Non-equilibrium theory of the allele frequency spectrum ... - ICMS

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The single locus diffusion forward equationWrite π(y, t) for the probability density of the 0-diffusion at frequencyy ∈ (0, 1) and time t > 0.The Kolomogorov forward equation∂∂π(y, t) = −∂t ∂y [a(y)π(y, t)] + 1 2∂y2[b(y)π(y, t)].holds with suitable boundary conditions.∂ 2
The diffusion frequency spectrum without mutationSuppose at time 0 that there are countably many loci at whichderived alleles are present, with respective frequencies x 1 , x 2 , . . ..Assume for now there is no further mutation from the ancestralstate.After passage to the diffusion limit, the frequency spectrum attime t is just the intensity measure of the point process thatcomes from starting independent copies of the 0-diffusion processat each of the x i and letting them run to time t.
Page 1 and 2: Non-equilibrium theory of the allel
Page 3 and 4: Equilibria under stable conditionsI
Page 5 and 6: What if population size or selectio
Page 7 and 8: Discrete time finite population mod
Page 9: The diffusion limit for a single lo
Page 13 and 14: Introducing mutation from the ances
Page 15 and 16: Introducing mutation from the ances
Page 17 and 18: Entrance boundary theory - continue
Page 19 and 20: Finding the boundary conditionSince
Page 21 and 22: Conclusion for the time-homogeneous
Page 23 and 24: Structured populationsThe same appr
Page 25 and 26: Example - continuedPut µ n (t) :=
Page 27 and 28: Recent human population growthAssum
Page 29 and 30: 10090f(x,0)f(x,0.3))8070f(x)6050403
Page 31 and 32: 43.5f i(n=20,t=0)f i(n=20,t=0.3))32
Page 33 and 34: 3f i(n=20,t=0)f i(n=20,t=0.3)2.52f
Page 35 and 36: *ReferencesBustamante, C. D., Wakel
Page 37 and 38: Nielsen, R., 2000. Estimation of po
Page 39: inference of selection and populati

Non-equilibrium theory of the allele frequency spectrum ... - ICMS

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