Population dynamics of cooperators engaging in public goods interactions in absence of defectors for different death rates d. Cooperators are unable to survive for and the only stable equilibrium is . However, for , 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 . For d=0, the equilibrium becomes unstable and the system converges to x=1. The dynamics is illustrated for N=5 and r=3.