Solve (4) numerically to find an interior fixed point (R,P)(0.8,0.9)(R^*,P^*)\approx(0.8,0.9)(R,P)(0.8,0.9) (computed from implicit equations). Evaluate Jacobian (9) and compute eigenvalues:
Off-diagonals: J120.811/(1+0.9)20.8/(1.92)0.22J_{12}\approx 0.8\cdot 1\cdot 1/(1+0.9)^2\approx 0.8/(1.9^2)\approx 0.22J120.811/(1+0.9)20.8/(1.92)0.22.
J210.911/(1+0.8)20.9/(1.82)0.28J_{21}\approx 0.9\cdot 1\cdot 1/(1+0.8)^2\approx 0.9/(1.8^2)\approx 0.28J210.911/(1+0.8)20.9/(1.82)0.28.
Diagonals: J11=0.810.05=0.04J_{11}=-0.8\cdot 1\cdot 0.05=-0.04J11=0.810.05=0.04, J22=0.910.05=0.045J_{22}=-0.9\cdot 1\cdot 0.05=-0.045J22=0.910.05=0.045.
So T0.085T\approx -0.085T0.085, (0.04)(0.045)0.220.280.00180.0616=0.0598<0\Delta\approx (-0.04)(-0.045)-0.22\cdot0.28 \approx 0.0018 - 0.0616 = -0.0598 <0(0.04)(0.045)0.220.280.00180.0616=0.0598<0. Negative determinant indicates a saddle (one positive eigenvalue) --- unstable interior equilibrium. By 8varying bR,bPb_R,b_PbR,bP or decreasing costs ccc, \Delta can become positive and TTT can cross zero, generating a Hopf.
This numeric shows how modest parameter changes (e.g., increasing coupling strength bbb or lowering cost ccc) can qualitatively change stability: from unstable saddle (no coadapted persistence) stable node (coadapted attractor) Hopf oscillation (sustained coevolutionary cycles). This is exactly the sort of bifurcation structure that maps to punctuated vs oscillatory molecular evolutionary dynamics.
6. Takeaways
a. The reduced CAS model exhibits multistability, sudden transitions (saddle-node), and oscillatory coevolution (Hopf) depending on biologically interpretable parameters: coupling strengths bR,Pb_{R,P}bR,P, selection scales sR,Ps_{R,P}sR,P, cost/self-damping cR,Pc_{R,P}cR,P, and mutation/turnover R,P\mu_{R,P}R,P.
b. Biological interpretation.
Strong coupling + low self-cost stable coadapted attractor (ribosome-like complex).
Strong coupling + moderate damping sustained Red Queen oscillations (ongoing molecular chase).
Varying mutation rates or demographic events (bottlenecks) change parameter landscapes and can trigger saddle-node transitions punctuated emergent innovation.
c. The analytic conditions derived from the Jacobian give clear, testable predictions for simulation (where full genotype spaces are modelled) and for empirical data: e.g., parameter regimes that produce long-lived attractors should show persistent covariation between RNA motifs and protein domains; regimes producing oscillations should show time-series covariance and periodic co-selection signatures.
B. Simulation results: synchronization and Red Queen cycles
To complement the analytical findings, we conducted numerical simulations of the coupled RNA--protein CAS model across a range of coupling strengths. These simulations reveal three qualitatively distinct dynamical regimes, corresponding to collapse, oscillatory Red Queen cycles, and stable coadapted attractors.
