. 2023 Aug 1;9(8):e18850. doi: 10.1016/j.heliyon.2023.e18850

Table 2.

Income matrix under corresponding strategies.

Equilibrium point	Jacobian matrix eigenvalues λ1; λ2; λ3	Real part notation	Stability condition
Case 1 (0, 0, 0)	$C_{m} - C_{i} - β P_{f 2} - R_{m}$ ; $C_{g 2} - C_{g 1} - α S_{f 1} + S_{f 2} - S_{c}$ ; $- L_{e}$	(-, -, -)	stable point
Case 2 (1, 0, 0)	$C_{i} - C_{m} + {β P}_{f 2} + R_{m}$ ; $C_{g 2} - C_{g 1} - {β P}_{f 2} - α S_{f 1} + S_{f 2} - S_{c}$ ; $C_{e} - β P_{e 2} - R_{i} + T_{e} - U_{c} + U_{i}$ ;	(+, -, -)	unstable point
Case 3 (0, 1, 0)	$C_{m} - C_{i} - R_{m}$ ; $C_{g 1} - C_{g 2} + {α S}_{f 1} - S_{f 2} + S_{c}$ ; $- S_{c}$	(-, -, -)	stable point
Case 4 (0, 0, 1)	$C_{m} - C_{i} - β P_{f 2} + R_{i} - R_{m} + T_{f}$ ; $C_{g 2} - C_{g 1} - {α S}_{f 1} + S_{f 2}$ ; $L_{e}$	(-, -, +)	unstable point
Case 5 (1, 1, 0)	$C_{i} - C_{m} + R_{m}$ ; $C_{g 1} - C_{g 2} + β P_{f 2} + α S_{f 1} - S_{f 2} + S_{c}$ ; $C_{e} - P_{e 1} - R_{i} - S_{c} + T_{e} - U_{c} + U_{i}$	(+, -, -)	unstable point
Case 6 (1, 0, 1)	$C_{i} - C_{m} + {β P}_{f 2} - R_{i} + R_{m} - T_{f}$ ; $C_{g 2} - C_{g 1} + P_{f 1} + P_{e 1} - β P_{e 2} - {β P}_{f 2} - α S_{f 1} + S_{f 2}$ ; ${β P}_{e 2} - C_{e} + R_{i} - T_{e} + U_{c} - U_{i}$	(+, +, +)	unstable point
Case 7 (0, 1, 1)	$C_{m} - C_{i} - P_{f 1} + R_{i} - R_{m} + T_{f}$ ; $C_{g 1} - C_{g 2} + α S_{f 1} - S_{f 2}$ $S_{c}$	(-,+, +)	unstable point
Case 8 (1, 1, 1)	$C_{i} - C_{m} + P_{f 1} - R_{i} + R_{m} - T_{f}$ ; $C_{g 1} - C_{g 2} - P_{f 1} - P_{e 1} + β P_{e 2} + {β P}_{f 2} + α S_{f 1} - S_{f 2}$ ; $β P_{e 2} - C_{e} + R_{i} - T_{e} + S_{c} + U_{c} - U_{i}$	(-, +, +)	unstable point