Transcript Fossils and thylacines: the statistics of species extinction

Cockroaches and
Thylacines: The Hazards of
Species Extinction
Christopher G. Small, University of Waterloo
(joint with Sheena Zhang & Grace Chiu)
Dedicated to
Eric Rowland Guiler
(1922—2008)

In this talk, I shall examine the problems
of species extinction at the macro- and
micro-levels.
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
At the macro-level, I shall consider the
statistics of extinction in the fossil record.
At the micro-level, I shall try to extract a few
facts from the limited data we have about the
thylacine population in the late nineteenth and
early twentieth centuries.
EXTINCTIONS
180
160
140
120
100
80
60
40
20
0
700
600
500
400
300
200
100
0
Time in millions of years before present
Extinctions of families over time
Time measured in millions of years (Ma) before present. Graph
shows the number of families of organisms which went extinct in
each time period. Source: The Fossil Record 2 Database, M. J.
Benton.
Number of extinctions (EXT)
180
160
140
120
100
80
60
40
20
0
0
5
10
15
20
25
30
35
Length of time interval (INT)
Extinctions (EXT) against time interval (INT):
Taxonomic families have become extinct at a rate of
m=5.858 families/Ma over the last 550 MY.
40
Pearson residuals of
extinctions
PRES
30
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
25
20

15

10
5

0
600
500
400
PRES =
300
200
100
EXT  mˆ INT
mˆ INT
0
-5

-10
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Permian-Triassic (251 Ma).
Ordovician-Silurian (444
Ma).
Triassic-Jurassic (200 Ma).
Devonian-Carboniferous
series of extinctions (360
Ma).
Cretaceous-Tertiary (65
Ma). Goodbye T-rex!
End Eocene event (33 Ma).
Holocene extinction (0 Ma).
Goodbye us?
Mass extinctions: standard?
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Variation in size of
extinctions far exceeds
“background noise.” I.e.,
extinctions of taxa are
clustered not independent.
Mass extinctions are
extreme but not
exceptional. (David Raup,
1991).
Mass extinctions are both
extreme and exceptional
(Richard Bambach & Andrew
Knoll, 2001).
CUMULATIVE EXTINCTIONS
4000
3500
3000
2500
2000
1500
1000
500
0
700
600
500
400
300
200
100
0
Extinctions have occurred at a fairly constant rate
on average over time despite occasional mass
extinctions.
CUMULATIVE ORIGINATIONS
6000
5000
4000
3000
2000
1000
0
700
600
500
400
300
200
100
0
New taxa have been appearing at a fairly
constant rate over time (with a slight increase in
speed over the last 50 Ma).
However ……………..
EXTANT NUMBER OF FAMILIES
2500
2000
1500
1000
500
0
700
600
500
400
300
200
100
0
Diversity over time (Ma)
The diversity of taxa (in this case families) at any given
time has been steadily increasing (with a slight drop at
the end of the Palaeozoic).
The hazards of extinction
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

The statistical approach to studying these
extinctions is to treat each species or family
as subject to an ongoing hazard of extinction.
This hazard is quantified by means of a
hazard function whose value can change over
time.
The probability of extinction will depend
upon the value of the hazard integrated over
time.
Hazards of extinction
 Let m (t ) denote the hazard function for extinction of a
randomly chosen taxon (species, family, phylum …) at
time t. That is, m (t) dt is the probability of extinction
of the taxon in a time interval of length dt given its
survival up to time t .
 For example, if the extinction rate is constant over
time, then m (t ) = m for all t, and the probability of
extinction by time t  t of a species extant at time t
is
1 e
 m t
.
Species extinction
sweepstakes (m=10-7)
Winners and losers
Winner? (> 108 years)
Loser? (4 x 106 years)
Estimating nonconstant
hazard functions

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
To estimate the hazard function, we would like to
observe extinctions in the fossil record and when
they happened.
So we would like to know the precise number of taxa
that are extant at different times.
Unfortunately, this is difficult:

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counts are binned into time periods,
dating of individual fossils is uncertain
pseudoextinction, Lazarus taxa, Elvis taxa.
At best we use proxy variables

at any time we can determine the taxa that appear
before and after that time.
STAGE
DURATION
MIDPOINT
ORIGINS
EXTINCTIONS
DIVERSITY
VENDIAN
40
590
4
1
16
CAERFAI
34
555.5
230
170
245
ST DAVID'S
19
527
68
52
143
MERIONETH
7
513.6
70
59
161
TREMADOC
17
501.5
62
25
164
ARENIG
17
484.5
142
34
281
LLANVIRN
7.5
472.3
74
35
321
LLANDEILO
4.5
466.3
57
17
343
CARADOC
21
453.5
108
55
434
4
441
54
99
433
LLANDOVERY
8.5
434.7
66
29
400
WENLOCK
6.5
427.2
63
40
434
ASHGILL
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Let
N (t ) be the number of taxa that are extant a
time t . Proxy variables will henceforth be represented
by using a tilde.
 A proxy for
~
N (t ) is N (t ) : the number of taxa
that appear in the fossil record both before and
after time t . Let


