## 5 Research on Evolutionary Game
### Evolutionary Stability Strategies (ESS)  
When the evolution is stable, the proportion of passenger group members adopting the induction strategy is \(x^{*}\). Because of the stability of ESS with small fluctuations, there is a neighborhood in which, when \(x<x^{*}\), \(\dfrac{\mathrm{d}x}{\mathrm{d}t}>0\), \(x\) can be further increased to \(x^{*}\) and, when \(x>x^{*}\), \(\dfrac{\mathrm{d}x}{\mathrm{d}t}<0\), \(x\) can be further reduced to \(x^{*}\)[15]. Therefore, when \(x=x^{*}\) the slope of \(\dfrac{\mathrm{d}x}{\mathrm{d}t}\) at \(x^{*}\) is negative; that is, when the proportion of passengers adopting the induction strategy is in a stable state, the conditions of \(\dfrac{\mathrm{d}x}{\mathrm{d}t}=0\) and \(\left(\dfrac{\mathrm{d}x}{\mathrm{d}t}\right)^{\prime}<0\) must be satisfied. The approximate phase diagram of the duplicative dynamic game equation is shown in Figure 2.  
The three intersection points of the phase curve and the horizontal axis in the phase diagram are the steady state points \(x_{1}^{*}\), \(x_{2}^{*}\), and \(x_{3}^{*}\). The determination of the specific location needs to be discussed according to the following three situations: \(x_{3}^{*}<0\), \(0<x_{3}^{*}<1\), and \(x_{3}^{*}>1\). The following results are obtained. 1. When \(\dfrac{q_{k_{2},i-1}^{\mathrm{on}}}{E_{k_{2}}}-\dfrac{q_{k_{1},i-1}^{\mathrm{ on}}}{E_{k_{1}}}<\dfrac{q_{i}^{\mathrm{wait}}}{2E_{k_{1}}}-\dfrac{q_{i}^{ \mathrm{wait}}}{E_{k_{2}}}\), \(x_{3}^{*}<0\) and \(\left(\dfrac{\mathrm{d}x_{1}^{*}}{\mathrm{d}t}\right)^{\prime}<0\), so \(x^{*}=x_{1}^{*}=0\). 2. When \(\dfrac{q_{i}^{\mathrm{wait}}}{2E_{k_{1}}}-\dfrac{q_{i}^{\mathrm{wait}}}{E_{k_{ 2}}}<\dfrac{q_{k_{2},i-1}^{\mathrm{on}}}{E_{k_{2}}}-\dfrac{q_{k_{1},i-1}^{ \mathrm{on}}}{E_{k_{1}}}<\dfrac{q_{i}^{\mathrm{wait}}}{E_{k_{1}}}-\dfrac{q_{i} ^{\mathrm{wait}}}{2E_{k_{2}}}\), \(0<x_{3}^{*}<1\) and \(\left(\dfrac{\mathrm{d}x_{3}^{*}}{\mathrm{d}t}\right)^{\prime}<0\), so \(x^{*}=x_{3}^{*}\). 3. When \(\dfrac{q_{k_{2},i-1}^{\mathrm{on}}}{E_{k_{2}}}-\dfrac{q_{k_{1},i-1}^{\mathrm{ on}}}{E_{k_{1}}}>\dfrac{q_{i}^{\mathrm{wait}}}{E_{k_{1}}}-\dfrac{q_{i}^{ \mathrm{wait}}}{2E_{k_{2}}}\), \(x_{3}^{*}>1\), and \(\left(\dfrac{\mathrm{d}x_{2}^{*}}{\mathrm{d}t}\right)^{\prime}<0\), so \(x^{*}=x_{2}^{*}=1\). In the above results, \(q_{k_{2},i-1}^{\mathrm{on}}/E_{k_{2}}\) and \(q_{k_{2},i-1}^{\mathrm{on}}/E_{k_{2}}\) represent the loading rate of \(k_{1}\) and \(k_{2}\), respectively, at time \(t_{0}\). \(q_{k_{2},i-1}^{\mathrm{on}}/E_{k_{2}}-q_{k_{1},i-1}^{\mathrm{on}}/E_{k_{1}}\) represents the difference between the loading rate of \(k_{2}\) and \(k_{1}\) at time \(t_{0}\). Its meaning can be understood as the difference in the degree of congestion in the two vehicles. The greater the difference, the greater the difference in the degree of congestion, which is represented by \(\Delta C\). The bus system center induces passengers to travel by bus \(k_{1}\), so the condition that \(\Delta C\) is greater than \(0\) should be satisfied. That is, the congestion degree of bus \(k_{1}\) should meet the condition that it is not greater than the congestion degree of \(k_{2}\). Through result (1), it can be determined that, when \(\Delta C<\dfrac{q_{i}^{\mathrm{wait}}}{2E_{k_{1}}}-\dfrac{q_{i}^{ \mathrm{wait}}}{E_{k_{2}}}\), the proportion of passenger group members receiving induction in \(0\) is ESS. In other words, during the period from \(t_{0}\) to \(t_{1}\), with the passage of time, the members of the passenger group will gradually give up choosing bus \(k_{1}\) as their travel plan. Through result (2), it can be found that, when \(\dfrac{q_{i}^{\mathrm{wait}}}{2E_{k_{1}}}-\dfrac{q_{i}^{\mathrm{wait}}}{E_{k_ {2}}}<\Delta C<\dfrac{q_{i}^{\mathrm{wait}}}{E_{k_{1}}}-\dfrac{q_{i}^{\mathrm{ wait}}}{2E_{k_{2}}}\), the proportion of passenger group members receiving induction in \(x_{3}^{*}\) is ESS. In other words,  
Figure 2: Phase diagram of duplicative dynamic equation.