# Exploring and Modeling the Incentive Strategies of New Energy Ship Application in Yangtze River
## 3 Model Analysis
### Analysis of the Evolution and Stability Strategies of Shipping Enterprises and Governments  
The evolutionary stability of shipping enterprises and governments can be analyzed by the local stable stability of the Jacobian matrix of evolved systems. The Jacobian matrix is as follows:  
\[J= \begin{bmatrix}\frac{\partial F(x)}{\partial x}&\frac{\partial F(x)}{ \partial y}\\ \frac{\partial F(y)}{\partial x}&\frac{\partial F(y)}{\partial y}\end{bmatrix} = \begin{pmatrix}\text{a}_{11}&\text{a}_{12}\\ \text{a}_{21}&\text{a}_{22}\end{pmatrix}  \tag{11}\]  
With  
\[a_{11}=(1-2x)\{y[g(m)s+T_{1}+T_{2}]+(B_{1}-C_{1}-T-U_{0})-(B_{2}-C_{2}-T)\} \tag{12}\]  
\[a_{12}=x(1-x)[g(m)s+T_{1}+T_{2}] \tag{13}\]  
\[a_{21}=y(1-y)[R_{2}-T_{1}-T_{2}-g(m)s] \tag{14}\]  
\[a_{22}=(1-2y)\{x[R_{2}-T_{1}-T_{2}-g(m)s]+T_{2}\} \tag{15}\]  
Then, Det(J) and Tr(J) are, respectively:  
\[Det(J)=(1-2x)\{y[g(m)s+T_{1}+T_{2}]+(B_{1}-C_{1}-T-U_{0})-(B_{2}-C_{2}-T)\}\{ x[R_{2}-T_{1}-T_{2}-g(m)s]+T_{2}\} \tag{16}\]  
\[\text{tr}(J)=(1-2x)\{y[g(m)s+T_{1}+T_{2}]+(B_{1}-C_{1}-T-U_{0})-(B_{2}-C_{2}-T )\}+(1-2y)\{x[R_{2}-T_{1}-T_{2}-g(m)s]+T_{2}\} \tag{17}\]  
If the equilibrium point satisfies Det(J) \(>0\) and tr(J) \(<0\), it indicates that the equilibrium point is an evolutionary stability strategy of the system [41; 42]. Its evolutionary equilibrium strategy (ESS) is shown in Table 3.  
Figure 3: Phase diagrams for government strategies. **Inference 1. **_From Table 2 condition 2 can be seen, when \(B_{1}-C_{1}-U_{0}>B_{2}-C_{2}\), \(x=1\). This indicates that when the net benefit gained by the enterprise from choosing NESs is greater than that from choosing TFSs, the enterprise's focus is on upgrading the green level and reducing the cost, and the government's strategic choices will not have an impact on the enterprise's behavior, and the enterprise always tends to choose new energy ship application; when the government subsidizes and the net benefit gained by the government is positive, the government chooses to subsidize the enterprise, while if the government's net benefit is negative, and then it does not subsidize. In order for the government to subsidize, it needs to satisfy \(R_{2}>g(m)s+T_{1}\), where \(R_{2}\) depends on the government's expectations, and the higher the government's expectation of new energy ship applications, the larger \(R_{2}\) is; at the same time, the decrease in \(g(m)s\) and \(T_{1}\) will affect the enterprise's choice, which, in turn, will affect the application of NESs; thus, the government needs to subsidize the enterprise's development of NESs and give full play to the positive effect of the government's subsidy on NESs._  
**Inference 2. **_From Table 2, condition 3 and condition 4 can be seen, when \(B_{1}-C_{1}+g(m)s-(T-T_{1})-U_{0}<B_{2}-B_{2}-(T+T_{2})\), (0,1) is the only evolutionarily stable strategy; that is, at this time regardless of the size of the \(R_{2}\) and \(g(m)s+T_{1}\) comparison results, the government will choose to subsidize the enterprise; but at this time, the enterprise's behavior is affected by the joint impact of costs and benefits, and the enterprise's choice of new energy ship applications under the government subsidies to obtain the net benefit is still less than the choice of traditional energy ships, and the enterprise will also choose traditional energy ships even if the government subsidizes them. The main reason for this phenomenon is that the government subsidy cannot make up for the increased cost of the enterprise choosing new energy ship applications, which makes the strategy of the enterprise and the government tend to the equilibrium point._  
**Inference 3.