## 2 Lotka-Volterra Based Modeling of Competing Relationships
### Analysis of Multi-Mode Bus Competition
\begin{array}{l}x_{1}(2)-x_{1}(1)\quad=a_{1}(\frac{x_{1}(2)+x_{1}(1 )}{2})-b_{1}(\frac{x_{1}(2)+x_{1}(1)}{2})^{2}-c_{1}(\frac{x_{1}(2)+x_{1}(1)}{ 2})(\frac{x_{2}(2)+x_{2}(1)}{2})-\\ \quad\quad\quad\quad\quad\quad\quad d_{1}(\frac{x_{1}(2)+x_{1}(1)}{2})(\frac {x_{3}(2)+x_{3}(1)}{2})\\ \quad\quad\quad\quad\quad\vdots\\ x_{1}(n)-x_{1}(n-1)\quad=a_{1}(\frac{x_{1}(n)+x_{1}(n-1)}{2})-b_{1}(\frac{x_ {1}(n)+x_{1}(n-1)}{2})^{2}-c_{1}(\frac{x_{1}(n)+x_{1}(n-1)}{2})(\frac{x_{2}(n) +x_{2}(n-1)}{2})-\\ \quad\quad\quad\quad\quad\quad\quad d_{1}(\frac{x_{1}(n)+x_{1}(n-1)}{2})(\frac {x_{3}(n)+x_{3}(n-1)}{2})\\ \end{array} \right. \tag{9}\]  
which is expressed in matrix form as  
model with the simultaneous introduction of competitive influence effect coefficient \(\alpha_{ij}\) and cooperative influence effect coefficient \(\beta_{ij}\), and its synergistic evolution equations are  
\[\left\{ \begin{array}{l}\frac{dx_{1}}{dt}=f(x_{1},x_{2})=r_{1}x_{1} \bigg{(}1-\frac{k_{1}x_{1}}{K_{1}}-\frac{\alpha_{12}x_{2}}{K_{2}}+\frac{\beta_{ 12}x_{2}}{K_{2}}\bigg{)}\\ \frac{dx_{2}}{dt}=g(x_{1},x_{2})=r_{2}x_{2}\bigg{(}1-\frac{k_{2}x_{2}}{K_{2}}- \frac{\alpha_{21}x_{1}}{K_{1}}+\frac{\beta_{21}x_{2}}{K_{2}}\bigg{)}\end{array} \right. \tag{12}\]  
where \(x_{1}\), \(x_{2}\) are the existing demand for conventional public transportation and rail transit, as a function of time; \(\frac{dx_{1}}{dt}\), \(\frac{dx_{2}}{dt}\) are the growth rate of the two modes at the moment t; \(r_{1}\), \(r_{2}\) are the natural growth rate of the volume of traffic, respectively; \(K_{1}\), \(K_{2}\) are the scale thresholds of the transportation volume of the two modes under the level of economic development of the cold region in a certain period of time and are constants; \(k_{1}\), \(k_{2}\) are the coefficients of influence of the two modes themselves; \(\alpha_{12}\), \(\alpha_{21}\) are the coefficients of influence of the regional average competition between the two modes under the level of economic development of the cold region in a certain period of time; \(\beta_{12}\), \(\beta_{21}\) are the coefficients of influence of the regional average cooperation between the two modes under the level of economic development of the cold region in a certain period of time; and \(\alpha_{12},\alpha_{21},\beta_{12},\beta_{21}\geq 0\). The four equilibrium points in the Equation are \(M_{0}(0,0)\), \(M_{1}(0,K_{2})\), \(M_{2}(K_{1},0)\), \(M_{3}\big{(}\frac{K_{1}(1+\beta_{12}-\alpha_{12})}{1-(\beta_{12}-\alpha_{12} )(\beta_{21}-\alpha_{21})},\frac{K_{2}(1+\beta_{21}-\alpha_{21})}{1-(\beta_{12 }-\alpha_{12})(\beta_{21}-\alpha_{21})}\big{)}\), and the evolutionary outcome depends on the interrelationships of \(\alpha_{12}\), \(\alpha_{21}\), \(\beta_{12}\), and \(\beta_{21}\). (2) Stability analysis of competing synergies  
The stability of the four equilibrium points can be judged by the Jacobi matrix if \(\det(J)\
eq 0\), the four equilibrium points satisfy \(\det(J)>0\) and \(-tr(J)=\Big{(}\frac{\partial f}{\partial x_{1}}+\frac{\partial g}{\partial x_{ 2}}\Big{)}>0\), which are the stabilization points. The Jacobi matrix is  
\[J= \begin{bmatrix}\gamma_{1}\bigg{(}1-\frac{2x_{1}}{K_{1}}-\alpha_{12}\frac{x_ {2}}{K_{2}}+\beta_{12}\frac{x_{2}}{K_{2}}\bigg{)}&\gamma_{1}\frac{x_{1}}{K_{2} }(-\alpha_{12}+\beta_{12})\\ \gamma_{2}\frac{x_{2}}{K_{1}}(-\alpha_{21}+\beta_{21})&\gamma_{2}\bigg{(}1- \frac{2x_{1}}{K_{2}}-\alpha_{21}\frac{x_{1}}{K_{1}}+\beta_{21}\frac{x_{1}}{K_ {1}}\bigg{)}\end{bmatrix}