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slide-1
SLIDE 1

■♥tr♦❣r❡ss✐♦♥ ♦❢ ❛ ❜❧♦❝❦ ♦❢ ❣❡♥♦♠❡ ✇✐t❤ ♠❛♥② ✇❡❛❦❧② s❡❧❡❝t❡❞ ✈❛r✐❛♥ts

❍✐♠❛♥✐ ❙❛❝❤❞❡✈❛

■❙❚ ❆✉str✐❛ ❙❛❝❤❞❡✈❛ ❛♥❞ ❇❛rt♦♥✱ ■♥tr♦❣r❡ss✐♦♥ ♦❢ ❛ ❜❧♦❝❦ ♦❢ ❣❡♥♦♠❡ ✉♥❞❡r ✐♥✜♥✐t❡s✐♠❛❧ s❡❧❡❝t✐♦♥✱ ●❡♥❡t✐❝s ✭✷✵✶✽✮✳ ❙❛❝❤❞❡✈❛ ❛♥❞ ❇❛rt♦♥✱ ❘❡♣❧✐❝❛❜✐❧✐t② ♦❢ ✐♥tr♦❣r❡ss✐♦♥ ✉♥❞❡r ❧✐♥❦❡❞✱ ♣♦❧②❣❡♥✐❝ s❡❧❡❝t✐♦♥✱ ●❡♥❡t✐❝s ✭✷✵✶✽✮✳

✶ ✴ ✶✷

slide-2
SLIDE 2

▼♦st tr❛✐ts ❛r❡ ❤✐❣❤❧② ♣♦❧②❣❡♥✐❝

❋r♦♠✿ ❨❛♥❣ ❡t ❛❧✱ ◆❛t✉r❡ ❘❡✈ ●❡♥❡t✱ ✷✵✶✶

O(✶✵✺) ❝♦♠♠♦♥ ❙◆Ps→ ❝❛✉s❛❧ ❡✛❡❝ts ♦♥ ❤❡✐❣❤t✳ ❬❇♦②❧❡ ❡t ❛❧✱ ✷✵✶✼❪ ✐♥✜♥✐t❡s✐♠❛❧ ♠♦❞❡❧ ✇✐t❤ ❧✐♥❦❛❣❡✳

✷ ✴ ✶✷

slide-3
SLIDE 3

■♥✜♥✐t❡s✐♠❛❧ ♠♦❞❡❧ ✇✐t❤ ❧✐♥❦❛❣❡

■♥❤❡r✐t❛♥❝❡ ♦❢ ❜❧♦❝❦s ♦❢ ❣❡♥♦♠❡✿

  • yO

zO yP zP

P ③❖

②❖ ②P ③P ❱✵②❖

②❖ ②P

❬❘♦❜❡rts♦♥✱ ✶✾✼✼❪

❱✵ ❤❡t❡r♦❣❡♥❡✐t② ❛❧♦♥❣ t❤❡ ❣❡♥♦♠❡✳

✸ ✴ ✶✷

slide-4
SLIDE 4

■♥✜♥✐t❡s✐♠❛❧ ♠♦❞❡❧ ✇✐t❤ ❧✐♥❦❛❣❡

■♥❤❡r✐t❛♥❝❡ ♦❢ ❜❧♦❝❦s ♦❢ ❣❡♥♦♠❡✿

  • yO

zO yP zP

P(③❖) = N

  • ②❖

②P ③P, ❱✵②❖

  • ✶ − ②❖

②P

❬❘♦❜❡rts♦♥✱ ✶✾✼✼❪

❱✵ ≈ ❤❡t❡r♦❣❡♥❡✐t② ❛❧♦♥❣ t❤❡ ❣❡♥♦♠❡✳

✸ ✴ ✶✷

slide-5
SLIDE 5

❆♣♣r♦❛❝❤✐♥❣ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❧✐♠✐t

✈❛r✐❛♥❝❡ σ✷

infinitesimal limit r

▲ ❧♦❝✐ ✭✉♥❝♦rr❡❧❛t❡❞ ❡✛❡❝ts✮ s♣r❡❛❞ ✉♥✐❢♦r♠❧②✳ ❱✵ = σ✷/r → ❣❡♥✐❝ ✈❛r✐❛♥❝❡ ♣❡r ✉♥✐t ♠❛♣ ❧❡♥❣t❤ ❲❤❡♥ ❝❛♥ s❡❧❡❝t✐♦♥ ❵s❡❡✬ ❞✐s❝r❡t❡ ❧♦❝✐❄

