The behavior of hypercycle spirals in a two-dimensional cellular
automaton
model is analyzed. Each of these
spirals can be approximated by an Archimedean spiral and the center,
width, and phase of the hypercycle spiral change according to
Brownian motion. A barrier exists between two spirals if the phase
synchronization hypothesis is taken into account, and the occurrence
rate of pair decay (the simultaneous disappearance of two spirals)
can be explained by the random walk simulation with the barrier.
Adjacent species violation is shown to be a necessary condition for
creating new spirals. A hypercycle system can live long if the
spirals in the system are somewhat unstable, since new spirals
cannot emerge when existing spirals are too stable.