accounting for uncertainty when estimating pleistocene megafauna extinction times

Corey J. A. Bradshaw1,2, Barry W. Brook1, Chris S. M. Turney3, Alan Cooper1,4

1THE ENVIRONMENT INSTITUTE, University of Adelaide; 2South Australian Research & Development Institute; 3School of Geography, University of Exeter; 4Australian Centre for Ancient DNA

• extinction must be inferred from record of sightings/collections

• when a species becomes increasingly rare before extinction, might persist unseen for many years

• so the time of last sighting often poor estimate of extinction date

Roberts & Solow 2003 Nature 426:245

present pastx x x xx x x?? xx x x

• optimal linear estimation• joint distribution of k same Weibull form regardless of parent

distribution• estimated extinction time q

• L: symmetric k×k matrix

• n: Estimated shape parameter of joint Weibull distribution of k

Roberts & Solow 2003 Nature 426:245

present pastx x x xx x x

qxx x x

kTTTT ...321

k

i

iiTa1

q̂

eeea t 111 LL

ijji

ji

i

iiij

,

ˆ

ˆ2ˆ2

n

nn

2

1 11

1log1

1ˆ

k

i i

k

TT

TT

kn

CI

11000 12000 13000 14000

YBP

11000 12000 13000 14000

YBP

• maximum likelihood to account for radio carbon dating error

• assume true ages U independent/uniformly distributed over (b1,g1) where b1 = extinction date

• PDF of Xj:

Solow et al. 2006 PNAS 103:7351

present pastx x x xx x x

b1 xx x x

jjj UX

11

11

)(bg

g

b

jj

j

xx

xf

11000 12000 13000 14000

YBP

• but... previous sighting rate important• length of period since last sighting informative• given previous sighting rate(n/tn), probability of next sighting

• where p drops below threshold with increasing T-tn, TE inferred

McInerny et al. 2006 Conserv Biol 20:562

present pastx x x xx x x

TE xx x x

ntT

nt

np

1

5 10 15 20 25 30

10

90

01

10

00

11

10

01

12

00

11

30

01

14

00

11

50

0

samples

Te

• but... TE depends on number of samples in ‘final’ period• declining influence of dates within time since last sighting• sequentially recalculated TE, weighting by cumulative distance

from T1

present pastx x x xx x x

TE xx x xT1

5 10 15 20

12

00

01

21

00

12

20

01

23

00

12

40

01

25

00

samples

Te

2 4 6 8 10 12 14

16

00

01

70

00

18

00

01

90

00

20

00

0

samples

Te

2 4 6 8 10 12 14 16

27

40

02

76

00

27

80

02

80

00

28

20

0

samples

Te

2 4 6 8 10 12

25

50

02

60

00

26

50

02

70

00

27

50

0

samples

Te

2 4 6 8 10

23

00

02

40

00

25

00

02

60

00

27

00

02

80

00

29

00

0

samples

Te

2 4 6 8 10 12 14

29

50

03

00

00

30

50

03

10

00

samples

Te

5 10 15 20 25 30

10

90

01

10

00

11

10

01

12

00

11

30

01

14

00

11

50

0

samples

Te

• but... TE depends on number of samples in ‘final’ period• declining influence of dates within time since last sighting• sequentially recalculated TE, weighting by cumulative distance

from T1

present pastx x x xx x x

TE xx x xT1x x x xx x x xx x x

• now simply combine methods with Gaussian resampler within carbon date errors for each record

10000 11000 12000 13000 14000 15000

YBP

Solow

10588

11112

9907

10000 11000 12000 13000 14000 15000

YBP

weighted McInerney

11097

11514

10662

11097

11514

10662

11097

11514

10662

11097

11514

10662

• now simply combine methods with Gaussian resampler within carbon date errors for each record