✹ ✴ ✶✷

slide-6
SLIDE 6

❙♣r❡❛❞ ♦❢ ❛ ❣❡♥♦♠❡ ✇✐t❤ ❧✐♥❦❡❞✱ ✇❡❛❦❧② s❡❧❡❝t❡❞ ✈❛r✐❛♥ts

NATIVE NATIVE

INTRODUCED

(trait value 0)

✜t♥❡ss ∝ ❡β③ ◆❖ ♥❡✇ ♠✉t❛t✐♦♥s

■♥tr♦❣r❡ss✐♦♥ ❞✉❡ t♦ ✇❡❛❦❧② s❡❧❡❝t❡❞✱ ❧✐♥❦❡❞ ❧♦❝✐ ✈s✳ ♥❡✉tr❛❧ ✐♥tr♦❣r❡ss✐♦♥❄ ❲❤❛t ❞❡t❡r♠✐♥❡s ❵s✉❝❝❡ss✬ ♦❢ ❞✐✛❡r❡♥t ❢r❛❣♠❡♥ts ♦❢ ❛ ❜❧♦❝❦❄

✺ ✴ ✶✷

slide-7
SLIDE 7

■♥✐t✐❛❧ s♣r❡❛❞ ♦❢ ✐♥tr♦❞✉❝❡❞ ❣❡♥♦♠❡✿ s♥❛♣s❤♦ts

▼♦❞❡❧ ❛s ❛ ❜r❛♥❝❤✐♥❣ ♣r♦❝❡ss✿ ❛t ♠♦st ✶ ✐♥tr♦❞✉❝❡❞ ❢r❛❣♠❡♥t ♣❡r ✐♥❞✐✈✐❞✉❛❧

15 35 0.125 0.25 individuals map position β√V0 = 0.1 t=20

  • 0.3
  • 0.2
  • 0.1

0.1 trait value

− →

45 90 0.125 0.25 individuals map position β√V0 = 0.1 t=80

  • 0.2
  • 0.1

0.1 trait value ✻ ✴ ✶✷

slide-8
SLIDE 8

■♥✐t✐❛❧ s♣r❡❛❞ ♦❢ ✐♥tr♦❞✉❝❡❞ ❣❡♥♦♠❡✿ s♥❛♣s❤♦ts

▼♦❞❡❧ ❛s ❛ ❜r❛♥❝❤✐♥❣ ♣r♦❝❡ss✿ ❛t ♠♦st ✶ ✐♥tr♦❞✉❝❡❞ ❢r❛❣♠❡♥t ♣❡r ✐♥❞✐✈✐❞✉❛❧

15 35 0.125 0.25 individuals map position β√V0 = 0.1 t=20

  • 0.3
  • 0.2
  • 0.1

0.1 trait value

− →

45 90 0.125 0.25 individuals map position β√V0 = 0.1 t=80

  • 0.2
  • 0.1

0.1 trait value 20 40 0.125 0.25 individuals map position β√V0 = 0.1 t=20

  • 0.2

0.2 0.4 trait value

− →

350 700 0.125 0.25 individuals map position β√V0 = 0.1 t=80

  • 0.2

0.2 0.4 trait value ✻ ✴ ✶✷

slide-9
SLIDE 9

❘❛t❡ ♦❢ ✐♥tr♦❣r❡ss✐♦♥

1 100 104

20 40 60 80 100 Average number of surviving blocks time t y0 = 0.25 z0 = 0

β√V0 = 0.0 β√V0 = 0.05 β√V0 = 0.1 β√V0 = 0.2

❊①♣♦♥❡♥t✐❛❧❧② ❢❛st ✈s✳ ∼ t s♣r❡❛❞ ❲❡❛❦❧② s❡❧❡❝t❡❞ ✈❛r✐❛♥ts

?

