Group Theory in Homogeneous Populations (Rescuing Darwin from the mud)

Considerable recent work on the evolution of behaviour has been set in structured populations. An interesting “cancellation” result is known for structures, such as lattices, cycles and island models, which are homogeneous in the sense that the population “looks the same” from every site. In such populations all proximate or immediate fitness effects on others (for example, payoffs in a game or contest) play no role in the evolution of the behaviour. The altered competitive effects of such behaviour exactly cancel the proximate fitness effects. In mathematics, the internal symmetry which drives this result is powerfully described by the theory of mathematical groups and recent work has used this theory to clarify and extend a number of existing results. I review this body of work here. Mathematics Subject Classification (2000). 05C57, 91A43, 92D50.