  Cyclic dominance of three species is a commonly occurring interaction
dynamics, often denoted the rock-paper-scissors (RPS) game. Such type of
interactions is known to promote species coexistence. Here, we generalize
recent results of Reichenbach et al. (e.g. Nature 448, 1046 (2007)) of a
four-state variant of RPS. We show that spiral formation takes place only
without a conservation law for the total density. Nevertheless, in general fast
diffusion can destroy species coexistence. We also generalize the four-state
model to slightly varying reaction rates. This is shown both analytically and
numerically not to change pattern formation, or the effective wave length of
the spirals, and therefore does not alter the qualitative properties of the
cross-over to extinction.