≈ ♥❡✉tr❛❧ ✈❛r✐❛♥ts❄ ◆♦t ✐❢ ❧✐♥❦❡❞

✼ ✴ ✶✷

slide-10
SLIDE 10

▼❡❞✐✉♠✲s✐③❡❞ ❢r❛❣♠❡♥ts ❞r✐✈❡ ❡①♣♦♥❡♥t✐❛❧ ✐♥tr♦❣r❡ss✐♦♥

❋r❛❣♠❡♥t ✇✐t❤ ♠❛♣ ❧❡♥❣t❤ ②∗ ❛♥❞ tr❛✐t ✈❛❧✉❡ ③∗✿ ❚②♣✐❝❛❧ ③∗ ∼ ±√❱✵②∗✱ ❛ss✉♠✐♥❣ ✉♥❝♦rr❡❧❛t❡❞ ❧♦❝✐✳ β③∗ > ②∗ = ⇒ s✉❝❝❡ss❢✉❧ ❢r❛❣♠❡♥t t❤❛t ✐♥❝r❡❛s❡s ✐♥t❛❝t ❡①♣♦♥❡♥t✐❛❧❧②✳ ❙✉❝❝❡ss❢✉❧ ❢r❛❣♠❡♥ts✿ t②♣✐❝❛❧ ♠❛♣ ❧❡♥❣t❤ ❛♥❞ ❣r♦✇t❤ r❛t❡ ∝ β✷❱✵✳ β → str❡♥❣t❤ ♦❢ s❡❧❡❝t✐♦♥ ♦♥ tr❛✐t❀ ❱✵ = σ✷/r → ✈❛r✐❛♥❝❡ ♣❡r ✉♥✐t ♠❛♣ ❧❡♥❣t❤

✽ ✴ ✶✷

slide-11
SLIDE 11

■♥tr♦❣r❡ss✐♦♥ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ✈❛r✐❛♥ts✿ ✐♥✐t✐❛❧ ✈s✳ ❧♦♥❣✲t❡r♠

✽✵ ❝♦♣✐❡s ♦❢ ✹✵❝▼ ❜❧♦❝❦ ✇✐t❤ ✷✵✹✽ ❧♦❝✐ ✭βσ = ✵.✵✵✵✼✮ ✐♥tr♦❞✉❝❡❞ ✐♥t♦ ♣♦♣✉❧❛t✐♦♥ ✭◆ = ✹✵✵✵✮✳

❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ❢♦r s❤♦rt✲t❡r♠ ❞②♥❛♠✐❝s ↓ ❊①♣❡❝t❡❞ ❣r♦✇t❤ r❛t❡ ♦❢ ✈❛r✐❛♥t ≈ β③ − ② ❢♦r ❢❛st❡st✲❣r♦✇✐♥❣ ❝♦♥t❛✐♥✐♥❣ ❢r❛❣♠❡♥t✳

✾ ✴ ✶✷

slide-12
SLIDE 12

■♥tr♦❣r❡ss✐♦♥ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ✈❛r✐❛♥ts✿ ✐♥✐t✐❛❧ ✈s✳ ❧♦♥❣✲t❡r♠

✽✵ ❝♦♣✐❡s ♦❢ ✹✵❝▼ ❜❧♦❝❦ ✇✐t❤ ✷✵✹✽ ❧♦❝✐ ✭βσ = ✵.✵✵✵✼✮ ✐♥tr♦❞✉❝❡❞ ✐♥t♦ ♣♦♣✉❧❛t✐♦♥ ✭◆ = ✹✵✵✵✮✳

❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ❢♦r s❤♦rt✲t❡r♠ ❞②♥❛♠✐❝s ↓ ❊①♣❡❝t❡❞ ❣r♦✇t❤ r❛t❡ ♦❢ ✈❛r✐❛♥t ≈ β③ − ② ❢♦r ❢❛st❡st✲❣r♦✇✐♥❣ ❝♦♥t❛✐♥✐♥❣ ❢r❛❣♠❡♥t✳

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.002 0.004 0.006 0.008 0.01 0.012

Fraction of replicates with introgressed allele

expected growth rate of introgressed allele N = 4000 t = 600

  • 0.002
  • 0.001

0.0 0.001 0.002

allelic effect

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.002 0.004 0.006 0.008 0.01 0.012

Fraction of replicates with introgressed allele

expected growth rate of introgressed allele N = 4000 t = 24000

  • 0.002
  • 0.001

0.0 0.001 0.002

allelic effect

✾ ✴ ✶✷

slide-13
SLIDE 13

▲♦♥❣✲t❡r♠ ✐♥tr♦❣r❡ss✐♦♥✿ r♦❧❡ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❡✛❡❝ts

0.2 0.4 0.6 0.8

  • 0.001

0.001 0.002

Fraction of replicates with introgressed allele

selective effect of introgressed allele N = 4000 t = 24000 0.2 0.4 0.02 0.04 0.06 map position along genome 0.2 0.4 0.6 0.8

