Fig. 2

The nonlinear public goods function influences growth rate differences and long-term stable states. a–c For β > σ, differences in intrinsic growth rates can have a maximum at a fraction of cooperators between 0 and 1. The number and positions of equilibria then critically depends on neighborhood size, and saddle-node bifurcations are possible. These patterns are also reflected in the individual-based model, revealing large regions of coexistence (shown in c). d–f For β < σ, differences in intrinsic growth rates tend to be monotonically increasing, and we can observe unstable coexistence and a form of a transcritical bifurcation pattern. The individual based model switches sharply from all-C to all-D (shown in f). We used α_C,D  =  1.0, δ_C,D  =  0.1, κ  =  0.5, and K  =  1000. Stochastic simulation results were obtain from 200 independent simulations started from 90 C and 10 D cells