  Cyclic dominant systems, like rock-paper-scissors game, are frequently used
to explain biodiversity in nature, where mobility, reproduction and
intransitive competition are on stage to provide the coexistence of
competitors. A significantly new situation emerges if we introduce an apex
predator who can superior all members of the mentioned three-species system. In
the latter case the evolution may terminate into three qualitatively different
destinations depending on the apex predator decaying ratio $q$. In particular,
the whole population goes extinct or all four species survive or only the
original three-species system remains alive as we vary the control parameter.
These solutions are separated by a discontinuous and a continuous phase
transitions at critical $q$ values. Our results highlight that cyclic dominant
competition can offer a stable way to survive even in a predator-prey-like
system that can be maintained for large interval of critical parameter values.
