  Stable mixtures of cooperators and defectors are often seen in nature. This
fact is at odds with predictions based on linear public goods games under weak
selection. That model implies fixation either of cooperators or of defectors,
and the former scenario requires a level of group relatedness larger than the
cost/benefit ratio, being therefore expected only if there is either kin
recognition or a very low cost/benefit ratio, or else under stringent
conditions with low gene flow. This motivated us to study here social evolution
in a large class of group structured populations, with arbitrary
multi-individual interactions among group members and random migration among
groups. Under the assumption of weak selection, we analyze the equilibria and
their stability. For some significant models of social evolution with
non-linear fitness functions, including contingent behavior in iterated public
goods games and threshold models, we show that three regimes occur, depending
on the migration rate among groups. For sufficiently high migration rates, a
rare cooperative allele A cannot invade a monomorphic population of asocial
alleles N. For sufficiently low values of the migration rate, allele A can
invade the population, when rare, and then fixate, eliminating N. For
intermediate values of the migration rate, allele A can invade the population,
when rare, producing a polymorphic equilibrium, in which it coexists with N.
The equilibria and their stability do not depend on the details of the
population structure. The levels of migration (gene flow) and group relatedness
that allow for invasion of the cooperative allele leading to polymorphic
equilibria with the non-cooperative allele are common in nature.
