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Stable mixtures of cooperators and defectors are often seen in nature. Thisfact is at odds with predictions based on linear public goods games under weakselection. That model implies fixation either of cooperators or of defectors,and the former scenario requires a level of group relatedness larger than thecost/benefit ratio, being therefore expected only if there is either kinrecognition or a very low cost/benefit ratio, or else under stringentconditions with low gene flow. This motivated us to study here social evolutionin a large class of group structured populations, with arbitrarymulti-individual interactions among group members and random migration amonggroups. Under the assumption of weak selection, we analyze the equilibria andtheir stability. For some significant models of social evolution withnon-linear fitness functions, including contingent behavior in iterated publicgoods games and threshold models, we show that three regimes occur, dependingon the migration rate among groups. For sufficiently high migration rates, arare cooperative allele A cannot invade a monomorphic population of asocialalleles N. For sufficiently low values of the migration rate, allele A caninvade the population, when rare, and then fixate, eliminating N. Forintermediate 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 thepopulation structure. The levels of migration (gene flow) and group relatednessthat allow for invasion of the cooperative allele leading to polymorphicequilibria with the non-cooperative allele are common in nature.

Invasion, polymorphic equilibria and fixation of a mutant social allele in group structured populations