Abstract
We consider diploid bi-parental analogues of Cannings models: in a population of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from parents $i$ and $j$, where $V=(V_{i,j})_{1\le i \neq j \le N}$ is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an $n$-sample of genes towards a ($\Xi $-)coalescent. This complements Mohle and Sagitov’s (2001) result for the haploid case and sharpens the profile of Mohle and Sagitov’s (2003) study of the diploid case, which focused on fixed couples, where each row of $V$ has at most one non-zero entry. We apply the convergence result to several examples, in particular to two diploid variations of Schweinsberg’s (2003) model, leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.
Highlights
Introduction and main resultsFor haploid population models, in which every individual has one parent, coalescent processes have been used widely in order to describe the ancestralCoalescent results for diploid exchangeable population models structure of a sample of n genes when the total population size N is sufficiently large.The purpose of this work is to extend the coalescent theory to general diploid population models, in which individuals carry two copies of each gene which they inherit from two distinct parental individuals
We derive the diploid analogue of Möhle and Sagitov’s classification of the ancestral processes in exchangeable haploid population models [22]
There has been a tremendous development in coalescent theory
Summary
In which every individual (gene) has one parent (gene), coalescent processes have been used widely in order to describe the ancestral. In order to consider a suitable scaling for the large population limit a key quantity is the probability that two genes (picked at random) from two distinct individuals, which are chosen randomly without replacement from the same generation, have a common ancestor (gene) in the previous generation. In our model this quantity is given by cN. Let ξn,N (m) := cd ξn,N (m) ∈ En be the ancestral partition of the n sampled genes m generations in the past, irrespective of the grouping into diploid individuals.
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