From the model in Section 4.A (notation consistent with that section),
F1(T,E,P)=TSE(E)BL(1T)BPP,F2(T,E,P)=EEE+PPTT,F3(T,E,P)=PP(1PPmax)+ESE(E)TTKK(E,)P.\begin{aligned} F_1(T,E,P) &= \alpha_T S_E(E)\,B_L\,(1-T) - B_P P,\\ F_2(T,E,P) &= \mu_E - \lambda_E E + \phi_P P - \phi_T T,\\ F_3(T,E,P) &= \rho_P P\Big(1-\tfrac{P}{P_{\max}}\Big) + \eta_E S_E(E) - \delta_T T - \kappa_K K(E,\Theta) P. \end{aligned}F1(T,E,P)F2(T,E,P)F3(T,E,P)=TSE(E)BL(1T)BPP,=EEE+PPTT,=PP(1PmaxP)+ESE(E)TTKK(E,)P.
(If your exact expressions for some coefficients differ, replace them symbolically; the structure below is generic.)
Differentiate to obtain the Jacobian entries Jij=Fi/xjJ_{ij} = \partial F_i/\partial x_jJij=Fi/xj evaluated at X=(T,E,P)X^*=(T^*,E^*,P^*)X=(T,E,P).
Calculate the (analytical) partials:
SE(E)S_E(E)SE(E) derivative:
SE(E)=ddEEE+=E1(E+)2.S_E'(E) \;=\; \frac{d}{dE}\frac{E^\kappa}{E^\kappa + \theta^\kappa} \;=\; \frac{\kappa\,\theta^\kappa\,E^{\kappa-1}}{(E^\kappa+\theta^\kappa)^2}.SE(E)=dEdE+E=(E+)2E1.
Entries:
J11=TF1=TSE(E)BL,J12=EF1=TBL(1T)SE(E),J13=PF1=BP,\begin{aligned} J_{11} &= \partial_T F_1 = -\alpha_T\,S_E(E^*)\,B_L,\\[4pt] J_{12} &= \partial_E F_1 = \alpha_T\,B_L\,(1-T^*)\,S_E'(E^*),\\[4pt] J_{13} &= \partial_P F_1 = -B_P, \end{aligned}J11J12J13=TF1=TSE(E)BL,=EF1=TBL(1T)SE(E),=PF1=BP, J21=TF2=T,J22=EF2=E,J23=PF2=P,\begin{aligned} J_{21} &= \partial_T F_2 = -\phi_T,\\[4pt] J_{22} &= \partial_E F_2 = -\lambda_E,\\[4pt] J_{23} &= \partial_P F_2 = \phi_P, \end{aligned}J21J22J23=TF2=T,=EF2=E,=PF2=P, J31=TF3=T,J32=EF3=ESE(E)KPK(E,),J33=PF3=P(12PPmax)KK(E,).\begin{aligned} J_{31} &= \partial_T F_3 = -\delta_T,\\[4pt] J_{32} &= \partial_E F_3 = \eta_E S_E'(E^*) - \kappa_K P^* \, K'(E^*,\Theta),\\[4pt] J_{33} &= \partial_P F_3 = \rho_P\Big(1 - 2\frac{P^*}{P_{\max}}\Big) - \kappa_K K(E^*,\Theta). \end{aligned}J31J32J33=TF3=T,=EF3=ESE(E)KPK(E,),=PF3=P(12PmaxP)KK(E,).
Notes:
If KKK (elite cooptation) does not depend on EEE in your parameterization, then K(E)=0K'(E^*)=0K(E)=0 and J32=ESE(E)J_{32}=\eta_E S_E'(E^*)J32=ESE(E).
All coefficient symbols (e.g. T,E,P,E,T,K,BL,BP\alpha_T,\lambda_E,\rho_P,\eta_E,\delta_T,\kappa_K,B_L,B_PT,E,P,E,T,K,BL,BP) are functions of the leadership vector \Theta via the mappings in Section 2.B--2.C; therefore JJJ is an explicit function of \Theta and XX^*X.
Compactly:
J(X)=(J11J12J13J21J22J23J31J32J33).J(X^*) = \begin{pmatrix} J_{11} & J_{12} & J_{13}\\[4pt] J_{21} & J_{22} & J_{23}\\[4pt] J_{31} & J_{32} & J_{33} \end{pmatrix}.J(X)=J11J21J31J12J22J32J13J23J33.
3. Characteristic polynomial and stability (Routh--Hurwitz)
