## 4 Model Stability Analysis
### Analysis of System
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Equilibrium Point** & **Eigenvalue \(\mathbf{\lambda_{1}},\mathbf{\lambda_{2}},\mathbf{\lambda_{3}}\)** & **Sign or Condition** & **Stability** \\ \hline \(\mathrm{E_{1}}(0,0,0)\) & \(-C_{\mathrm{co}}+C_{\mathrm{cf}}+C_{\mathrm{cfr}}+C_{\mathrm{cfs}},C_{\mathrm{ p}}-C_{\mathrm{cfr}},-C_{\mathrm{g}}+F_{\mathrm{c}}+F_{\mathrm{c}}\) & \((-,-,+)\) & Unstable \\ \(\mathrm{E_{2}}(1,0,0)\) & \(C_{\mathrm{co}}-C_{\mathrm{cf}}-C_{\mathrm{cfr}}-C_{\mathrm{cfs}},C_{\mathrm{ p}},-C_{\mathrm{g}}+F_{\mathrm{p}}\) & \((+,+,\times)\) & Unstable \\ \(\mathrm{E_{3}}(0,1,0)\) & \(-C_{\mathrm{co}}+C_{\mathrm{cf}}+C_{\mathrm{cfs}}+R_{\mathrm{c}}-C_{\mathrm{ p}}+C_{\mathrm{cfr}},-C_{\mathrm{g}}+F_{\mathrm{c}}\) & \((-,+,\times)\) & Unstable \\ \(\mathrm{M_{p}}\) & \(-C_{\mathrm{co}}+C_{\mathrm{cf}}+C_{\mathrm{cfs}}+F_{\mathrm{c}}+C_{\mathrm{ g}}+F_{\mathrm{c}},C_{\mathrm{p}}-C_{\mathrm{cfr}}+F_{\mathrm{p}}+\) & \((-,-,-)\) & Unstable \\ \(\mathrm{E_{4}}(0,0,1)\) & \(-C_{\mathrm{co}}+C_{\mathrm{cf}}+C_{\mathrm{cfr}}+C_{\mathrm{cfs}}+F_{\mathrm{c }},C_{\mathrm{p}}-C_{\mathrm{cfr}}+F_{\mathrm{p}}+\) & \((-,-,-)\) & ESS \\ \(\mathrm{M_{p}}\), \(C_{\mathrm{g}}-F_{\mathrm{c}}-F_{\mathrm{p}}-T_{\mathrm{g}}\) & Condition 1 & ESS \\ \(\mathrm{E_{5}}(1,1,0)\) & \(C_{\mathrm{co}}-C_{\mathrm{cf}}-C_{\mathrm{cfs}}-R_{\mathrm{c}}-C_{\mathrm{ pg}}-C_{\mathrm{pg}}-M_{\mathrm{p}}\) & \((-,-,-)\) & ESS \\ \(\mathrm{E_{6}}(1,0,1)\) & \(C_{\mathrm{co}}-C_{\mathrm{cf}}-C_{\mathrm{cfr}}-C_{\mathrm{cfs}}-F_{\mathrm{c }}\,C_{\mathrm{p}}+F_{\mathrm{p}}^{\mathrm{p}}+M_{\mathrm{p}}\), \(C_{\mathrm{g}}\) & \((\times,+,\times)\) & Unstable \\ \(-F_{\mathrm{p}}\) & \((\times,+,\times)\) & Unstable \\ \(\mathrm{E_{7}}(0,1,1)\) & \(-C_{\mathrm{co}}+C_{\mathrm{cf}}+C_{\mathrm{cfs}}+F_{\mathrm{c}}+R_{\mathrm{c }}-C_{\mathrm{p}}+C_{\mathrm{cfr}}-F_{\mathrm{p}}\) & \((+,\times,\times)\) & Unstable \\ \(-M_{\mathrm{p}}\), \(C_{\mathrm{g}}-F_{\mathrm{c}}+M_{\mathrm{p}}\) & \((-,+,\times)\) & Unstable \\ \(\mathrm{E_{8}}(1,1,1)\) & \(C_{\mathrm{co}}-C_{\mathrm{cf}}-C_{\mathrm{cfs}}-R_{\mathrm{c}}-F_{\mathrm{c }}-F_{\mathrm{p}}-M_{\mathrm{p}}-C_{\mathrm{p}}\), & \((-,-,+)\) & Unstable \\ \(\mathrm{C_{g}}+M_{\mathrm{p}}\) & \(a_{1,\lambda_{2}}=\lambda_{3}=\) & \((-,0,0)\) & Unstable \\ \(i\) & Condition 2 & Uncertain \\ \(a_{2,\lambda_{2}}=\lambda_{3}=\) & \((-,0,0)\) & Uncertain \\ \(\mathrm{E_{10}}(x_{1},0,z_{2})\) & \(\sqrt{x_{1}(1-x_{1})\,F_{\mathrm{c}}z_{2}(1-z_{2})(F_{\mathrm{c}}+T_{\mathrm{ g}})}*i\) & Condition 3 & Uncertain \\ \(\mathrm{E_{12}}(x_{2},y_{2},0)\) & \(a_{2,\sqrt{x_{2}(1-x_{2})(R_{\mathrm{c}}-C_{\mathrm{cfr}})y_{2}(1-y_{2})C_{ \mathrm{cfr}}}}\) & \((\times,+,-)\) & Unstable \\ \(\mathrm{E_{13}}(x_{3},y_{3},1)\) & \(\sqrt{x_{3}(1-x_{3})(R_{\mathrm{c}}-C_{\mathrm{cfr}})y_{3}(1-y_{3})C_{ \mathrm{cfr}}}\) & Condition 3 & Unstable \\ \hline \hline \end{tabular} Note: \(\times\) indicates that the symbol is undefined; \(x_{1}\), \(x_{2}\), \(x_{3}\), \(y_{1}\), \(y_{2}\), \(y_{3}\), \(z_{1}\), \(z_{2}\) are the coordinates of the corresponding equilibrium points; if conditions are not satisfied, it is unstable or meaningless. Condition 1: \(-C_{\mathrm{co}}+C_{\mathrm{cf}}+C_{\mathrm{cfr}}+C_{\mathrm{cfs}}+F_{\mathrm{c }}<0\), \(F_{\mathrm{cp}}+M_{\mathrm{p}}-C_{\mathrm{cfr}}+C_{\mathrm{p}}<0\), according to Table 3, conditions 2 and 3 are not satisfied, so the equilibrium points \(\mathrm{E_{9}}\) and \(\mathrm{E_{10}}\) are meaningless. Conditions 4 and 5 are met, so equilibrium points \(\mathrm{E_{12}}\) and \(\mathrm{E_{13}}\) are unstable points. Condition 1 is satisfied, so only \(\mathrm{E_{4}}\) and \(\mathrm{E_{5}}\) are asymptotic stable points. \(\square\)  
\end{table} 
Table 4: Stability analysis of equilibrium points. **Proof 8**. : When \(F_{\rm c}>C_{\rm co}-C_{\rm cf}-C_{\rm cfr}-C_{\rm cfs}\) and \(F_{\rm p}+M_{\rm p}>C_{\rm cfr}-C_{\rm p}\), according to Table 3, conditions 1 and 5 are not satisfied, which means that \(\rm E_{4}\) is an unstable point, and \(\rm E_{13}\) is meaningless.