Structured coalescents, coagulation equations and multi-type branching processes

Abstract: Consider a structured population consisting of $d$ colonies, with migration rates that are proportional to a positive parameter $K$. We sample $N_K$ individuals distributed evenly across the $d$ colonies and trace their ancestral lineages back. Within a colony, we assume that pairs of ancestral lineages coalesce at a constant rate, as in a Kingman's coalescent. We identify each ancestral lineage with the set, or block, of its descendants in the sample and we encode the state of the system using a $d$-dimensional vector of empirical measures; the $i$-th component records the blocks present in colony $i$ and the initial location of the lineages composing each block. We are interested in the asymptotic behaviour of the process of empirical measures as $K\to\infty$. We consider two regimes: the critical sampling regime, where $N_K \sim K$ and the large sampling regime where $N_K \gg K$. After an appropriate time-space scaling, we show that the process of empirical measures converges to the solution of a $d$-dimensional coagulation equation. In the critical sampling regime, the solution can be represented in terms of a multi-type branching process. In the large sampling regime, the solution can be represented in terms of the entrance law of a multi-type continuous state branching process.