## 2 Lotka-Volterra Based Modeling of Competing Relationships
### Analysis of Multi-Mode Bus Competition
 \tag{13}\]  
Substituting the four equilibrium points \(M_{0}\), \(M_{1}\), \(M_{2}\) and \(M_{3}\) into Equation (13) and calculating \(\det(J)\) and \(-tr(J)\), the stability or stability conditions can be found, as shown in Table 2.  
According to Table 2, \(M_{0}(0,0)\) is not a stabilization point, when \(\alpha_{12}-\beta_{12}>1\) and \(\alpha_{21}-\beta_{21}<1\), \(\det(J)>0\), \(-tr(J)>0\), \(M_{1}(0,K_{2})\) is the stabilization point; when \(\alpha_{21}-\beta_{21}>1\) and \(\alpha_{12}-\beta_{12}<1\), \(\det(J)>0\), \(-tr(J)>0\), \(M_{2}(K_{1},0)\) is the stabilization point; and when \(\alpha_{12}-\beta_{12}<1\) and \(\alpha_{21}-\beta_{21}<1\), \(\det(J)>0\), \(-tr(J)>0\), \(M_{3}\big{(}\frac{K_{1}(1+\beta_{12}-\alpha_{12})}{1-(\beta_{12}-\alpha_{12} )(\beta_{21}-\alpha_{21})},\)\(\frac{K_{2}(1+\beta_{21}-\alpha_{21})}{1-(\beta_{12}-\alpha_{12} )(\beta_{21}-\alpha_{21})}\big{)}\) is the stabilization point. The conventional bus and rail transit competition and cooperation evolution trend is shown in Figure 1; different competition, cooperation impact effect coefficients, and \begin{table}
\begin{tabular}{c c c} \hline \hline \(M_{i}\) & \(\det(J)\) & -\(tr(J)\) & Stability Condition \\ \hline \(M_{0}\) & \(\gamma_{1}\gamma_{2}\) & \(-(\gamma_{1}+\gamma_{2})\) & instability \\ \(M_{1}\) & \(-\gamma_{1}\gamma_{2}(1+\beta_{12}-\alpha_{12})\) & \(\gamma_{2}-\gamma_{1}(1+\beta_{12}-\alpha_{12})\) & \(\alpha_{12}-\beta_{12}>1,\alpha_{21}-\beta_{21}<1\) \\ \(M_{2}\) & \(-\gamma_{1}\gamma_{2}(1+\beta_{21}-\alpha_{21})\) & \(\gamma_{1}-\gamma_{2}(1+\beta_{21}-\alpha_{21})\) & \(\alpha_{21}-\beta_{21}>1,\alpha_{12}-\beta_{12}<1\) \\ \(M_{3}\) & \(\frac{\gamma_{1}\gamma_{2}(1+\beta_{12}-\alpha_{12})(1+\beta_{12}-\alpha_{21})}{1-( \beta_{12}-\alpha_{12})(\beta_{21}-\alpha_{21})}\) & \(\frac{\gamma_{1}(1+\beta_{12}-\alpha_{12})+(1+\beta_{12}-\alpha_{21})}{1-(\beta_ {12}-\alpha_{12})(\beta_{21}-\alpha_{21})}\) & \(\alpha_{12}-\beta_{12}<1,\alpha_{21}-\beta_{21}<1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stability analysis of co-evolution of co-opetition. conventional bus and rail transit competition and cooperation evolution trends tend to have different equilibrium points. As shown in Figure 1a, when \(\alpha_{12}-\beta_{12}>1\) and \(\alpha_{21}-\beta_{21}<1\), the regional characteristics are similar to Figure 1a, and the trend of competitive evolution eventually converges to the equilibrium point \(M_{2}\). At this point, the competitive impact of mutual cooperation and rail transit on the market is smaller than that of the conventional buses on the market, and the conventional buses will gradually occupy the whole market. As shown in Figure 1c, when \(\alpha_{12}-\beta_{12}<1\) and \(\alpha_{21}-\beta_{21}<1\), similarly, the trend of competitive evolution eventually converges to the equilibrium point \(M_{3}\). At this point, the competition between conventional buses and rail transit is more moderate, and together with the cooperative relationship between them, they can promote each other to reach an equilibrium state in the market. As shown in Figure 1d, when \(\alpha_{12}-\beta_{12}>1\) and \(\alpha_{21}-\beta_{21}>1\), \(M_{3}\) is the saddle point, and there is no stabilizing point for competing with each other.