Using (5):
EEE+PPT(1BPTBLSE(E)P)=0.\mu_E - \lambda_E E^* + \phi_P P^* - \phi_T\left(1 - \dfrac{B_P}{\alpha_T B_L S_E(E^*)} P^*\right) = 0.EEE+PPT(1TBLSE(E)BPP)=0.
Rearrange to isolate E\mu_EE:
E=EEPP+TTBPTBLSE(E)P(6)\boxed{ \; \mu_E \;=\; \lambda_E E^* \;-\; \phi_P P^* \;+\; \phi_T \;-\; \phi_T \dfrac{B_P}{\alpha_T B_L S_E(E^*)} P^* \; } \tag{6}E=EEPP+TTTBLSE(E)BPP(6)
Equation (6) will be our expression for E\mu_EE at equilibrium as a function of (E,P)(E^*,P^*)(E,P) and parameters \Theta. In bifurcation analysis E\mu_EE is the primary control parameter; this equation describes the locus of equilibria in (E,P,E)(E,P,\mu_E)(E,P,E) space.
(b) Substitute into protest steady state (4c)
Plug (5) into (4c):
0=PP(1PPmax)+ESE(E)T(1BPTBLSE(E)P)KK(E,)P.0 = \rho_P P^*\Big(1-\frac{P^*}{P_{\max}}\Big) + \eta_E S_E(E^*) - \delta_T\left(1 - \frac{B_P}{\alpha_T B_L S_E(E^*)} P^*\right) - \kappa_K K(E^*,\Theta) P^*.0=PP(1PmaxP)+ESE(E)T(1TBLSE(E)BPP)KK(E,)P.
Collect terms to make a relationship between PP^*P and EE^*E. Write it as:
PP(1PPmax)intrinsic logisticKK(E,)Pcooptation damping+ESE(E)T+TBPTBLSE(E)P=0.(7)\underbrace{\rho_P P^*\Big(1-\frac{P^*}{P_{\max}}\Big)}_{\text{intrinsic logistic}} \; - \; \underbrace{\kappa_K K(E^*,\Theta) P^*}_{\text{cooptation damping}} \; + \; \eta_E S_E(E^*) \; - \; \delta_T \; + \; \delta_T \dfrac{B_P}{\alpha_T B_L S_E(E^*)} P^* \;=\; 0. \tag{7}intrinsic logisticPP(1PmaxP)cooptation dampingKK(E,)P+ESE(E)T+TTBLSE(E)BPP=0.(7)
Rearrange grouping terms in powers of PP^*P:
