  The n-person Prisoner's Dilemma is a widely used model for populations where
individuals interact in groups. The evolutionary stability of populations has
been analysed in the literature for the case where mutations in the population
may be considered as isolated events. For this case, and assuming simple
trigger strategies and many iterations per game, we analyse the rate of
convergence to the evolutionarily stable populations. We find that for some
values of the payoff parameters of the Prisoner's Dilemma this rate is so low
that the assumption, that mutations in the population are infrequent on that
timescale, is unreasonable. Furthermore, the problem is compounded as the group
size is increased. In order to address this issue, we derive a deterministic
approximation of the evolutionary dynamics with explicit, stochastic mutation
processes, valid when the population size is large. We then analyse how the
evolutionary dynamics depends on the following factors: mutation rate, group
size, the value of the payoff parameters, and the structure of the initial
population. In order to carry out the simulations for groups of more than just
a few individuals, we derive an efficient way of calculating the fitness
values. We find that when the mutation rate per individual and generation is
very low, the dynamics is characterised by populations which are evolutionarily
stable. As the mutation rate is increased, other fixed points with a higher
degree of cooperation become stable. For some values of the payoff parameters,
the system is characterised by (apparently) stable limit cycles dominated by
cooperative behaviour. The parameter regions corresponding to high degree of
cooperation grow in size with the mutation rate, and in number with the group
size.
