VI. Introduction to Branching Modelsfoote/MODEL/2007/notes2007...for birth-death models. Clades with higher origination and extinction rates are expected to have greater volatility

GEOS 33001/EVOL 33001 11 October 2007 Page 1 of 23

VI. Introduction to Branching Models

1 Overview

1.1 Homogeneous vs. heterogeneous models

1.1.1 Homogeneous models: all taxa governed by same rates at all times.

1.1.2 Simplicity-reality trade-off

1.1.3 Alternative risk models

• Temporal variance

• Variance among taxa

• Dependence on taxon age


1.2 Hierarchical vs. non-hierarchical models

1.2.1 Several models are hierarchical in the sense that they track theaggregate behavior of lower-level entities (species) within higher-levelentities (clades).

1.2.2 Others more explicitly hierarchical in the sense that they include anadditional process that gives rise to new clades.

1.2.3 Because we will allow the origin of new clades from existing clades, theexisting clades are rendered paraphyletic—what Raup (1985,Paleobiology 11:42-52) referred to as paraclades.


1.3 Pure birth vs. birth-death models

1.3.1 Pure birth unrealistic, but useful for training intuition

1.3.2 For some problems (e.g. dealing with extant clades lacking fossil record)net diversification rate is of greatest interest, and pure birth model is auseful first pass.

1.4 Forward vs. inverse problems

1.4.1 Forward: prediction of system behavior, given model assumptions

1.4.2 Inverse: estimation of model parameters

1.5 Analytical approaches vs. simulation

1.5.1 Analytical approach useful when feasible

1.5.2 Simulation often necessary when model not analytically tractable

2 Pure-birth (Yule) model

(Yule, 1924, Phil. Trans. R. Soc. Lond. B 213:21-87.

2.1 Historic background

2.2 Model structure

2.2.1 Discrete time steps

2.2.2 p=Probability of branching per unit time; let p′ = (1− p)

2.2.3 Nt denotes the number of species in the clade at time t; N0 = 1 at t = 0.

2.3 Model outcome

2.3.1 After one time unit:

• Pr{N1 = 1} = p′

• Pr{N1 = 2} = p

2.3.2 After two time units:

• Pr{N2 = 1} = p′2


• There are two ways to get to N2 = 2:

Pr(N2 = 2) = pp′2 (if N1 = 2)

+ p′p (if N1 = 1)

= pp′(p′+ 1)

• Pr{N2 = 3} = 2p2p′ (if N1 = 2)

• Pr{N2 = 4} = pp2 = p3 (if N1 = 2)

2.3.3 We can proceed like this, but it gets tedious. To simplify, switch tocontinuous time

• Let p be the instantaneous rate per lineage-million-years. If one million years issubdivided into n fine increments of 1/n m.y. each, then the probability of multipleevents in a fine increment of time will be negligible. Thus, the probability of noevents in 1 m.y. will be equal to (1− p/n)n. As n →∞, this converges to e−p.

• In general, the probability that a process acting at a rate p over a time t will yieldzero events is equal to e−pt (Poisson distribution).

• Thus, for the Yule process, the proportion of monotypic taxa after time t is expectedto be equal to e−pt.

• Applying the same approach to N = 2, 3, ... etc., we end up with the probability ofhaving a standing diversity of exactly n species after time t (where n ≥ 0, since thereis no extinction):

Pn,t = e−pt(1− e−pt)(n−1)

• This is one of the classic Hollow Curves.

• Note that, for any p and t, the maximal probability is that the clade will bemonotypic. The probability drops off as n increases, and the curve becomes flatter ast increases.

• Expectation: E(n, t) = Pn,t = ept, which is intuitively reasonable.

• Variance: V (n, t) = ept(ept − 1).

• Note that the variance grows exponentially with time. This implies that an enormousrange of seemingly different diversity trajectories may be probabilistically consistentwith a single underlying growth parameter. This same principle holds for birth-deathmodels.


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• Note also that the variance increases with the branching rate. Again, this is also truefor birth-death models. Clades with higher origination and extinction rates areexpected to have greater volatility in diversity (and empirically this has beenverified—Gilinsky 1994, Paleobiology 20:445-458). Such clades are also more likely tocrash to total extinction just by chance (ammonoids as a classic example).

• Because variance (volatility) increases with the rate of the process, it is also expectedto be higher at lower taxonomic levels.

From Gilinsky (1994), who measures volatility as average absoluteproportional change in diversity per m.y. (see his eqn. 1).


3 Birth-death model

3.1 Model structure

3.1.1 p is the instantaneous branching rate per lineage-million-years (Lmy)

3.1.2 q is the instantaneous extinction rate per Lmy

0

2

4

6

8

10

Number of lineage−million−years (Lmy) within a time interval

Tim

e (m

.y.)

