Figure 2. Evolution while individuals are arranged in groups (group-evolution step).

(A) Population average of cooperator fraction, x, as a function of time t. Depending on the average initial group size, n₀, three different scenarios arise: decrease of cooperation (red line, n₀ = 30), transient increase of cooperation (green line, n₀ = 6, increase until cooperation time t_c) and permanently enhanced cooperation (blue line, n₀ = 4). These three scenarios arise from the interplay of two mechanisms. While the group-growth mechanism, due to faster growth of more cooperative groups, can cause a maximum in the fraction of cooperators for short times, the group-fixation mechanism, due to a larger maximum size of purely cooperative groups, assures cooperation for large times. Both mechanisms become less efficient with increasing initial group sizes and are not effective in the deterministic limit (dashed black line, solution of Eq. (S7) for N₀ = 6) as the rely on fluctuations. (B) The strength of the group-growth mechanism decreases with an increasing initial fraction of cooperators. This is illustrated by comparing the time evolution for three different initial fractions of cooperators and a fixed initial group size n₀ = 5. After a fixed time, here t = 3.03, the fraction of cooperators is larger than the initial one for x₀ = 0.2, equal to it for x₀ = 0.5, and eventually becomes smaller than the initial value, as shown for x₀ = 0.8. (C) Change of the average group size, n = Σ_iv_i /M. At the beginning the groups grow exponentially, while they later saturate to their maximum group size. As this maximum size depends on the fraction of cooperators, the average group size declines with the loss in the level of cooperation (n₀ = 6, green line). The deterministic solution for the same set of parameters which does not account for fluctuations (dashed black line, solution of Eq. (S7)) describes this behavior qualitatively. s = 0.1, p = 10.