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

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Entrance boundary theory – continuedThe ↓-diffusion is the 0-diffusion conditioned to hit 0 before 1.It has the Doob h-transform semigroup(Pt ↓ (x, dy) = 1 − s(x)) −1 (P t (x, dy) 1 − s(y)).s(1)s(1)From Williams (1974), the ↑-diffusion started at 0 and killed atthe last time it visits y > 0 is the time-reversal of the ↓-diffusionstarted at y and killed when it first hits 0.Write (Q ↓ t ) t≥0 := semigroup of killed ↓-diffusion.
Finding the boundary conditionSince s(y) ≈ y for y close to 0,lim yφ t (y) = lim s(y)φ t (y)y↓0 y↓0= limy↓0s(y)= θ 2 limy↓0= θ 2 limy↓0= θ 2 limy↓0= θ 2 limy↓0∫ t0∫ t0∫ ∞0∫ ∞0∫ t0θ λ t−s (dy)ds2 dyPt−s ↑ (0, dy)dyP ↑ s (0, dy)dyP ↑ s (0, dy)dydsdsdsQ ↓ s(y, dy)ds.dy
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 and 10: The diffusion limit for a single lo
Page 11 and 12: The diffusion frequency spectrum wi
Page 13 and 14: Introducing mutation from the ances
Page 15 and 16: Introducing mutation from the ances
Page 17: Entrance boundary theory - continue
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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