  The Prisoner's Dilemma Process on a graph $G$ is an iterative process where
each vertex, with a fixed strategy (cooperate or defect), plays the game with
each of its neighbours. At the end of a round each vertex may change its
strategy to that of its neighbour with the highest pay-off. Here we study the
spread of cooperative and selfish behaviours on a toroidal grid, where each
vertex is initially a cooperator with probability $p$. When vertices are
permitted to change their strategies via a randomized asynchronous update
scheme, we find that for some values of $p$ the limiting density of cooperators
may be modelled as a polynomial in $p$. Theoretical bounds for this density are
confirmed via simulation.
