  Non-transitive dominance and the resulting cyclic loop of three or more
competing species provide a fundamental mechanism to explain biodiversity in
biological and ecological systems. Both Lotka-Volterra and May-Leonard type
model approaches agree that heterogeneity of invasion rates within this loop
does not hazard the coexistence of competing species. While the resulting
abundances of species become heterogeneous, the species who has the smallest
invasion power benefits the most from unequal invasions. Nevertheless, the
effective invasion rate in a predator and prey interaction can also be modified
by breaking the direction of dominance and allowing reversed invasion with a
smaller probability. While this alteration has no particular consequence on the
behavior within the framework of Lotka-Volterra models, the reactions of
May-Leonard systems are highly different. In the latter case, not just the
mentioned "survival of the weakest" effect vanishes, but also the coexistence
of the loop cannot be maintained if the reversed invasion exceeds a threshold
value. Interestingly, the extinction to a uniform state is characterized by a
non-monotonous probability function. While the presence of reversed invasion
does not fully diminish the evolutionary advantage of the original predator
species, but this weakened effective invasion rate helps the related prey
species to collect larger initial area for the final battle between them. The
competition of these processes determines the likelihood in which uniform state
the system terminates.
