## 2 Lotka-Volterra Based Modeling of Competing Relationships
### Analysis of Multi-Mode Bus Competition
\begin{array}{l}\frac{dx_{1}}{dt}=x_{1}r_{1}\left(1-\frac{\alpha_{11 }x_{1}}{K_{1}}-\frac{\alpha_{21}x_{2}}{K_{2}}-\frac{\alpha_{31}x_{3}}{K_{3}} \right)\\ \frac{dx_{2}}{dt}=x_{2}r_{2}\left(1-\frac{\alpha_{12}x_{1}}{K_{1}}-\frac{ \alpha_{22}x_{2}}{K_{2}}-\frac{\alpha_{32}x_{3}}{K_{3}}\right)\\ \frac{dx_{3}}{dt}=x_{3}r_{3}\left(1-\frac{\alpha_{13}x_{1}}{K_{1}}-\frac{ \alpha_{23}x_{2}}{K_{2}}-\frac{\alpha_{33}x_{3}}{K_{3}}\right)\end{array} \right. \tag{5}\]  
where \(\frac{\alpha_{ij}}{K_{i}}\) is the competition coefficient of system \(i\) to system \(j\), when \(i=j\) represents the mode's own blocking coefficient; \(\frac{dx_{1}}{dt}\), \(\frac{dx_{2}}{dt}\), \(\frac{dx_{3}}{dt}\), \(K_{1}K_{2}K_{3}\) are the growth rate of the travel volume of the three travel modes of conventional buses, rail transit, and cabs collectively; and \(K_{1}\), \(K_{2}\), \(K_{3}\) are the scale threshold of the conventional buses, rail transit, and cabs' traffic volume. For solving the parameters of the Lotka-Volterra model, the related literature uses the nonlinear least squares method to estimate the model parameters, but this method is sensitive to the initial values of the parameters. Considering the first-order linear differential equation properties and data availability of the Lotka-Volterra competing model constructed in this paper which considers the urban transit subsystems in three cold regions, this thesis introduces the idea of gray direct modeling in the gray system model and discretizes the model, on the basis of which the model parameters are solved. The model is first simplified by applying the following transformations to the model parameters:  
\[\left\{ \begin{array}{l}a_{1}=r_{1},b_{1}=\frac{\alpha_{11}r_{1}}{K_{1}},c_{1 }=\frac{\alpha_{21}r_{1}}{K_{2}},d_{1}=\frac{\alpha_{31}r_{1}}{K_{3}}\\ a_{2}=r_{2},b_{2}=\frac{\alpha_{12}r_{2}}{K_{1}},c_{2}=\frac{\alpha_{22}r_{2}} {K_{2}},d_{2}=\frac{\alpha_{32}r_{2}}{K_{3}}\\ a_{3}=r_{3},b_{3}=\frac{\alpha_{13}r_{3}}{K_{1}},c_{3}=\frac{\alpha_{23}r_{3}} {K_{2}},d_{3}=\frac{\alpha_{33}r_{3}}{K_{3}}\\ \end{array} \right. \tag{6}\]  
After parametric transformations, the Lotka-Volterra model constructed above is transformed as follows:  
\[\left\{ \begin{array}{l}\frac{dx_{1}}{dt}=a_{1}x_{1}-b_{1}{x_{1}}^{2}-c_{1}x_ {1}x_{2}-d_{1}x_{1}x_{3}\\ \frac{dx_{2}}{dt}=a_{2}x_{2}-b_{2}x_{1}x_{2}-c_{2}{x_{2}}^{2}-d_{2}x_{2}x_{3}\\ \frac{dx_{3}}{dt}=a_{3}x_{3}-b_{3}x_{1}x_{3}-c_{3}x_{2}x_{3}-d_{3}{x_{3}}^{2} \\ \end{array} \right. \tag{7}\]  
As a result, the previous solution for \(\alpha_{ij},K_{i},r_{i}\) is simplified to the solution for \(a_{i}\), \(b_{i}\), \(c_{i}\), and \(d_{i}\). On this basis, based on the gray direct modeling method, the first-order linear differential equation of the model is discretized, and the model is transformed into the first differential equation of the model with \(x_{i}(t)-x_{i}(t-1)\) as the whitening value of \(\frac{dx_{i}}{dt}\) and \(\frac{[x_{i}(t)+x_{i}(t-1)]}{2}\) as the background value of \(\frac{dx_{i}}{dt}\) as follows:  
\[ \begin{array}{l}x_{1}(t)-x_{1}(t-1)=a_{1}(\frac{x_{1}(t)+x_{1}(t-1)}{2})-b _{1}(\frac{x_{1}(t)+x_{1}(t-1)}{2})^{2}-c_{1}(\frac{x_{1}(t)+x_{1}(t-1)}{2})( \frac{x_{2}(t)+x_{2}(t-1)}{2})-\\ d_{1}(\frac{x_{1}(t)+x_{1}(t-1)}{2})(\frac{x_{3}(t)+x_{3}(t-1)}{2}),t=2,3, \ldots n\\ \end{array}  \tag{8}\]  
Bringing the data at time \(t=2,3,\ldots,n\) into the above equation yields an n-dimensional system of linear equations:  
\[\left\{