The picture above shows an example of such spirals. The two papers referenced above contain many futher literature references on other systems in which these spirals were observed, and also give pictures of spirals in their systems. It seems that the reason why these spirals are formed up to now is regarded as somewhat of a mystery. I believe that I can explain why these spirals are formed. It seems likely that in the other, more complicated, systems where spiral appear, the same reason is behind spiral formation.
Parasites kill off hosts and reproduce at the expense of hosts, thus parasites ''chase'' the hosts.
Square chessboard, of any size bigger than ca. Each square contains either a host, or a parasite, or is empty. As in very many cellular automata, the ''neighbours'' of a square are defined as the 8 neighbouring squares. As in all cellular automata, the simulation proceeds in turns. Fill the corresponding square in the new situation with a host.