Dynamics of Social Systems: Cooperation and Free-Riding
  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. 
Yiping Ma, Mirta B. Gordon, Jean-Pierre Nadal