By Alison Etheridge
This paintings displays 16 hours of lectures added through the writer on the 2009 St Flour summer time institution in likelihood. It presents a swift creation to quite a number mathematical versions that experience their origins in theoretical inhabitants genetics. The types fall into sessions: forwards in time versions for the evolution of frequencies of alternative genetic varieties in a inhabitants; and backwards in time (coalescent) versions that hint out the genealogical relationships among members in a pattern from the inhabitants. a few, just like the classical Wright-Fisher version, date correct again to the origins of the topic. Others, just like the a number of merger coalescents or the spatial Lambda-Fleming-Viot approach are even more fresh. All percentage a wealthy mathematical constitution. organic phrases are defined, the types are conscientiously stimulated and instruments for his or her examine are awarded systematically.
Read or Download Some Mathematical Models from Population Genetics: École d'Été de Probabilités de Saint-Flour XXXIX-2009 PDF
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This paintings displays 16 hours of lectures added through the writer on the 2009 St Flour summer season tuition in chance. It offers a swift advent to quite a number mathematical types that experience their origins in theoretical inhabitants genetics. The versions fall into periods: forwards in time versions for the evolution of frequencies of alternative genetic kinds in a inhabitants; and backwards in time (coalescent) versions that hint out the genealogical relationships among contributors in a pattern from the inhabitants.
Extra resources for Some Mathematical Models from Population Genetics: École d'Été de Probabilités de Saint-Flour XXXIX-2009
Sample text
J j The correction to E[(δ pi )2 ] is of O(1/N 2 ). This gives the following lemma. 1 (Multi-allele Wright–Fisher diffusion with mutation).
Now 1 0 m(p)d p = 1 0 Cp2ν2 −1 (1 − p)2ν1−1 d p = C Γ (2ν1 )Γ (2ν2 ) Γ (2(ν1 + ν2 ) (where Γ is Euler’s Gamma function) and so the stationary distribution is just ψ (p) = Γ (2(ν1 + ν2 )) 2ν2 −1 p (1 − p)2ν1−1 . 15) Ethier and Kurtz (1986), Chap. 1 gives a direct proof of uniqueness of this stationary distribution. ✷ The stationary distribution gives us some understanding of the longterm balance between the competing forces of mutation (which maintains genetic diversity) and genetic drift (which removes variation from the population).
Let us write E for the space of possible allelic types for individuals in our population. The Moran model for a population of size N is then simply a continuous time Markov chain on E N and its infinitesimal generator, KN , evaluated on a function f : E N → R, is given by KN f (x1 , x2 , . . , xN ) = N ∑ Ai f (x1 , x2 , . . , xN ) i=1 + 1 N N ∑ ∑ [Φi j f (x1 , . . , xN ) − f (x1, . . 8) where Φi j f (x1 , . . , xN ) is the function obtained from f by replacing x j by xi . The operator Ai is the generator of the mutation process, A, acting on the ith coordinate.