  Cyclic dominance is frequently believed to be a mechanism that maintains
diversity of competing species. But this delicate balance could also be fragile
if some of the members is weakened because an extinction of a species will
involve the annihilation of its predator hence leaving only a single species
alive. To check this expectation we here introduce a fourth species which
chases exclusively a single member of the basic model composed by three
cyclically dominant species. Interestingly, the coexistence is not necessarily
broken and we have detected three consecutive phase transitions as we vary only
the invasion strength of the fourth pestilent species. The resulting phases are
analyzed by different techniques including the study of the Hamming distance
density profiles. Some of our observations strengthen previous findings about
cyclically dominant system, but they also offer new revelations and
counter-intuitive phenomenon, like supporting pestilent species may result in
its extinction, hence enriching our understanding about these very simple but
still surprisingly complex systems.
