Bifurcations analysis of oscillating hypercycles.

We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species (n>4) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium points collide (named QSS). It has also been reported that, for values below QSS, the system goes to extinction. In this paper, we use a suitable Poincaré map associated to the hypercycle system to analyze the dynamics in the bistability regime, where both oscillatory dynamics and extinction are possible. The stable periodic orbit is identified, together with an unstable periodic orbit. In particular, we are able to unveil the vanishing mechanism of the oscillatory dynamics: a saddle-node bifurcation of periodic orbits as the replication quality factor, Q, undergoes a critical fidelity threshold, QPO. The identified bifurcation involves the asymptotic extinction of all hypercycle members, since the attractor placed at the origin becomes globally stable for values Q<QPO. Near the bifurcation, these extinction dynamics display a periodic remnant that provides the system with an oscillating delayed transition. Surprisingly, we found that the value of QPO is slightly higher than QSS, thus identifying a gap in the parameter space where the oscillatory dynamics has vanished while the unstable equilibrium points are still present. We also identified a degenerate bifurcation of the unstable periodic orbits for Q=1.