Figure 4.

Co-evolutionary dynamics of cooperation and the environment under moderate selection intensity. From panel I to panel IV, the PGG is fixed to be played in state s¹, while in state s², the IPGG, dSH, PGG and dSD are played, respectively. Under replicator dynamics [12,60,62], the gradients of selection in these four games are shown in each panel, respectively. Blue solid circles are used to depict stable equilibria, while open blue circles are used to depict unstable equilibria. The direction of evolution is indicated by arrows. The phase graphs in each panel show the co-evolutionary dynamics of the time proportion of the PGG and the average proportion of C players for different value intervals of the tipping point ${\overline{x}}_C$. Corresponding to the value interval $0<{\overline{x}}_C<1$, the first row in panel I shows the persistent oscillations of cooperation and the environment. The bottom right sub-figure in panel I shows the linear relation between the average abundance of C players and the average time proportion of the PGG, which suggests that condition (3.3) is still valid for relatively moderate selection intensity. The first row in panel II uses the parameter condition [62] under which there is a stable and an unstable interior equilibrium for the dSH under replicator dynamics (the bottom left), whereas the second row uses that under which there is a unique interior unstable equilibrium (the bottom right). The population structure is a complete graph. Parameter values: N = 400, d = 5, β = 2, $\mathcal{C}=c=1$, ${\mathcal{B}}_s=12$, r_s = 3 for all panels, except in panel II, where r_s = 4 and T = 3 for the first row, and r_s = 5.01 and T = 4 for the second row.