Table 3 Stability Analysis of the Equilibrium Point.

From: Authentic or spurious: an evolutionary game analysis of manufacturing enterprises’ greenwashing behavior under dynamic government regulation

Equilibrium points

Eigenvalues of the Jacobian matrix

Stability conditions

(0, 0, 0)

\(\begin{array}{c}{\lambda }_{1}={C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+\alpha {R}_{1}+\alpha {R}_{2}\\ {\lambda }_{2}=-{C}_{4} < 0\\ {\lambda }_{3}=M-I-{C}_{5}\end{array}\)

When \(\begin{array}{c}\alpha \,<\, \frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}}{{R}_{1}+{R}_{2}}\\ {C}_{5} \,>\, M-I\end{array}\), ESS

(0, 1, 0)

\(\begin{array}{c}{\lambda }_{1}={C}_{2}-{C}_{1}+P+Q+{R}_{1}-{R}_{2}+\alpha {R}_{1}+\alpha {R}_{2}\\ {\lambda }_{2}={C}_{4} > 0\\ {\lambda }_{3}=H-{C}_{5}+M\end{array}\)

Instability

(0, 0, 1)

\(\begin{array}{c}{\lambda }_{1}={C}_{2}-{C}_{1}+L+M+{R}_{1}-{R}_{2}+\alpha {R}_{1}+\alpha {R}_{2}\\ {\lambda }_{2}=F-{C}_{4}+G\\ {\lambda }_{3}={C}_{5}+I-M\end{array}\)

When \(\begin{array}{c}\alpha \,<\, \frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-M-L}{{R}_{1}+{R}_{2}},\\ C4 \,>\, F+G,C5 \,<\, M-I\end{array}\),ESS

(0, 1, 1)

\(\begin{array}{c}{\lambda }_{1}={C}_{2}-{C}_{1}+L+M+P+\\ Q+{R}_{1}-{R}_{2}+\alpha {R}_{1}+\alpha {R}_{2}\\ {\lambda }_{2}={C}_{4}-F-G\\ {\lambda }_{3}={C}_{5}-H-M\end{array}\)

When \(\begin{array}{c}\alpha \,<\, \frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-L-M-P-Q}{{R}_{1}+{R}_{2}},\\ C4 \,<\, F+G,C5 \,<\, H+M\end{array}\),ESS

(1, 0, 0)

\(\begin{array}{c}{\lambda }_{1}={C}_{1}-{C}_{2}-{R}_{1}+{R}_{2}-\alpha {R}_{1}-\alpha {R}_{2}\\ {\lambda }_{2}=J-{C}_{4}-P\\ {\lambda }_{3}=-{C}_{5}-I-L < 0\end{array}\)

When \(\begin{array}{c}\alpha \,>\, \frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}}{{R}_{1}+{R}_{2}},\\ C4 \,>\, J-P\end{array}\), ESS

(1, 1, 0)

\(\begin{array}{c}{\lambda }_{1}={C}_{1}-{C}_{2}-P-Q-{R}_{1}+{R}_{2}-\alpha {R}_{1}-\alpha {R}_{2}\\ {\lambda }_{2}={C}_{4}+P-J\\ {\lambda }_{3}=H-{C}_{5}-L\end{array}\)

When \(\begin{array}{c}\alpha \,>\, \frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-P-Q}{{R}_{1}+{R}_{2}},\\ {C}_{4} \,<\, J-P,{C}_{5} \,>\, H-L\end{array}\), ESS

(1, 0, 1)

\(\begin{array}{c}{\lambda }_{1}={C}_{1}-{C}_{2}-L-M-{R}_{1}+{R}_{2}-\alpha {R}_{1}-\alpha {R}_{2}\\ {\lambda }_{2}=F-{C}_{4}+G-P+J\\ {\lambda }_{3}={C}_{5}+I+L > 0\end{array}\)

Instability

(1, 1, 1)

\(\begin{array}{c}{\lambda }_{1}={C}_{1}-{C}_{2}-L-M-P-\\ Q-{R}_{1}+{R}_{2}-\alpha {R}_{1}-\alpha {R}_{2}\\ {\lambda }_{2}={C}_{4}-F-G+P-J\\ {\lambda }_{3}={C}_{5}-H+L\end{array}\)

When \(\begin{array}{c}\alpha \,>\, \frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-L-M-P-Q}{{R}_{1}+{R}_{2}},\\ {C}_{4} \,<\, F+G+J-P,{C}_{5} \,<\, H-L\end{array}\),ESS

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