  We consider (N,r) games of predation with N species and r < N prey and
predators, acting in a cyclic way. Further basic reactions include
reproduction, decay and diffusion over a regular grid, without a hard
constraint on the occupation number per site, so in a "bosonic" implementation.
For special combinations of N and r and appropriate parameter choices we
observe games within games, that is, different coexisting games, depending on
the spatial resolution. As concrete and simplest example we analyze the (6,3)
game. Once the players segregate from a random initial distribution, domains
emerge, which effectively play a (2,1)-game on the coarse scale of domain
diameters, while agents inside the domains play (3,1) (rock-paper-scissors),
leading to spiral formation with species chasing each other. As the (2,1)-game
has a winner in the end, the coexistence of domains is transient, while agents
inside the remaining domain coexist, until demographic fluctuations lead to
extinction of all but one species in the very end. This means that we observe a
dynamical generation of multiple space and time scales with emerging
re-organization of players upon segregation, starting from a simple set of
rules on the smallest scale and changed rules from the coarser perspective.
These observations are based on Gillespie simulations. We discuss the
deterministic limit derived from a van Kampen expansion. The linear stability
analysis predicts the number of forming domains, their composition in terms of
species; it also explains the instability of interfaces between domains, which
drives their extinction; spiral patterns are identified as motion along
heteroclinic cycles. The numerical solutions reproduce the observed patterns of
the Gillespie simulations including even extinction events, so that the
mean-field analysis here is very conclusive, which is due to the specific
implementation of rules.
