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.