~
~
~
N (t ) = N (t )  N (t  t )
~
~
~
 ln N (t ) = ln N (t )  ln N (t  t )
~
Let E (t ) be the cumulative number of taxa that
appear before time t but not after time t . Define
~
E (t ) similarly.
~
Define O (t ) be the number of taxa that appear
~

O
(t ) similarly.
after time t but not before t , and
Generalized Birth and Death Process
Assumptions (D. Stoyan, 1980):
 Let m (t ) be the hazard function for a
taxon to become extinct at time t .
 Similarly, m (t ) = m (t  t )  m (t )
 Let  (t ) be the rate of speciation of a
given taxon into new taxa at time t .
 Similarly,  (t ) =  (t  t )   (t ) .

Given all the information about extant
taxa at time t , the expected number of
extinctions between t and t  t is
s
t  t
 ( x)m ( x) dx

t
E E (t ) =  m ( s ) N (t ) e
ds
.
t
 Given all the information about extant
taxa at time t , the expected number of
originations between t and t  t is
s
t  t
 ( x)m ( x) dx

t
E O(t ) =   ( s ) N (t ) e
ds
.
t
.
 When t is small these integrals
reduce to
E E (t ) = m (t ) N (t )
e
[ (t )  m (t )] t
1
 (t )  m (t )
and
E O(t ) =  (t ) N (t )
e
[ (t )  m (t )] t
 (t )  m (t )
1
 So estimators for m and  are found by
substituting proxies and solving
~
~ e
E (t ) = mˆ (t ) N (t )
[ˆ (t )  mˆ (t )] t
1
ˆ (t )  mˆ (t )
and
~
~ e
ˆ
O (t ) =  (t ) N (t )
[ˆ (t )  mˆ (t )] t
ˆ (t )  mˆ (t )
1
 These simultaneous equations have
solutions
~
~
E (t )  ln N (t )
mˆ (t ) = ~
N (t )
t
and
~
~

O
(
t
)

ln
N
(t )
ˆ
 (t ) = ~
N (t )
t
Hazard of extinction over
cenozoic
0.1
0.09
0.08
0.06
0.05
0.04
Hazard
0.07
0.03
0.02
0.01
0
70
50
30
10
Time in Ma before present
-10
An approximate 95% conditional confidence
interval for the hazard function can be
obtained from the delta method and a
variance stabilizing transformation to be