15000 16000 17000 18000 19000 20000

0e

+0

01

e-0

42

e-0

43

e-0

44

e-0

45

e-0

46

e-0

4

YBP

Pr

uniform

linearsigmoidal

exponential

logarithmic

‘true’extinction

15000 15200 15400 15600 15800 16000

15000 15200 15400 15600 15800 16000

15000 15200 15400 15600 15800 16000

15000 15200 15400 15600 15800 16000

15000 15200 15400 15600 15800 16000

YBP

uniform

linear

sigmoidal

exponential

logarithmic

uniform linear sigmoidal exponential logarithmic0.0001

0.001

0.01

0.1

1

10

100

S&RSolowMMwMw-rs

Method

2

uniform linear sigmoidal exponential logarithmic0

5

10

15

S&RSolowwM-rs

Method

CV

uniform linear sigmoidal exponential logarithmic0.0

0.2

0.4

0.6

0.8

1.0

S&RSolowwM-rs

Method

Pr(

overl

ap

)

incre

asin

g a

ccu

racy

incre

asin

g p

recis

ion

incre

asin

g o

verl

ap

uniform linear sigmoidal exponential logarithmic0.0001

0.001

0.01

0.1

1

10

100

SolowMw-rs

Method

2

uniform linear sigmoidal exponential logarithmic0.0

0.2

0.4

0.6

0.8

1.0

SolowwM-rs

Method

Pr(

ove

rla

p)

Glacials, Interglacials, Stadials and Interstadials

stadial

interstadial

Interglacial

Glacial

Interglacial

Extracting An Ice Core

Annual Layers In Ice Core

Cariaco Basin Bathymetry

• water exchange with the open Caribbean Sea is restricted

• intense seasonal productivity and high sedimentation rate

• anoxic below 300 m

• limited bioturbation (post-depositional mixing of sediments by marine life)

10000 20000 30000 40000 50000

YBP

BisonX.Eur

SaigaAnt.Eur

Mammoth.Eur BisonPris.Eur

CaveBear.Eur

Neand.Eur Muskox.Eur

Mammoth.NA

Equus.NA CaveLion.NA

StiltHorse.NA

ArctSim.NA

Cervalces.NA

SaigaAnt.NA

Dasypus.NA

Camelops.NA

Bootherium.NA

0 10000 20000 30000 40000 50000

01

00

00

20

00

03

00

00

40

00

05

00

00

calibrated AMS

ca

lib

rate

d M

w-r

sMw-rs ~ -83+ 0.98 × AMS

10000 20000 30000 40000 50000

YBP

HOL IS1 IS2 IS3 IS4 IS5 IS6 IS7 IS8 IS9IS10 IS11 IS12

BisonX.Eur

SaigaAnt.Eur

Mammoth.Eur BisonPris.Eur

CaveBear.Eur

Neand.Eur Muskox.Eur

Mammoth.NA

Equus.NA CaveLion.NA

StiltHorse.NA

ArctSim.NA

Cervalces.NA

SaigaAnt.NA

Dasypus.NA

Camelops.NA

Bootherium.NA

UrsArc.NA.Occ

CavBear.NA.Occ

BisonCau.Eur.Occ

10000 15000 20000 25000 30000 35000 40000

YBP

IS1 IS2 IS3 IS4 IS5 IS6 IS7 IS8 IS9 IS10

Mammoth Equus S.Horse

Bison

C.BearSF.Bear

Neand

10000 15000 20000 25000 30000 35000 40000

YBP

IS1 IS2 IS3 IS4 IS5 IS6 IS7 IS8 IS9 IS10

Mammoth Equus S.Horse

Bison

C.BearSF.Bear

Neand

Interstadials – OXCAL wPDFs

extinctions - unconstrained

P(rand overlap = 0.12)

extinctions - constrained

P(rand overlap) = 0.09

combined ext/app P(rand overlap) = 0.13

10000 15000 20000 25000 30000 35000 40000

YBP

IS1 IS2 IS3 IS4 IS5 IS6 IS7 IS8 IS9 IS10

U.arctosWapiti

10000 15000 20000 25000 30000 35000 40000

YBP

IS1 IS2 IS3 IS4 IS5 IS6 IS7 IS8 IS9 IS10

Mammoth Equus S.Horse

Bison

C.BearSF.Bear

Neand

Interstadials – OXCAL raw dates

extinctions - raw dates

P(rand overlap = 0.06)

appearances – raw dates

combined ext/app P(rand overlap) = 0.11

10000 15000 20000 25000 30000 35000 40000

YBP

S1.GS1YD S2.H1 S3.C1 S4.C2 S5.C3 S6.H2 S7.C4 S8.GS4C5S9.H3 S10.GS6C6S11.GS7C7S12.GS8C8 S13.H4 S14.GS10C9S15.GS11C10

Mammoth Equus S.Horse

Bison

C.BearSF.Bear

Neand

Stadials – OXCAL raw dates

extinctions - raw dates

P(rand overlap = 0.46)

combined ext/app P(rand overlap) = 0.27