  • 0.002
  • 0.001

0.001 0.002

Fraction of replicates with introgressed allele

selective effect of introgressed allele N = 4000 t = 24000

❖✇♥ ❡✛❡❝t → ✜♥❡✲s❝❛❧❡ ✈❛r✐❛t✐♦♥ ❛t ❧♦♥❣ t✐♠❡s ✐♥ ❧❛r❣❡ ♣♦♣✉❧❛t✐♦♥s✳

✶✵ ✴ ✶✷

slide-14
SLIDE 14

❈♦♥❝❧✉s✐♦♥s

▲✐♥❦❡❞ ✈❛r✐❛♥ts ❝r✉❝✐❛❧ t♦ ♣♦❧②❣❡♥✐❝ ❛❞❛♣t❛t✐♦♥✳ ❊①♣♦♥❡♥t✐❛❧ s♣r❡❛❞ ♦❢ ❣❡♥❡t✐❝ ♠❛t❡r✐❛❧ ❡✈❡♥ ✉♥❞❡r ✐♥✜♥✐t❡s✐♠❛❧ s❡❧❡❝t✐♦♥✳ ❙✉❝❝❡ss❢✉❧ ❢r❛❣♠❡♥ts ❛r❡ ♠❡❞✐✉♠✲s✐③❡❞✳ ❚✇♦ ♣❤❛s❡s ♦❢ ✐♥tr♦❣r❡ss✐♦♥✿ s❤♦rt✲t❡r♠ ✭❤✐t❝❤❤✐❦✐♥❣ ✇✐t❤ s✐♥❣❧❡ s✉❝❝❡ss❢✉❧ ❢r❛❣♠❡♥ts✮ ✈s✳ ❧♦♥❣✲t❡r♠ ✭♣❛rt✐❛❧ ♣✉r❣✐♥❣ ♦❢ ❞❡❧❡t❡r✐♦✉s ✈❛r✐❛♥ts✮✳ ■♥❞✐✈✐❞✉❛❧ ✈❛r✐❛♥ts ✇✐t❤ ❤✐❣❤ ✐♥tr♦❣r❡ss✐♦♥ ♣r♦❜❛❜✐❧✐t② ♠❛② ♥♦t ❜❡ ❛❞❛♣t✐✈❡✳

✶✶ ✴ ✶✷

slide-15
SLIDE 15

❯♥❞❡rst❛♥❞✐♥❣ ❤❡t❡r♦❣❡♥❡♦✉s ❣❡♥♦♠✐❝ ❧❛♥❞s❝❛♣❡s

◆❡✉tr❛❧ ❇❡♥❡✜❝✐❛❧ ❉❡❧❡t❡r✐♦✉s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 200 400 600 800 1000

Introgression probability

position of locus along genome S1,L = 0

A

0.2 0.4 0.6 0.8 200 400 600 800 1000 position of locus along genome S1,L = 0.05

B

0.05 0.1 0.15 0.2 0.25 0.3 0.35 200 400 600 800 position of locus along genome S1,L = −0.05

C

✷✲t✐❡r❡❞ ❞❡s❝r✐♣t✐♦♥✿ ❲❡❛❦ ♣❛✐r✇✐s❡ ❛ss♦❝✐❛t✐♦♥s ❜❡t✇❡❡♥ ❧✐♥❦❛❣❡ ❜❧♦❝❦s❄

✶✷ ✴ ✶✷

slide-16
SLIDE 16

❯♥❞❡rst❛♥❞✐♥❣ ❤❡t❡r♦❣❡♥❡♦✉s ❣❡♥♦♠✐❝ ❧❛♥❞s❝❛♣❡s

◆❡✉tr❛❧ ❇❡♥❡✜❝✐❛❧ ❉❡❧❡t❡r✐♦✉s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 200 400 600 800 1000

Introgression probability

position of locus along genome S1,L = 0

A

0.2 0.4 0.6 0.8 200 400 600 800 1000 position of locus along genome S1,L = 0.05

B

0.05 0.1 0.15 0.2 0.25 0.3 0.35 200 400 600 800 position of locus along genome S1,L = −0.05

