Substitute into (9a) to get a single scalar equation in EE^*E (and parameters \Theta):
A2(E)(A1(E)2A2(E))2+A1(E)(A1(E)2A2(E))+A0(E)=0.A_2(E^*)\left(-\frac{A_1(E^*)}{2A_2(E^*)}\right)^2 + A_1(E^*)\left(-\frac{A_1(E^*)}{2A_2(E^*)}\right) + A_0(E^*) = 0.A2(E)(2A2(E)A1(E))2+A1(E)(2A2(E)A1(E))+A0(E)=0.
This simplifies to the discriminant condition:
(E)=A1(E)24A2(E)A0(E)=0.(10)\boxed{ \; \Delta(E^*) \;=\; A_1(E^*)^2 \;-\; 4 A_2(E^*) A_0(E^*) \;=\; 0. \; } \tag{10}(E)=A1(E)24A2(E)A0(E)=0.(10)
Thus the fold locus is given by all EE^*E solving (10). For each such EE^*E we can compute PP^*P via (9b) and TT^*T via (5), and then find E\mu_EE from (6). The resulting E\mu_EE is the candidate crit()\mu_{\mathrm{crit}}(\Theta)crit() where the fold occurs.
Remark: Because AiA_iAi depend on SE(E)S_E(E^*)SE(E), K(E)K(E^*)K(E), etc., (10) is generally a nonlinear equation in EE^*E that must be solved numerically or symbolically (if closed forms are available). But it is explicit and amenable to algebraic manipulation (e.g., resultant elimination) if required.
5. Jacobian at equilibrium --- explicit partial derivatives
To analyze local stability and bifurcation nondegeneracy, compute the Jacobian matrix J=F/XJ=\partial F/\partial XJ=F/X evaluated at the equilibrium X=(T,E,P)X^*=(T^*,E^*,P^*)X=(T,E,P).
From (1)--(3) the Jacobian entries Jij=Xi/XjJ_{ij}=\partial \dot X_i/\partial X_jJij=Xi/Xj are:
J=(TTETPTTEEEPETPEPPP)(T,E,P).J = \begin{pmatrix} \partial_T \dot T & \partial_E \dot T & \partial_P \dot T\\[4pt] \partial_T \dot E & \partial_E \dot E & \partial_P \dot E\\[4pt] \partial_T \dot P & \partial_E \dot P & \partial_P \dot P \end{pmatrix}_{(T^*,E^*,P^*)}.J=TTTETPETEEEPPTPEPP(T,E,P).
Compute each entry:
