This is very much a working paper. It presents some preliminary results having to do with the long run dynamics of certain types of demographic processes. A population is heterogeneous with regard to its preferences for two alternatives A and B. If the choice of alternatives displays increasing returns, i.e. the more one of the alternatives is chosen the more attractive it becomes, the long run properties of the system are, in general, not predictable. It may, however. have fixed points to one of which the system will converge. The stability of the fixed points depends very much on the correlation of the distribution of original preferences. As this is work in progress, suggested directions for future research are presented.