  Rock-scissors-paper game, as the simplest model of intransitive relation
between competing agents, is a frequently quoted model to explain the stable
diversity of competitors in the race of surviving. When increasing the number
of competitors we may face a novel situation because beside the mentioned
unidirectional predator-prey-like dominance a balanced or peer relation can
emerge between some competitors. By utilizing this possibility in the present
work we generalize a four-state predator-prey type model where we establish two
groups of species labeled by even and odd numbers. In particular, we introduce
different invasion probabilities between and within these groups, which results
in a tunable intensity of bidirectional invasion among peer species. Our study
reveals an exceptional richness of pattern formations where five quantitatively
different phases are observed by varying solely the strength of the mentioned
inner invasion. The related transition points can be identified with the help
of appropriate order parameters based on the spatial autocorrelation decay, on
the fraction of empty sites, and on the variance of the species density.
Furthermore, the application of diverse, alliance-specific inner invasion rates
for different groups may result in the extinction of the pair of species where
this inner invasion is moderate. These observations highlight that beyond the
well-known and intensively studied cyclic dominance there is an additional
source of complexity of pattern formation that has not been explored earlier.
