sR,sP>0s_R,s_P>0sR,sP>0 scale selection strength for RNA and protein respectively;
R,P0\mu_R,\mu_P\ge 0R,P0 represent net mutational/turnover loss rates;
FRF_RFR, FPF_PFP are normalized effective fitness contributions that encode interdependence.
A simple and biologically motivated choice for the coupling that retains nonlinearity is
FR(R,P)=aR+bRP1+P,FP(R,P)=aP+bPR1+R,(2)F_R(R,P) = a_R + b_R\,\frac{P}{1+\kappa P},\qquad F_P(R,P) = a_P + b_P\,\frac{R}{1+\kappa R}, \tag{2}FR(R,P)=aR+bR1+PP,FP(R,P)=aP+bP1+RR,(2)
with constants aR,P0a_{R,P}\ge0aR,P0 (intrinsic baseline function), bR,P0b_{R,P}\ge0bR,P0 (coupling gains), and >0\kappa>0>0 a saturation constant that prevents unbounded gains at high partner abundance (captures diminishing returns / cost). This form models the idea that RNA fitness increases with supportive proteins but saturates; likewise for protein fitness with RNA.
Thus the full reduced system is
R=R(sR(aR+bRP1+P)R),P=P(sP(aP+bPR1+R)P).(3)\begin{aligned} \dot R &= R\Big( s_R\big(a_R + b_R\frac{P}{1+\kappa P}\big) - \mu_R \Big),\\[4pt] \dot P &= P\Big( s_P\big(a_P + b_P\frac{R}{1+\kappa R}\big) - \mu_P \Big). \end{aligned} \tag{3}RP=R(sR(aR+bR1+PP)R),=P(sP(aP+bP1+RR)P).(3)
This two-variable model is sufficient to analyze fixed points, linear stability, and the primary bifurcations that generate attractors and oscillations in the CAS.
2. Fixed points
Fixed points (R,P)(R^*,P^*)(R,P) satisfy R=0, P=0\dot R=0,\ \dot P=0R=0, P=0. Trivial solutions include (0,0)(0,0)(0,0) and boundary solutions where one species is extinct. Nontrivial interior equilibria satisfy
sR(aR+bRP1+P)=R,sP(aP+bPR1+R)=P.(4)s_R\Big(a_R + b_R\frac{P^*}{1+\kappa P^*}\Big) = \mu_R,\qquad s_P\Big(a_P + b_P\frac{R^*}{1+\kappa R^*}\Big) = \mu_P. \tag{4}sR(aR+bR1+PP)=R,sP(aP+bP1+RR)=P.(4)
Each equation can be inverted numerically to yield P(R)P^*(\mu_R)P(R) and R(P)R^*(\mu_P)R(P). Existence of a biologically relevant positive interior fixed point requires the right hand sides be in the achievable ranges of the left hand functions; for instance R<sR(aR+bR)\mu_R < s_R(a_R + b_R)R<sR(aR+bR) is necessary (since P/(1+P)1/P/(1+\kappa P)\le 1/\kappaP/(1+P)1/ if scaled; with our normalization simply P/(1+P)(0,1/)P/(1+\kappa P)\in(0,1/\kappa)P/(1+P)(0,1/) and similarly for the other).