~
~
~
E (t )  2 E (t )  ln N (t )
~
N (t )
t
Comments
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Most extinctions are “mass extinctions.”
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Models for extinctions must explain the homogeneity of extinctions and
originations in the face of increasing diversity of taxa.
Research may be concentrating too much on taxa and not enough on ecological niches (D.
H. Erwin, 2006).
We should be modelling hazards and not simply fitting them.
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The Generalised BD model is useful for hazards, but does not fit other aspects of data
well.
Current work is investigating modelling the extinction rate as a nonnegative
strongly stationary process.
Without modelling there are no null hypotheses: 26 Ma cycles of extinction?
(Raup and Sepkoski,1984). Statistical artefact? (Stigler and Wagner).
The error analysis of proxy variables is nontrivial.
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It is formally equivalent to the problem of estimating the support of a distribution.
C. Marshall (1994), but more work necessary!
Decline of the thylacine
Possible sources of
thylacine decline:
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Hunting and trapping (the latter for zoos
worldwide).
Destruction of habitat.
Competition with wild dogs.
Disease (esp. reports of a “distemper-like”
disease, 1910).
Thylacines presented for govt. bounty & thylacines killed
at Woolnorth
180
20
160
18
140
16
120
100
14
12
10
80
60
8
6
40
4
20
2
0
0
Tasmania
Woolnorth
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This time series is the outcome of a number of factors including
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It is difficult to extract information about the thylacine from such
data sets because they have information about both humans and
thylacines.
These factors are confounded. To extract information about the
thylacine demographics alone, we must either
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
thylacine demographics
human demographics, and
socio-economic factors associated with the settlement of Tasmania at
the time.
find additional ancillary information about the socio-economic activity,
or
find a mathematical model for the socio-economic activity.
Both approaches can be viable, but the second is difficult because,
to put it simply, human beings rarely obey the equations imposed on
them.
Van Diemen’s Land Company at
Woolnorth
Records at VDL Company,
Woolnorth (E. Guiler, 1985)
1899
26 Aug.
Tom went to the Mount to look after a tiger with his
dogs.
2 Nov.
Sent some men to hunt tiger out of Studland Bay run.
11 Nov.
All hands in a.m. hunting a tiger out of the Forest.
Set snares for a tiger on Saltwater Creek fence.
6 Feb.
Tracks seen in Forest.
27 May
Chasing tiger in the Forest.
17 July
Tiger at Studland Bay, at the Knolls.
1891
1 Aug.
Laid poison at Harcus for hunters’ dogs.
1892
26 Aug.
Two men to Studland Bay to shift tiger.
1890
1893
No comments.
Records at VDL Company,
Woolnorth (continued)
1898
1899
1900
20 Feb.
One tiger caught, no locality given.
20 July
One tiger caught, McCabe’s Paddock
31 Dec.
Snaring in the Forest.
3 July
Saw two tigers at Swan Bay.
6 July
Caught two tigers in Forest and Three Sticks
22 July
Tiger scaring on Three Sticks and Studland Bay
23 Nov.
One tiger caught, probably at the Mount.
24 Jan.
One tiger caught, locality not stated.
8 Feb.
Tiger scaring at Three Sticks
Hunting Record at Woolnorth
(1898—1906)
Hunting Record in Woolnorth
Thylacines caught
Hunting attempts
35
30
25
20
15
10
5
0
1898 1899 1900
1901 1902 1903
1904 1905 1906
Year
We model the number of potential encounters of thylacines in
year t as binomial with parameters n(t ) and p , where n(t ) is
the (unknown) number of thylacines in the area at time t and p
is the (unknown) probability of killing per individual. Let h(t ) be
the probability that a thylacine hunt at time t is “successful.”
h(t ) = Pr{ Bin (n(t ), p)  0 }
= 1  Pr{ Bin (n(t ), p) = 0 }
 1  e  p n(t ) .
Therefore, n(t ) will be
n(t )  
ln [1  h(t )]
.
p
Suppose we model the rate of decline of n(t ) as
d E n(t )
=  m E n(t ) ,
dt
which implies that ln n(t )   m t  C0 . Then
ln   ln [1  h(t )]  = m t  C0  ln p  error
= m t  C  error.
Therefore, we model using the regression equation:


ln  ln [1  h(t j )] =  m t j  C   j
which can be fit by usual least square methods.
Rate of thylacines h(t) successfully be hunted
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1898
1899
1900
original
1901
1902
1903
Year
3 Step smoothed
1904
1905
regression
1906
1
0.9
0.8
Thylacine Population around Woolnorth
0.7
0.6
0.5
Population estimate
Exponential fit from regression
0.4
0.3
0.2
0.1
0
1898
1899
1900
1901
1902
Year
1903
1904
1905
1906
Comments
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The data suggest that by the beginning of the
twentieth century the decline in the thylacine
population was substantial.
While a “distemper-like” disease may have
contributed to thylacine decline, the evidence is
that the thylacine was disappearing before this.
Habitat, dogs and hunting are the main factors to
be considered.
References
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Bambach, R. K. & Knoll, H. (2001). “Is there a separate class of `mass
extinctions?’” GSA Annual Meeting, 5—8 November 2001.
Erwin, Douglas H. (2006). Extinction: How Life on Earth Nearly Ended 250
Million Years Ago. Princeton University.
Guiler, Eric R. (1985). Thylacine: The Tragedy of the Tasmanian Tiger,
Oxford University.
Marshall, C. R. (1994). “Confidence intervals on stratigraphic ranges:
partial relaxation of the assumption of randomly distributed fossil
horizons.” Paleobiology 20, 459—469.
Raup, D. M. (1991). Extinction: Bad Genes or Bad Luck? Norton.
Raup, D. M. & Sepkoski, J. J. Jr. (1984). “Periodicity of extinction in the
geologic past.” Proc. Nat. Acad. Sci. USA 81, 801—805.
Stigler, S. M. & Wagner, M. J. (1987). “A substantial bias in nonparametric
tests for periodicity in geophysical data.” Science 13, 940—945.
Stoyan, D. (1980). “Estimation of transition rates of inhomogeneous birthdeath processes with a paleontological application.” Elektronische
Informationsverabeitung u. Kybernetic 16, 647—649.