Fig. 4

Differences in neighborhood size (public good sensitivity) can destabilize all-C but increase the coexistence regime. In all panels: K = 1000, δ₁ = δ₂ = 0.1, γ = 1.0/day, σ = 2.0, β = 5.0 and κ = 0.5/day. a Temporal trajectories of the fraction of cooperators for two different values of n_C, where we considered a resident population of defectors (x_D(0)/K = 1−δ/α) that is invaded by a small population of cooperators (x_C(0) = 10), which move toward coexistence. The time to reach coexistence critically depends on n_D − n_C. b For fixed n_D = 30 the saddle-node bifurcation pattern of Fig. 3b (where n_D = n_C) changes such that the all-C state becomes entirely unstable, and we see a shift in the critical value of n_C above which cooperation is lost. c Smaller values of, e.g., n_D = 15, act against defectors. Then we predict an increase in range of values of n_C for which cooperators and defectors stably coexist