Globally optimal associations in dispersal and exploitation strategy for Model 1. Axes: P measures the impact of the public good on individual fitness, and s is the individual cost to cooperators in contributing to the public good. σ * = y*/(z* + y*) indexes the tendency of cooperators to disperse (σ* > 0.5) or cheats to disperse (σ* < 0.5). Thick curves demarcate areas of parameter space yielding different levels of σ, whereas thin lines show areas in with either y* = 1 or z* = 1. Caption a: k = 1.2, n = 0.1; caption b: k = 10, n = 0.1; caption c: k = 1.2, n = 0.9; caption d: k = 10, n = 0.9. Note that for legibility, very thin areas parallel to thick lines are omitted, in which 0.5 < σ* < 1 for caption c, and 0 < σ* < 1 for caption d. Unless otherwise noted, dispersal rates are greater than zero and less than unity. Other parameters: c = e = 0.2, Q = 0.2. See main text for numerical methods.