"Coalescents with Migration in the Moderate Regime"Mellis, Sophia-MarieMulti-type models have recently experienced renewed interest in the stochastic modeling of evolution. This is partially due to their mathematical analysis often being more challenging than their single-type counterparts; an example of this is the site-frequency spectrum of a colony-based population with moderate migration. In this talk, we model the genealogy of such a population via a multi-type coalescent starting with $N_{K}$ colored singletons with $d \geq 2$ possible colors (colonies). The process is described by a continuous-time Markov chain with values on the colored partitions of the colored integers in $\{1, \ldots, N_{K}\}$; blocks of the same color coalesce at rate $1$, while they are also allowed to change color at a rate proportional to $K$ (migration). Given this setting, we study the asymptotic behavior, as $K\to\infty$ at small times, of the vector of empirical measures, whose $i$-th component keeps track of the blocks of color $i$ at time $t$ and the initial coloring of the integers composing each of these blocks. We show that, in the proper time-space scaling, it converges to a multi-type branching process (case $\frac{N_{K}}{K} \underset{K\to\infty}{\longrightarrow} 1$) or a multi-type Feller diffusion (case $N_{K} \gg K$). Using this result, we derive an applicable representation of the site-frequency spectrum. This is joint work with Fernando Cordero, Sebastian Hummel and Emmanuel Schertzer. |
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