Effects of parameters on optimal dispersal levels for Model 1. Effects of public good production (P), frequency of cooperators (n) and effective group size (k). Caption a: overall dispersal d*; Caption b: investment in cooperator dispersal y*; Caption c investment in cheater dispersal z*. Thin line: k = 1.2, n = 0.1; dashed line: k = 10, n = 0.1; thick line: k = 1.2, n = 0.9; thick dashed line: k = 10, n = 0.9. Other parameters: c = e = 0.2, Q = 0.2, s = 0.6.