Fig. 1.
Dynamics of the system with cooperators, defectors and punishers in the simplex S3 for different mutation rates. The arrows show the drift term 𝒜k(x) of the Fokker–Planck equation, white circles are stable fixed points in the limit M → ∞. The discontinuities are a consequence of the strong selection. Blue corresponds to fast dynamics and red to slow dynamics close to the fixed points of the system. The system does typically not access the gray shaded area, because the minimum average fraction of each type because of mutations is μ/3. (A) For vanishing mutation probability (μ → 0), there is only 1 stable fixed point in the defector corner. (B) For μ = 0.2, there are 2 stable fixed points, one close to the cooperator corner and one close to the defector corner. The population noise can drive the system from the vicinity of one of these points to the other, which makes an analytical description of the dynamics difficult. (C) For μ = 0.5, there is only a single stable fixed point, which is closest to the cooperator corner—thus, cooperators prevail for high mutation rates. We use the position of this fixed point as an estimate for the average abundance of the strategies, see Fig. 2 [parameters: M = 100, N = 5, r = 3, c = 1; β = 1, γ = 0.3; σ = 1, graphical output based on the Dynamo software (37)].