Abstract
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.
Highlights
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
Populations are usually divided into several subpopulations separated by geographical distance
For a large class of spatial models, including games in phenotype space[28], games on sets[29], games on islands[30], and games in group-structured populations[31], the two-strategy selection and the multi-strategy selection under weak selection can be characterized by a single parameter[32] and two parameters[33], respectively
Summary
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. When a game between two strategies (say 1, 2) described by the payoff matrix (aij)2 × 2 (aij is the payoff of an individual using i when interacting with an individual using j) is considered[32], strategy 1 is more abundant than strategy 2 on average under weak selection if σa11 + a12 > a21 + σa[22] It indicates that the multi-strategy selection is the sum of two competition terms. The parameters σ1 and σ2 quantify the effect of the population structure on the pairwise competition and the competition of all strategies with equal frequency, respectively They quantify the dependence of the multi-strategy selection on these two competition terms, respectively
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