Inclusive fitness analysis on mathematical groups.

Recent work on the evolution of behaviour is set in a structured population, providing a systematic way to describe gene flow and behavioural interactions. To obtain analytical results one needs a structure with considerable regularity. Our results apply to such "homogeneous" structures (e.g., lattices, cycles, and island models). This regularity has been formally described by a "node-transitivity" condition but in mathematics, such internal symmetry is powerfully described by the theory of mathematical groups. Here, this theory provides elegant direct arguments for a more general version of a number of existing results. Our main result is that in large "group-structured" populations, primary fitness effects on others play no role in the evolution of the behaviour. The competitive effects of such a trait cancel the primary effects, and the inclusive fitness effect is given by the direct effect of the actor on its own fitness. This result is conditional on a number of assumptions such as (1) whether generations overlap, (2) whether offspring dispersal is symmetric, (3) whether the trait affects fecundity or survival, and (4) whether the underlying group is abelian. We formulate a number of results of this type in finite and infinite populations for both Moran and Wright-Fisher demographies.