24

6 2 3 5

2 1 5

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2 3

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3.1.3 See Appendix from Raup (1985) (reproduced below) for some of themost important predictions of this model.

(Note that Raup uses λ and µ where we use p and q.)

3.1.4 Of most use to us will be equations for lineage survival (A1-A4),survival of a paraclade (A11-A14), paraclade diversity (A15-A18), andtotal progeny (A28-A30).




4 Discrete-time (“MBL”) simulation of clade histories

4.1 Historically extremely important in consciousness-raisingabout role of stochastic processes in paleontological data.

4.2 Because variance depends on magnitude of rates, it isessential that simulations be empirically scaled (see Stanleyet al. (1981) Paleobiology 7:115).

4.3 With discrete-time modeling, time steps should be shortenough that one can neglect the probability of multipleevents resulting from the corresponding continuous-timeprocess within a time step.

4.3.1 How would you determine whether, for a given p and q, the time stepsare short enough to yield reasonable approximation to continuousprocess?

5 Inverse Methods, 1: Long-term average rates of

species within a clade

5.1 Analysis of “exact” durations, not binned into age classes

5.1.1 Assume exponential survivorship (consequence of time-homogeneousextinction)

5.1.2 Assumption often not critical: If q varies over time and q(T ) denotes the

time-specific rate, then survivorship depends on e−R t0 q(T )dT rather than

e−qt, i.e. it depends on the mean extinction rate. What is critical is thatrate not be age-dependent.

5.1.3 Also assume effectively infinite window of observation (so truncationeffects can be ignored).

5.1.4 Density function: f(t) = qe−qt (“relative probability”)

• We already saw this as waiting time to first event.

5.1.5 Distribution: F (t) =∫ t

0f(t)dt = 1− e−qt

5.1.6 Probability of survival at least to time t:Pr{dur ≥ t} = e−qt


0 10 20 30 40

0.0

0.2

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1.0

Duration (arbitrary time units)

Pro

port

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with

dur

atio

n >

T

A

0 10 20 30 40

0.01

0.02

0.05

0.10

0.20

0.50

1.00

Duration (arbitrary time units)

Pro

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with

dur

atio

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T (

log

scal

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B


5.1.7 Maximum-likelihood estimate of q:

• Let there be n observed durations T1, ..., Tn.

• Likelihood: L(q) =∏

i qe−qTi

• Support (log-likelihood): S =∑

i{ln(q)− qTi}

• dSdq

=∑

i 1/q −∑

i Ti = n/q −∑

i Ti

• Maximize S, so set dSdq

to zero.

• This implies q̂ = n/∑

i Ti, i.e. the ML estimate of q is the inverse of the meanobserved duration.

5.1.8 This approach generally requires that durations be stratigraphically wellresolved, because we want the “exact” duration. The next approachallows durations that are discretely binned.

5.2 Dynamic Survivorship Analysis (see Alroy mammal example)

Survivorship Table for North American Cenozoic Mammal Species.Cumulative Cumulative

Duration (m.y.) Number Proportion Number Proportion0-1 1718 0.584 2941 1.0001-2 429 0.146 1223 0.4162-3 331 0.113 794 0.2703-4 200 0.068 463 0.1574-5 91 0.031 263 0.0895-6 64 0.022 172 0.0586-7 38 0.013 108 0.0377-8 31 0.011 70 0.0248-9 14 0.0048 39 0.0139-10 6 0.0020 25 0.008510-11 7 0.0024 19 0.006511-12 2 0.0007 12 0.004112-13 5 0.0017 10 0.003413-14 3 0.0010 5 0.001714-15 1 0.0003 2 0.000716-17 1 0.0003 1 0.0003SOURCE: Alroy, 1994


5.2.1 Assume age classes spaced at ∆t

5.2.2 Raw numbers (dx):Pr(x−∆t ≤ T < x) = e−q(x−∆t) − e−qx = e−qx(eq∆t − 1).

• Since (eq∆t − 1) is a constant, the proportions decay exponentially even though theyare discretely binned.

5.2.3 Cumulative numbers (lx):Pr(T ≥ x) = e−qx.

5.2.4 Both raw and cumulative counts/proportions decay exponentially, so wecan calculate slope of regression of ln(lx) or ln(dx) against age (duration)x and take magnitude of this slope as estimate of q.

Data from Alroy (1994); slopes (solid) are -0.458 (cumulative) and -0.476 (raw).

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5.2.5 Another approach is to use maximum likelihood (always on raw counts,never on cumulative numbers!):

• Likelihood: L(q) =∏m

x=1(PDx)dx

• Here m is the number of age classes, and PDx is the probability of falling in age classx if extinction rate is q: PDx = e−q(x−1) − e−qx. Note that this assumes q is expressedas a rate per unit time interval, where the intervals correspond to age classes.

