Figure 5

Evolution of the probability of playing rewarding cooperation in a simulated population of N _T = 400 individuals interacting in groups of size n = 5. Solid lines show the average phenotypic value of the population, gray dots show trait values of 10 individuals randomly sampled every N _T time steps. Each set of solid line and gray dots represent one realization of the stochastic process starting with a different initial condition where the population is monomorphic for trait value z ₀. Dotted red lines represent the analytical prediction for the value of the convergence unstable interior point z ^*. Parameters: w = 10, μ = 0.01, ν = 0.05. Left panels (a,c): well-mixed population updated with a Moran death-birth process (κ ≈ −0.0025). Right panels (b,d): square lattice with periodic boundary conditions and von Neumann neighborhood, i.e., each node is connected to North, East, South, and West neighbors (κ ≈ 0.2462). Top row panels (a,b): r ₁ = 4.5, r ₂ = 4.5. Bottom row panels (c,d): r ₁ = 1.1, r ₂ = 8.0. In all cases, the analytical model predicts bistable evolutionary dynamics with a single convergence unstable equilibrium z ^* dividing the basins of attraction of the two stable equilibria z = 0 and z = 1. (a) z ^* ≈ 0.5309. (b) z ^* ≈ 0.2472. (c) z ^* ≈ 0.606. (d) z ^* ≈ 0.632.