Fig. 3: Peer punishment can resolve the social dilemma of public goods game in structured populations.

The traditional replicator dynamics produce results for well-mixed populations, and our framework allows for exploring the dynamics in structured populations. a and b The state space is bifurcated by x^(DE) and \({{{{{{{{\bf{x}}}}}}}}}_{\star }^{(CE)}\). The final state, either D or (C + E)_V, is determined by the initial conditions. Under mild punishment (β = 0.7), a structured population hinders cooperation by reducing the state space leading to the (C+E)_V outcome. c and d Conversely, with strong punishment (β = 5), structured populations consistently result in the extinction of defection, thereby resolving the social dilemma in public goods games. In contrast, the state space in well-mixed populations remains divided into two distinct basins. e As the punishment strength β increases, \({x}_{D}^{(DE)}\) increases, expanding the initial space leading to the (C+E)_V outcome. In well-mixed populations, \({x}_{D}^{(DE)}\to 1\) as β → ∞, and the basin leading to defection cannot be completely eliminated. However, in structured populations, \({x}_{D}^{(DE)}\to k(k+1)/[(k-2)(k+3)] \, > \, 1\) when β > β^⋆, invariably resulting in the extinction of defection. f The diagram of the different effects of punishment in well-mixed versus structured populations. Structured populations are advantageous in promoting cooperation under strong punishment but are less effective when the punishment is mild. Input parameters: r = 3, c = 1, α = 0.7, k = 4.