  We consider a two-dimensional model of three species in rock-paper-scissors
competition and study the self-organisation of the population into fascinating
spiraling patterns. Within our individual-based metapopulation formulation, the
population composition changes due to cyclic dominance (dominance-removal and
dominance-replacement), mutations, and pair-exchange of neighboring
individuals. Here, we study the influence of mobility on the emerging patterns
and investigate when the pair-exchange rate is responsible for spiral waves to
become elusive in stochastic lattice simulations. In particular, we show that
the spiral waves predicted by the system's deterministic partial equations are
found in lattice simulations only within a finite range of the mobility rate.
We also report that in the absence of mutations and dominance-replacement, the
resulting spiraling patterns are subject to convective instability and
far-field breakup at low mobility rate. Possible applications of these
resolution and far-field breakup phenomena are discussed.
