The stationary distribution of a sample from the Wright–Fisher diffusion model with general small mutation rates
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The stationary distribution of a sample taken from a Wright–Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright–Fisher diffusion. The approach is different from Burden and Tang (Theor Popul Biol 112:22–32, 2016; Theor Popul Biol 113:23–33, 2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate \(\lambda \).
KeywordsCoalescent tree Small mutation rates Wright–Fisher diffusion
Mathematics Subject Classification92B99 92D15
This research was done when Robert Griffiths visited the Mathematical Sciences Instutite, Australian National University in November and December 2017. He thanks the Instutite for their support and hospitality.
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