Abstract
Consider the Markov process taking values in the partitions of $\mathbb{N} $ such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate $d$. This is a special case of exchangeable fragmentation-coalescence process, called Kingman’s coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of independent diffusions. Moreover, we introduce a new process valued in the partitions of $\mathbb{Z} $ called Kingman’s coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate according to a Poisson process of intensity $d$. By coupling Kingman’s coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman’s coalescent with erosion to $\{1, \dots , n\}$ converges as $n\to \infty $ to the total progeny of a critical binary branching process.
Highlights
1.1 MotivationIn evolutionary biology, speciation refers to the event when two populations from the same species lose the ability to exchange genetic material, e.g. due to the formation of a past present{1, 3}, {2} {1, 2, 3} {1, 2}, {3} {1, 2, 3}new geographic barrier or accumulation of genetic incompatibilities
Even if speciation is usually thought of as irreversible, related species can often still exchange genetic material through exceptional hybridization, migration events or sudden collapse of a geographic barrier [19]. This can lead to the transmission of chunks of DNA between different species, a phenomenon known as introgression, which is currently considered as a major evolutionary force shaping the genomes of groups of related species [17]
We suppose that the species are monomorphic, i.e., that all individuals in the same species carry the same alleles at all genes, and that their dynamics is given by a Moran model: at rate one for each pair of species (s1, s2), species s2 dies, s1 gives birth to a new species, replicates its genome and sends it into the daughter species
Summary
Speciation refers to the event when two populations from the same species lose the ability to exchange genetic material, e.g. due to the formation of a. The gene that has been transmitted during this event is removed from its original species and placed in the migrant’s original species Such events occur at rate (N − 1)d for each gene, and the migrant species is chosen uniformly in the population. Setting the introgression rate to dN = d/N and letting N → ∞, introgression events occur at rate d for each gene At each such event the gene is sent to a new species that does not contain any of the other n − 1 ancestral gene lineages, i.e., it is placed in a singleton block. This is the description of Kingman’s coalescent with erosion, that we more formally introduce
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