Intermediate-Range Migration Furnishes a Narrow Margin of Efficiency in the Two-Strategy Competition

It is well-known that the effects of spatial selection on the two-strategy competition can be quantified by the structural coefficient σ under weak selection. We here calculate the accurate value of σ in group-structured populations of any finite size. In previous similar models, the large population size has been explicitly required for obtaining σ, and here we analyze quantitatively how large the population should be. Unlike previous models which have only involved the influences of the longest and the shortest migration rang on σ, we consider all migration ranges together. The new phenomena are that an intermediate range maximizes σ for medium migration probabilities which are of the tiny minority and the maximum value is slightly larger than those for other ranges. Furthermore, we find the ways that migration or mutation changes σ can vary significantly through determining analytically how the high-frequency steady states (distributions of either strategy over all groups) impact the expression of σ obtained before. Our findings can be directly used to resolve the dilemma of cooperation and provide a more intuitive understanding of spatial selection.