Abstract
Two-dimensional (2D) hypercycles have been shown to generate spiral patterns, which may protect the hypercycle from parasites that would be fatal to the hypercycle in a homogeneous spatial distribution. We perform numerical experiments on a partial differential equations hypercycle model and show that scroll rings are formed and are not stable: They contract by a power law and disappear within finite time. Similar results are obtained with a 3D cellular automaton hypercycle model. For the 3D hypercycle the final state is homogeneously oscillating, except with initial conditions creating plane waves or 2D spirals. This indicates that the mechanism which may protect the 2D hypercycle from parasites is not applicable to 3D hypercycles. The contraction of the scroll rings is analogous to what has been observed and calculated for other phenomena and models, of which one is the Belousov-Zhabotinsky reaction, described by several mathematical models.
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