They do a mathematical analysis of a PD game with evolutionary dynamics where there is a probability m of assortment above and beyond random. In addition to a frequency-dependent game, they also analyze a density-dependent game in which population size can be below the environmental carrying capacity, so there is a nonzero probability of no interaction.
The most important result from this is that, once cooperators have established themselves in an empty habitat, he invasion of defectors is possible only if:
{m/[m + (1 – m)x]}*b < c,
where x is the frequency of individuals that are cooperators. This implies that a cooperative population with low density can prevent the invasion of defectors. Which is the main cool result.
The other cool thing, which is not al that important, is this figure portraying a spatial simulation of their model:
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