 0.000020x_{2}{}^{2}+0.000031x_{2}x_{3}\\ \frac{dx_{3}}{dt}&=0.656746x_{3}+0.000001x_{1}x_{3}- 0.000055x_{2}x_{3}-0.000043x_{3}{}^{2}\end{aligned} \right.\]  
Based on the above model, the coefficients of the three modes of transportation are solved according to Equation (5), and the matrix of the coefficients of passenger transportation among the three modes of transportation is obtained:  
\[ \begin{bmatrix}0.000019&-0.000007&-0.000003\\ 0.000093&0.000067&0.000083\\ 0.000128&-0.000108&0.000065\end{bmatrix} \]  
where \(C_{ij}\) is \(\frac{\alpha_{ij}}{k_{i}}\), the competing coefficient of the \(i\)th travel mode to the \(j\)th travel mode, and when \(i=j\), it denotes the self-influence coefficient of that travel mode. Based on the results of the above model calculations, the three modes of travel in Harbin City are analyzed in terms of competing relationships. The coefficient of competitiveness of bus travel to rail travel is \(-0.000007\), which shows a facilitating effect: the coefficient of competitiveness of rail to bus is \(0.000093\), which shows an inhibiting effect. The two competing relationships show a biased symbiosis. The coefficient of competitiveness of bus travel to cab travel is \(-0.000003\), which shows a facilitating effect; the coefficient of competitiveness of cab travel to bus is \(0.000128\), which shows an inhibiting effect. The two competing relationships show a biased symbiosis. The coefficient of competitiveness of rail travel to cab travel is \(0.000083\), which shows an inhibiting effect; the coefficient of competitiveness of cab travel to bus is \(-0.000108\), which shows a facilitating effect. The two competing relationships show a biased symbiosis. From the above coefficient matrix analysis, it is known that rail transit and cab travel modes have an inhibiting effect on the development of conventional transportation. Inrecent years, with the development of the economy and the rapid development of rail transit in major cities in China, rail transit in the city of medium- and long-distance transportation has the advantages of rapidity, punctuality, and comfort, so the medium- and long-distance passenger transportation occupies a dominant position and impacts on the conventional public transport market. Relative to public transportation, cabs do not have fixed stations, provide point-to-point services, do not have fixed routes, in accordance with the needs of passengers, and have destinations for direct delivery, are more flexible and convenient, have better accessibility, suitable for individual rides or specific needs, and therefore, the development of urban cabs in short- and medium-distance passenger transportation will be an inhibitor of the development of conventional public transportation. The rail transit travel mode has an inhibiting effect on the development of cab travel. Rail transit has a large capacity, no right-of-way problem, and can ensure speed. However, the current ground traffic congestion is serious, and the original speed advantage of cabs is obviously lost.