Multiple interior equilibria (multistability) can arise because the left-hand side of (4) is nonlinear and saturating; intersections of the two implicit curves may produce 0, 1 or several positive solutions.
3. Linear stability --- Jacobian and eigenvalues
Linearize (3) about an equilibrium (R,P)(R^*,P^*)(R,P). Compute partial derivatives:
RR=(sRFR(R,P)R)+RsRFRR(R,P),PR=RsRFRP(R,P),RP=PsPFPR(R,P),PP=(sPFP(R,P)P)+PsPFPP(R,P).\begin{aligned} \partial_R \dot R &= \Big( s_R F_R(R^*,P^*) - \mu_R\Big) + R^* s_R \frac{\partial F_R}{\partial R}\Big|_{(R^*,P^*)},\\[4pt] \partial_P \dot R &= R^* s_R \frac{\partial F_R}{\partial P}\Big|_{(R^*,P^*)},\\[6pt] \partial_R \dot P &= P^* s_P \frac{\partial F_P}{\partial R}\Big|_{(R^*,P^*)},\\[4pt] \partial_P \dot P &= \Big( s_P F_P(R^*,P^*) - \mu_P\Big) + P^* s_P \frac{\partial F_P}{\partial P}\Big|_{(R^*,P^*)}. \end{aligned}RRPRRPPP=(sRFR(R,P)R)+RsRRFR(R,P),=RsRPFR(R,P),=PsPRFP(R,P),=(sPFP(R,P)P)+PsPPFP(R,P).
At an interior fixed point the first parentheses vanish by (4), simplifying the Jacobian JJJ to
J=(RsRFR,RRsRFR,PPsPFP,RPsPFP,P),(5)J = \begin{pmatrix} R^* s_R F_{R,R} & R^* s_R F_{R,P}\\[6pt] P^* s_P F_{P,R} & P^* s_P F_{P,P} \end{pmatrix}, \tag{5}J=(RsRFR,RPsPFP,RRsRFR,PPsPFP,P),(5)
where FR,R=RFRF_{R,R}=\partial_R F_RFR,R=RFR, FR,P=PFRF_{R,P}=\partial_P F_RFR,P=PFR, etc., evaluated at (R,P)(R^*,P^*)(R,P).
For our choice (2) the derivatives are
P(P1+P)=1(1+P)2,R(R1+R)=1(1+R)2,\frac{\partial}{\partial P}\!\Big(\frac{P}{1+\kappa P}\Big) \;=\; \frac{1}{(1+\kappa P)^2},\qquad \frac{\partial}{\partial R}\!\Big(\frac{R}{1+\kappa R}\Big) \;=\; \frac{1}{(1+\kappa R)^2},P(1+PP)=(1+P)21,R(1+RR)=(1+R)21,
and cross-derivatives with respect to the partner variable are zero for the self terms. Hence
FR,P=bR1(1+P)2,FR,R=0,FP,R=bP1(1+R)2,FP,P=0.\begin{aligned} F_{R,P} &= b_R\frac{1}{(1+\kappa P^*)^2},\qquad F_{R,R}=0,\\[4pt] F_{P,R} &= b_P\frac{1}{(1+\kappa R^*)^2},\qquad F_{P,P}=0. \end{aligned}FR,PFP,R=bR(1+P)21,FR,R=0,=bP(1+R)21,FP,P=0.
