## 3 Evolutionary Game Model for Carbon Emission Reduction
### Stability Analysis on System Equilibrium Point  
The equilibrium points that were obtained from the dynamic replicator system are not necessarily the ESS in the system. According to Lyapunov stability, which ensures the stability of an equilibrium point by observing that solutions starting out near this equilibrium point stay near the equilibrium point forever [42; 43], we judge their evolutionary stability strategy by analyzing the Jacobian matrix. Only the status in which the Jacobian determinant of a matrix \(\det(J)\) in the system is positive and trace \(tr(J)\) is negative means the equilibrium points the ESS. The \(\det(J)\) and \(tr(J)\) are shown as follows.  
\[J=\left[\begin{array}{c}\partial\dot{x}/\partial x,\,\partial\dot{x}/ \partial y\\ \partial\dot{y}/\partial x,\partial\dot{y}/\partial y\end{array}\right]=\left[ \begin{array}{c}(1-2x)[u_{M}(q_{1}-q_{2}-q_{3})y+u_{M}q_{2}-c_{M}]x(1-x)u_{ M}(q_{1}-q_{2}-q_{3})\\ y(1-y)u_{R}(q_{1}-q_{2}-q_{3})(1-2y)[u_{R}(q_{1}-q_{2}-q_{3})x+u_{R}q_{3}-c_{R}] \end{array}\right] \tag{11}\]  
\[\det(J)= (1-2x)(1-2y)[u_{M}(q_{1}-q_{2}-q_{3})y+u_{M}q_{2}-c_{M}][u_{R}(q_{1}-q_{2 }-q_{3})x+u_{R}q_{3}-c_{R}] \tag{12}\] \[-x(1-x)y(1-y)u_{M}u_{R}(q_{1}-q_{2}-q_{3})^{2}\] \[tr(J)=(1-2x)[u_{M}(q_{1}-q_{2}-q_{3})y+u_{M}q_{2}-c_{M}]+(1-2y)[u _{R}(q_{1}-q_{2}-q_{3})x+u_{R}q_{3}-c_{R}] \tag{13}\]  
**Proposition 2**: _For these points of evolution stability strategy, there exist the following results as well as preconditions._  
1. \(E_{1}(0,0)\) is the evolution stability strategy (ESS) of dynamic replicator system, if (i) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\), or (ii) \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}\) and \(u_{M}q_{1}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\) or (iii) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}<u_{R}q_{2}+c_{R}<u_{R}q_{1}\);
2. \(E_{2}(1,0)\) is the ESS of dynamic replicator system, if (i) \(c_{M}<u_{M}q_{2}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\), or (ii) \(c_{M}<u_{M}q_{2}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(c_{R}<u_{R}q_{3}<u_{R}q_{1}<u_{R}q_{2}+c_{R}\) or (iii) \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\);
3. \(E_{3}(0,1)\) is the ESS of dynamic replicator system, if (i) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\), or (ii) \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\), or (ii) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(c_{R}<u_{R}q_{3}<u_{R}q_{1}<u_{R}q_{2}+c_{R}\);
4. \(E_{4}(1,1)\) is the ESS of dynamic replicator system, if (i) \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\), or (ii) \(u_{M}q_{2}>c_{M}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\), \(u_{R}q_{2}+c_{R}<u_{R}q_{1}\), or (iii) \(u_{M}q_{2}>c_{M}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\);
5. \(E_{1}(0,0)\) and \(E_{4}(1,1)\) are the ESS of the dynamic replicator system, if \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}<c_{R}<u_{R}q_{2}+c_{R}<u_{R}q_{1}\);
6. \(E_{2}(1,0)\) and \(E_{3}(0,1)\) are the ESS of the dynamic replicator system, if \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(c_{R}<u_{R}q_{3}<u_{R}q_{1}<u_{R}q_{2}+c_{R}\).  
The proof is shown in the Appendix A.  
Proposition 2 points out the prerequisites under which the stability of five equilibrium points holds and these points become the evolutionary stable strategies. The aforementioned example of equilibrium point \(E_{1}(0,0)\) still holds in case of those conditions given, which is elaborated in subsequent analysis.