Dynamics and transitions in symmetric hypercycles: the role of the hypercycle size

Abstract

The symmetric hypercycle with error tail has been widely studied both numerically and analytically in the last years, but there are still many open questions. In this work we will try to give an answer for a few of these questions. Namely we reproduce in higher dimensions a study of the possible coincidence of saddle-node bifurcations for equilibrium points and periodic orbits already done for n=5, we finish the classification of the stability of equilibrium points in any dimension (the case n=4 was still not classified as far as we know) and we try to give a more clear idea of the behaviour of the system in low dimensions (n=2 and n=3). We also reproduce other important results concerning important properties of the hypercycle and give the necessary background in dynamical systems to understand these results. Finally we contrast the ODE model with a completely different approach to model hypercycles which consists in using cellular automata.