C

✷✲t✐❡r❡❞ ❞❡s❝r✐♣t✐♦♥✿ ❲❡❛❦ ♣❛✐r✇✐s❡ ❛ss♦❝✐❛t✐♦♥s ❜❡t✇❡❡♥ ❧✐♥❦❛❣❡ ❜❧♦❝❦s❄

✶✷ ✴ ✶✷

slide-17
SLIDE 17

❙✇❡❡♣ s✐❣♥❛t✉r❡s✿ ♦♥❡ ✈s✳ ♠✉❧t✐♣❧❡ ❛❞❛♣t✐✈❡ ✈❛r✐❛♥ts❄

0.2 0.4 0.6 0.8 1

  • 0.1
  • 0.05

0.05 0.1 Haplotype diversity Position along genome 0.2 0.4 0.6 0.8 1

  • 0.1
  • 0.05

0.05 Position along genome

0.5 1

0.2 0.4 0.6 0.8 1

  • 0.1
  • 0.05

0.05 Haplotype diversity Position along genome

0.5 1

✈s✳

✶✷ ✴ ✶✷

slide-18
SLIDE 18

❙✇❡❡♣ s✐❣♥❛t✉r❡s✿ ♦♥❡ ✈s✳ ♠✉❧t✐♣❧❡ ❛❞❛♣t✐✈❡ ✈❛r✐❛♥ts❄

0.2 0.4 0.6 0.8 1

  • 0.1
  • 0.05

0.05 0.1 Haplotype diversity Position along genome 0.2 0.4 0.6 0.8 1

  • 0.1
  • 0.05

0.05 Position along genome

0.5 1

0.2 0.4 0.6 0.8 1

  • 0.1
  • 0.05

0.05 Haplotype diversity Position along genome

0.5 1

?

✈s✳

? ?

✶✷ ✴ ✶✷

slide-19
SLIDE 19

❘❡s♣♦♥s❡ ❢r♦♠ st❛♥❞✐♥❣ ✈❛r✐❛t✐♦♥

❙♣r❡❛❞ ♦❢ ❛ s✐♥❣❧❡ ❣❡♥♦♠❡ ✐♥t♦ ❛ ❤❡t❡r♦❣❡♥❡♦✉s ♣♦♣✉❧❛t✐♦♥❄ ❲❤❡♥ ❞♦❡s s❡❧❡❝t✐♦♥ s❡❡ ❞✐s❝r❡t❡ ❧♦❝✐❄ ❍❡t❡r♦❣❡♥❡✐t② ❛❧♦♥❣ t❤❡ ❣❡♥♦♠❡ ✐♥ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❧✐♠✐t❄

✶✷ ✴ ✶✷

slide-20
SLIDE 20

❲✐❞❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❢r❛❣♠❡♥t ❧❡♥❣t❤s

10−5 0.001 0.1

0.03 0.06 0.09 0.12 0.15 Probability distribution P(l) fragment length l y0 = 0.25

β√V0 = 0.0 β√V0 = 0.1 β√V0 = 0.2

t = ✶✵✵ ❣❡♥❡r❛t✐♦♥s ❛❢t❡r t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ❡✈❡♥t✳

✶✷ ✴ ✶✷

slide-21
SLIDE 21

■♥tr♦❣r❡ss✐♦♥ ♦❢ ❛ ❜❡♥❡✜❝✐❛❧ ❜❧♦❝❦

▲♦♥❣ ✈s✳ s❤♦rt ✐♥tr♦❞✉❝❡❞ ❜❧♦❝❦s ❆✈❡r❛❣❡ ❧❡♥❣t❤ ♦❢ s✉r✈✐✈✐♥❣ ❢r❛❣♠❡♥ts✳

0.01 0.1 1 10 100 Average length of surviving blocks time t β√V0=0.1

y0=0.02 y0=0.05 y0=0.1 y0=0.25

▼❡❞✐✉♠✲s✐③❡❞ ❜❧♦❝❦s ❤❛✈❡ ❧♦♥❣❡st ❞❡s❝❡♥❞❛♥t ❢r❛❣♠❡♥ts✳

✶✷ ✴ ✶✷

slide-22
SLIDE 22

❚♦ ❜❡ ❛ ❧♦♥❣ ❜❧♦❝❦ ♦r ♥♦t t♦ ❜❡❄

❉❡♣❡♥❞s ♦♥ t❤❡ ✐♥✐t✐❛❧ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❜❧♦❝❦✳

100 1000 10000 100000 Average number of descendants at t=100 square root of map length √y0 z0=0.0 z0=0.2 z0=0.4 z0=0.6 z0=0.8 z0=1.0 ✶✷ ✴ ✶✷