Cooperation in group-structured populations with two layers of interactions

Recently there has been a growing interest in studying multiplex networks where individuals are structured in multiple network layers. Previous agent-based simulations of games on multiplex networks reveal rich dynamics arising from interdependency of interactions along each network layer, yet there is little known about analytical conditions for cooperation to evolve thereof. Here we aim to tackle this issue by calculating the evolutionary dynamics of cooperation in group-structured populations with two layers of interactions. In our model, an individual is engaged in two layers of group interactions simultaneously and uses unrelated strategies across layers. Evolutionary competition of individuals is determined by the total payoffs accrued from two layers of interactions. We also consider migration which allows individuals to move to a new group within each layer. An approach combining the coalescence theory with the theory of random walks is established to overcome the analytical difficulty upon local migration. We obtain the exact results for all "isotropic" migration patterns, particularly for migration tuned with varying ranges. When the two layers use one game, the optimal migration ranges are proved identical across layers and become smaller as the migration probability grows.