Figure 3. Evolution of the overall cooperator fraction under repeated regrouping.

After many iterations, k, of the evolutionary cycle, a stationary level of cooperation is reached. (A) For small population bottlenecks, n₀ ≤ 3, group-growth and group-fixation mechanisms are effective and lead to purely cooperative populations. Growth parameters, bottleneck size and the regrouping time are chosen according to the experiments by Chuang et al.²⁹, see supplementary information. Without any fitting parameters, our simulation results (colored lines) are in good agreement with the experimental data (black points). (B) For larger bottlenecks, n₀ = 5, and depending on the relative efficiency of the group-growth and group-fixation mechanism, two qualitatively different regimes can be distinguished. While the group-growth mechanism leads to stable coexistence of cooperators and free-riders (green lines), the group-fixation mechanism can lead to a pure state of either only cheaters (red line) or only cooperators (blue line). The relative impact of these mechanisms depends strongly on the regrouping time T. For short regrouping times (T_short = 2.5 < t_s, green lines), the group-growth mechanism is effective, while for sufficiently long regrouping times (T_long = 20 > t_s, blue and red lines) the group-fixation mechanism acts more strongly. (C) With parameters equal to (B), the detailed interplay of the group-growth and group-fixation mechanisms is summarized in a bifurcation diagram showing the stationary levels of cooperation as a function of the regrouping time T. Depending of the relative efficiency of both mechanism, four different regimes arise: pure cooperation, coexistence, intermediate, and bistability. The times T_short and T_long correspond to the green and red/blue lines shown in (B). Parameters are x₀ = 0.086, T = 3.1, s = 0.05 and p = 6.6 in (A); see also supplementary information. In (B), x₀ = {0.1 (green), x₀ = 0.9 (green)} and x₀ = {0.5 (red), x₀ = 0.6 (blue)} for T_short = 2.5 and T_long = 20, respectively. s = 0.1 and p = 10 in (B/C).