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Abstract:
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Establishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stable attractors governing coexistence and extinction of all hypercycle members. The nature of the coexistence attractor is largely determined by the size of the hypercycle. For instance, for $ 2$ -member hypercycles the coexistence attractor is a stable node. For larger dimensions more complex dynamics appear. Numerical results on so-called elementary hypercycles with $ n=3$ and $ n=4$ species revealed, respectively, coexistence via strongly- and weakly-damped oscillations. Stability conditions for these cases have been provided by linear stability and Lyapunov functions. Typically, linear stability analysis of $ 4$ -member hypercycles indicates two purely imaginary eigenvalues and two negative real eigenvalues. For this case, stability can not be fully characterized linearizing near the fixed point. In this letter we determine the stability of a non-elementary $ 4$ -member hypercycle which considers exponential and hyperbolic replication terms under mutation giving place to an error tail. Since Lyapunov functions are not available for this case, we use the center manifold theory to rigorously show that the system has a stable coexistence fixed point. Our results also show that this fixed point can not undergo a Hopf bifurcation, as supported by numerical simulations \linebreak previously reported. |
