Impact of migration on the multi-strategy selection in finite group-structured populations.

For large quantities of spatial models, the multi-strategy selection under weak selection is the sum of two competition terms: the pairwise competition and the competition of multiple strategies with equal frequency. Two parameters σ1 and σ2 quantify the dependence of the multi-strategy selection on these two terms, respectively. Unlike previous studies, we here do not require large populations for calculating σ1 and σ2 , and perform the first quantitative analysis of the effect of migration on them in group-structured populations of any finite sizes. The Moran and the Wright-Fisher process have the following common findings. Compared with well-mixed populations, migration causes σ1 to change with the mutation probability from a decreasing curve to an inverted U-shaped curve and maintains the increase of σ2 . Migration (probability and range) leads to a significant change of σ1 but a negligible one of σ2 . The way that migration changes σ1 is qualitatively similar to its influence on the single parameter characterizing the two-strategy selection. The Moran process is more effective in increasing σ1 for most migration probabilities and the Wright-Fisher process is always more effective in increasing σ2 . Finally, our findings are used to study the evolution of cooperation under direct reciprocity.