Structured coalescents, coagulation equations and multi-type branching processes

Consider a structured population consisting of $d$ colonies, with migration rates proportional to a positive parameter $K$. We sample $N_K$ individuals, distributed evenly across the $d$ colonies, and trace their ancestral lineages backward in time. Within each colony, we assume that any pair of ancestral lineages coalesces at a constant rate, as in Kingman's coalescent. We identify each ancestral lineage with the set, or block, of its sampled descendants, 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 $i$ together with the initial locations of the lineages composing each block. We are interested in the asymptotic behavior 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-sample regime, where $N_K \gg K$. After an appropriate time rescaling, 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-sample regime, the solution can be represented in terms of the entrance law of a multi-type continuous-state branching process.