Convergence to the coalescent in populations of substantially varying size. (English) Zbl 1049.92026
A haploid population with non-overlaping generations and varying size is considered. It is supposed that the distributions of offspring numbers given the sizes of two consecutive generations are exchangeable. The population sizes \(M_r\) (\(r\) being the generation number) form a Markov chain with transition probabilities \(\pi_{ij}=Pr(M_r=x_j N\,|\,M_{r-1}=x_i N)\).
It is shown that the time-scaled ancestral process \(Z_{\lfloor Nt\rfloor}\) converges weakly to \(R_{\lambda t}\), where \(R_t\) is the Kingman coalescent, and \(\lambda\) is a scaling factor representing the effective population size \(Ne\approx N\lambda^{-1}\). Namely, \(\lambda=\sum_{ij} v_i\pi_{ij}\sigma^2_{ij}x_j x^{-2}_i\), where \(v_j=Pr(M_r=x_j N)\), \(\sigma_{ij}=b_{ij}-x_i/x_j\), \(b_{ij}\) is the limit of second order moments of the offspring number for one individual conditioned by \(M_r=x_j N\), \(M_{r-1}=x_i N\) as \(N\to\infty\). The result is applied to the classical Wright-Fisher model, to a population with stationary independent size fluctuations, and to some other examples.
It is shown that the time-scaled ancestral process \(Z_{\lfloor Nt\rfloor}\) converges weakly to \(R_{\lambda t}\), where \(R_t\) is the Kingman coalescent, and \(\lambda\) is a scaling factor representing the effective population size \(Ne\approx N\lambda^{-1}\). Namely, \(\lambda=\sum_{ij} v_i\pi_{ij}\sigma^2_{ij}x_j x^{-2}_i\), where \(v_j=Pr(M_r=x_j N)\), \(\sigma_{ij}=b_{ij}-x_i/x_j\), \(b_{ij}\) is the limit of second order moments of the offspring number for one individual conditioned by \(M_r=x_j N\), \(M_{r-1}=x_i N\) as \(N\to\infty\). The result is applied to the classical Wright-Fisher model, to a population with stationary independent size fluctuations, and to some other examples.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 92D10 | Genetics and epigenetics |
| 92D15 | Problems related to evolution |
| 60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |
| 60F17 | Functional limit theorems; invariance principles |
Keywords:
population genetics; genealogical tree; random population size; effective size; Kingman coalescentReferences:
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