On multi-type Cannings models and multi-type exchangeable coalescents

A multi-type neutral Cannings population model with mutation and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type exchangeable coalescent with mutation sharing the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions into reproductive and mutational parts. The second part deals with a different but closely related multi-type Cannings model with mutation and fixed total population size but stochastically varying subpopulation sizes. The latter model is analyzed forward and backward in time with an emphasis on its behaviour as the total population size tends to infinity. Forward in time, multitype limiting branching processes arise for large population size. Its backward structure and related open problems are briefly discussed.