Table 4 Conditions for evolutionary stable strategies (ESSs) at various equilibrium points.
Equilibrium points | Eigenvalues | Conditions of ESSs |
|---|---|---|
E1(0,0,0) | λ1 = -α·g λ2 = Cr-Ir + 2Pc·Q3 λ3 = Cw-Cg + K-Rg + T·Pc·Q1+θ·Rg | Cr-Ir + 2Pc·Q3 < 0 Cw-Cg + K-Rg + T·Pc·Q1+θ·Rg < 0 |
E2(1,0,0) | λ1 = -α·g λ2 = Cg-Cw-K + Rg-T·Pc·Q1-θ·Rg λ3 = Cr-Ir + K + S + U + 2Pc·Q3 + T·Pc·Q1-T·Pc·Q2 | Cg-Cw-K + Rg-T·Pc·Q1-θ·Rg < 0 Cr-Ir + K + S + U + 2Pc·Q3 + T·Pc·Q1-T·Pc·Q2 < 0 |
E3(0,1,0) | λ1 = -Cr + Ir-2Pc·Q3 λ2 = L1-L2-Rc + Rw-α·g λ3 = Cw-Cg-Rg-S + T·Pc·Q2 + θ·Rg | -Cr + Ir-2Pc·Q3 < 0 L1-L2-Rc + Rw-α·g < 0 Cw-Cg-Rg-S + T·Pc·Q2 + θ·Rg < 0 |
E4(1,1,0) | λ1 = -Cw + Cg + Rg + S-T·Pc·Q2-θ·Rg λ2 = -Cr + Ir-K-S-U-2Pc·Q3-T·Pc·Q1 + T·Pc·Q2 λ3 = (1-α)(L1-L2-Rc + Rw)-α·g | -Cw + Cg + Rg + S-T·Pc·Q2-θ·Rg < 0 -Cr + Ir-K-S-U-2Pc·Q3-T·Pc·Q1 + T·Pc·Q2 < 0 (1-α)(L1-L2-Rc + Rw)-α·g < 0 |
E5(1,0,1) | λ1 = -α·g λ2 = Cg-Cw-K-M + Rg-T·Pc·Q1-θ·Rg λ3 = Cr-Ir + K + Rc-Rw + S + U + 2Pc·Q3 + T·Pc·Q1-T·Pc·Q2 | Cg-Cw-K-M + Rg-T·Pc·Q1-θ·Rg < 0 Cr-Ir + K + Rc-Rw + S + U + 2Pc·Q3 + T·Pc·Q1-T·Pc·Q2 < 0 |
E6(0,1,1) | λ1 = L2-L1 + Rc-Rw-α·g λ2 = Ir-Cr-Rc + Rw-U-2Pc·Q3 λ3 = Cw-Cg + M-Rg-S + T·Pc·Q2 + θ·Rg | L2-L1 + Rc-Rw-α·g < 0 Ir-Cr-Rc + Rw-U-2Pc·Q3 < 0 Cw-Cg + M-Rg-S + T·Pc·Q2 + θ·Rg < 0 |
E7(0,0,1) | λ1 = -α·g λ2 = Cr-Ir + Rc-Rw + U + 2Pc·Q3 λ3 = Cw-Cg + K + M-Rg + T·Pc·Q1 + θ·Rg | Cr-Ir + Rc-Rw + U + 2Pc·Q3 < 0 Cw-Cg + K + M-Rg + T·Pc·Q1 + θ·Rg < 0 |
E8(1,1,1) | λ1 = -Cw + Cg-M + Rg + S-T·Pc·Q2-θ·Rg λ2 = (1-α)(L2-L1 + Rc-Rw)-α·g λ3 = Ir-Cr-K-Rc + Rw-S-U-2Pc·Q3-T·Pc·Q1 + T·Pc·Q2 | -Cw + Cg-M + Rg + S-T·Pc·Q2-θ·Rg < 0 (1-α)(L2-L1 + Rc-Rw)-α·g < 0 Ir-Cr-K-Rc + Rw-S-U-2Pc·Q3-T·Pc·Q1 + T·Pc·Q2 < 0 |
E9(x*,y*,z*) | Saddle point | |
