  We study the mean field dynamics of a model introduced by Phan et al [Wehia,
2005] of a polymorphic social community. The individuals may choose between
three strategies: either not to join the community or, in the case of joining
it, to cooperate or to behave as a free-rider. Individuals' preferences have an
idiosyncratic component and a social component. Cooperators bear a fixed cost
whereas free-riders support a cost proportional to the number of cooperators.
We study the dynamics of this model analytically in the mean field
approximation for both parallel and sequential updating. As we vary one of the
parameters while keeping the other parameters fixed, the phase diagram
experiences a rich class of bifurcations. Noticeably, a limit cycle is shown to
exist in both parallel and sequential updating, under certain parameter
settings. A comparison of the analytical predictions with computer simulation
is also included.
