On multi-type Cannings models and multi-type exchangeable coalescents. (English) Zbl 07874522

Möhle, Martin

Theor. Popul. Biol. 156, 103-116 (2024).

Summary: A multi-type neutral Cannings population model with migration 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 coalescent sharing the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions of transition probabilities into parts concerning reproduction and migration respectively. The following section 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 behavior as the total population size tends to infinity. Forward in time, multi-type limiting branching processes arise for large population size. Its backward structure and related open problems are briefly discussed.

MSC:

92-XX	Biology and other natural sciences
60J90	Coalescent processes
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
92D15	Problems related to evolution
92D25	Population dynamics (general)

Keywords:

consistency; exchangeability; migration; multi-type branching process; multi-type coalescent process; mutation

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