Invasion, polymorphic equilibria and fixation of a mutant social allele   in group structured populations
  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. 
Roberto H. Schonmann, Renato Vicente, Robert Boyd
q-bio.PE