Figure 1.

Population dynamics of cooperators engaging in public goods interactions in absence of defectors for different death rates d. Cooperators are unable to survive for $d>d_{\text{max}}=(r-1)(N-1)N^{-N/(N-1)}$ and the only stable equilibrium is $x_0=0$. However, for $d<d_{\text{max}}$, the system undergoes a bifurcation and two interior equilibria appear: one stable branch at higher x (solid line) and one unstable branch at lower x (dashed line). Consequently, cooperators thrive at sufficiently high densities but go extinct otherwise, i.e. approach $x_0$. For d=0, the equilibrium $x_0$ becomes unstable and the system converges to x=1. The dynamics is illustrated for N=5 and r=3.