Now the trace T=RsRcRPsPcP<0T = - R^* s_R c_R - P^* s_P c_P <0T=RsRcRPsPcP<0 and determinant \Delta as in (7) (but with the same off-diagonals). The linear stability is determined by the signs of TTT and \Delta:
Stable node/focus if T<0T<0T<0 and >0\Delta>0>0 with discriminant T24>0T^2 - 4\Delta > 0T24>0 (real eigenvalues) or <0<0<0 (complex conjugates with negative real part damped oscillations).
A Hopf bifurcation occurs when a pair of complex conjugate eigenvalues crosses the imaginary axis, i.e. when T=0T=0T=0 while >0\Delta>0>0. Because T<0T<0T<0 generically, varying parameters (e.g., lowering cRc_RcR or cPc_PcP, increasing coupling bR,bPb_R,b_PbR,bP, or changing selection scales sR,Ps_{R,P}sR,P) can push TTT through zero and induce oscillatory instability.
A saddle-node bifurcation (fold) occurs when two equilibria collide and annihilate. In this reduced system, multiplicity of equilibria arises from the nonlinear saturating forms in (4); saddle-node bifurcations occur at parameter values where the implicit curves in (4) are tangent.
4. Conditions for Hopf and saddle-node: biological interpretation
Hopf (oscillatory Red Queen).
From (9) a necessary (not sufficient) condition for a Hopf bifurcation is that the trace become zero:
RsRcR+PsPcP=0(requires changing sign; hence parameter change).R^* s_R c_R + P^* s_P c_P = 0 \quad\Longrightarrow\quad \text{(requires changing sign; hence parameter change).}RsRcR+PsPcP=0(requires changing sign; hence parameter change).
Because R,P,s,c>0R^*,P^*,s_\cdot,c_\cdot>0R,P,s,c>0, trace is normally negative; however effective cRc_RcR or cPc_PcP can be reduced (e.g., by environmental context that reduces cost), or selection strengths sR,Ps_{R,P}sR,P can be increased (stronger dependency), making the trace less negative and eventually zero. Practically this means stronger mutual coupling (large bR,bPb_R,b_PbR,bP, large sss) and weak self-damping can induce sustained oscillations --- the molecular analogue of Red Queen cycles.
Saddle-node (punctuated transitions).
Nonlinear solving of (4) can produce multiple intersections; saddle-node bifurcations occur at parameter values (e.g., mutation rates R,P\mu_R,\mu_PR,P, coupling gains bbb, environmental coefficients in aaa) where the number of interior equilibria changes. Biologically, crossing a saddle-node corresponds to a sudden transition: a previously stable coadapted state disappears, forcing the system to jump to another attractor (punctuated change).
5. Worked numeric illustration
Choose illustrative parameter values (dimensionless units) to demonstrate regimes:
aR=aP=0.01, bR=bP=1.0, =1.0a_R=a_P=0.01,\ b_R=b_P=1.0,\ \kappa=1.0aR=aP=0.01, bR=bP=1.0, =1.0,
cR=cP=0.05, sR=sP=1.0c_R=c_P=0.05,\ s_R=s_P=1.0cR=cP=0.05, sR=sP=1.0,
R=P=0.02\mu_R=\mu_P=0.02R=P=0.02.
