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. 2019 Aug 29;9:12575. doi: 10.1038/s41598-019-49075-3

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Stationary density of cooperators ρ_C in dependence on the periodic time τ, obtained when both payoff pairs fall into the snowdrift quadrant (left panel). We have used (T₁ = 1.3, S₁ = 0.7) and (T₂ = 1.7, S₂ = 0.3), whereby the corresponding ρ_C values (if either payoff pair would be applied permanently) are marked by horizontal dashed lines for comparison. It can be observed that the ρ_C value of the game with seasonal variations is more or less exactly the average of the two games that are alternated, and this regardless of the value of τ. Only when τ is very small is the relaxation in both cases not fast enough and so the average diverges towards the snowdrift game with a slightly faster relaxation [governed by (T₁, S₁), more supportive of cooperation, marked by the green dashed line]. The right panel shows the time evolution of the cooperation level f_C in the stationary state, as obtained for different values of τ using the same payoff pairs as in the left panel. The applied values of τ are indicated alongside the corresponding lines. For a proper comparison we have used a normalized time scale in 1/τ units. It can be observed that the relaxation dynamics is always (in both periods) relatively fast, indicative of the fast emerging locally ordered role-separating checkerboard patterns that govern both instances of the applied snowdrift game.