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Bifurcations analysis of oscillating hypercycles.

Antoni Guillamon1, Ernest Fontich2, Josep Sardanyés3

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Summary

Hypercycle extinction dynamics are revealed through a saddle-node bifurcation. A critical fidelity threshold (QPO) triggers extinction, with a surprising parameter gap where oscillations cease but instability persists.

Keywords:
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Area of Science:

  • Theoretical Ecology
  • Evolutionary Dynamics
  • Complex Systems

Background:

  • Hypercycles exhibit complex dynamics, including stable periodic orbits for many species (n>4).
  • Extinction has been linked to a replication quality factor (QSS) where unstable equilibria collide.
  • A bistability regime exists where both oscillations and extinction are possible.

Purpose of the Study:

  • To analyze hypercycle dynamics in the bistability regime using a Poincaré map.
  • To identify the mechanism causing the vanishing of oscillatory dynamics.
  • To characterize the transition to extinction as a function of the replication quality factor (Q).

Main Methods:

  • Utilized a Poincaré map to study hypercycle system dynamics.
  • Identified stable and unstable periodic orbits.
  • Analyzed the system's behavior near a critical fidelity threshold (QPO).

Main Results:

  • Discovered a saddle-node bifurcation of periodic orbits at QPO, leading to extinction.
  • The origin becomes a globally stable attractor for Q < QPO.
  • A parameter gap exists where QPO > QSS, with vanished oscillations but present unstable equilibria.
  • Identified a degenerate bifurcation of unstable periodic orbits at Q=1.

Conclusions:

  • Oscillatory dynamics in hypercycles vanish via a saddle-node bifurcation at QPO.
  • Extinction dynamics exhibit a periodic remnant, creating an oscillating delayed transition.
  • The identified parameter gap (QPO > QSS) highlights novel extinction pathways in hypercyclic systems.