0=PPmax(P)2+(PKK+TBPTBLSE(E))P+(ESE(E)T).\begin{aligned} 0 &= -\frac{\rho_P}{P_{\max}}(P^*)^2 \;+\; \left(\rho_P - \kappa_K K + \delta_T \dfrac{B_P}{\alpha_T B_L S_E(E^*)}\right) P^* \;+\; \left(\eta_E S_E(E^*) - \delta_T\right). \end{aligned}0=PmaxP(P)2+(PKK+TTBLSE(E)BP)P+(ESE(E)T).
Define for compactness (all quantities evaluated at EE^*E when relevant):
A2(E)PPmax,A1(E)PKK(E,)+TBPTBLSE(E),A0(E)ESE(E)T.\begin{aligned} A_2(E^*) &\equiv -\frac{\rho_P}{P_{\max}},\\[4pt] A_1(E^*) &\equiv \rho_P - \kappa_K K(E^*,\Theta) + \delta_T \dfrac{B_P}{\alpha_T B_L S_E(E^*)},\\[4pt] A_0(E^*) &\equiv \eta_E S_E(E^*) - \delta_T. \end{aligned}A2(E)A1(E)A0(E)PmaxP,PKK(E,)+TTBLSE(E)BP,ESE(E)T.
Then (7) becomes the scalar quadratic in PP^*P:
A2(E)(P)2+A1(E)P+A0(E)=0.(8)\boxed{ \; A_2(E^*) (P^*)^2 + A_1(E^*) P^* + A_0(E^*) \;=\; 0. \; } \tag{8}A2(E)(P)2+A1(E)P+A0(E)=0.(8)
Interpretation: For each admissible EE^*E, the possible PP^*P satisfy (8). The number of real, nonnegative roots PP^*P of this quadratic (subject to 0PPmax0\le P^*\le P_{\max}0PPmax and that TT^*T from (5) lies in [0,1][0,1][0,1]) determines how many equilibria the full system has for that EE^*E. Multiplicity (double root) occurs when discriminant vanishes --- this is the fold condition.
4. Fold condition (saddle-node) in algebraic form
A saddle-node (fold) occurs when two equilibria coalesce; algebraically this is a double root of (8) with respect to PP^*P for some EE^*E. The double-root condition is:
\begin{aligned} F(P^*,E^*) &\equiv A_2(E^*) (P^*)^2 + A_1(E^*) P^* + A_0(E^*) = 0, \tag{9a}\\[4pt] \frac{\partial F}{\partial P}(P^*,E^*) &\equiv 2 A_2(E^*) P^* + A_1(E^*) = 0. \tag{9b} \end{aligned}
Equations (9a)--(9b) eliminate the PP^*P dependence: solving (9b) gives the candidate double root:
P=A1(E)2A2(E).P^* = -\dfrac{A_1(E^*)}{2 A_2(E^*)}.P=2A2(E)A1(E).