• Support: S(q) =∑m

x=1 dx ln PDx

• Maximize the support via numerical optimization.

5.2.6 Caveat with cumulative proportions: may exaggerate regularity ofsurvivorship relationship and hide non-exponential behavior.

2 4 6 8 10

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68

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Number of species drawn from uniform distribution on (1,20)

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2 4 6 8 10

2040

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Duration

Cum

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num

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of s

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5.3 Lyellian proportions

5.3.1 LT is the proportion of species sampled T million years ago that are stillextant today. Thus, under the exponential survivorship model weexpect LT = e−qT or q = − ln(LT )/T .

5.3.2 Steven M. Stanley’s application to molluscs (see appended figure andtable).

Neogene faunas from California and Japan (Stanley et al. 1980)

Lyellian Proportions for Species Within Neogene Molluscan Faunas

Lyellian Proportion Extinction Rate (per Lmy)Age of Fauna (m.y.) Bivalves Gastropods Bivalves Gastropods2 0.67 0.60 0.20 0.264 0.65 0.35 0.11 0.268 0.38 0.42 0.12 0.1110 0.46 0.25 0.08 0.1412 0.30 0.05 0.10 0.2516 0.38 0.15 0.06 0.1220 0.08 0.08 0.13 0.13SOURCE: Stanley et al., 1980


5.4 Cohort survivorship analysis

5.4.1 Problem with dynamic survivorship: It assumes a stable agedistribution.

(E.g. there cannot be any 100-million-year old species if one samples a clade that is only 50million years old.) For this and other reasons, it is useful to consider survival in real time(cohort survivorship analysis).

5.4.2 Example of cohort survivorship table (Alroy’s mammal data):


Cohort survivorship curves from Alroy’s (1994) mammal dataRange of slopes: −0.2 to −1.3; median slope −0.5

60 50 40 30 20 10 0

5

10

20

50

100

Time (millions of years before present)

Per

cent

sur

vivi

ng (

log

scal

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5.4.3 One simple approach: Fit data to individual cohort curves and averagethem (example with Alroy’s mammal data).

5.4.4 Or fit a single rate to all cohort curves by superimposing them.

• With binned data, assignment of first appearance is somewhat arbitrary.

• But we can avoid this problem by simply tracking cohort from the end of the intervalin which it starts (exponential survivorship is memoryless).

• This solution comes at a cost: If we adopt it, we don’t use data on single-intervalspecies. (But, as we’ll see later, this may be a good thing!)


5.5 What about origination? Analogous methods with reversesurvivorship.

5.5.1 NB: Census Method (tabulating ages of living taxa) has been used toestimate extinction rate, but, under the time-homogeneous model, theages of living taxa depend on origination rate.

5.6 Inference of branching and extinction probabilities fromdiversity trajectory (Gilinsky and Good 1991, Paleobiology17:145-166).

5.6.1 Assume discrete-time “generations”, during which diversity cannotmore than double.

5.6.2 p0, p1, and p2 are the probabilities of leaving 0, 1, or 2 descendantsduring a generation, i.e. the probabilities of extinction, persistence, andbranching.

5.6.3 This can be expressed as a Probability Generating Function,G(x) = p0x

0 + p1x1 + p2x

2 + ..., where the coefficient before xn is theprobability that the diversity will be equal to n. (We ignore the termsin powers higher than x2, since we are assuming “generations” in whichdiversity can at most double.)

5.6.4 Suppose we start with n = 1. Then:

• Pr(n = 1 → n = 0) = p0,

• Pr(n = 1 → n = 1) = p1,

and

• Pr(n = 1 → n = 2) = p2.

5.6.5 Now suppose we start with n = 2. With a little thought about thedifferent ways of going from one diversity to another, we can see that:

• Pr(n = 2 → n = 0) = p20

• Pr(n = 2 → n = 1) = 2p0p1

• Pr(n = 2 → n = 2) = p21 + 2p0p2

• Pr(n = 2 → n = 3) = 2p1p2 and

• Pr(n = 2 → n = 4) = p22


5.6.6 This can be expressed as a different generating function,G2(x) = p0x

0 + 2p0p1x1 + [p2

1 + 2p0p2]x2 + 2p1p2x

3 + p22x

4 + ..., where again thecoefficient before xn is the probability that diversity will be equal to n.

5.6.7 What makes this useful is that (as we could show with lots of tediousalgebra) G2(x) = G(x)2. Similarly, the generating function G3(x) that wewould write out for starting with n = 3 is equal to G(x)3, and so on. (Inpractice, the expansion of these generating functions is done with aFourier transform; see Gilinsky and Good for details.)

5.6.8 NB: These probabilities can also be calculated using thecontinuous-time equations from Raup 1985 (eqs. A11-A14 [for n = 0]and A15-A18 [for n ≥ 1]).

5.6.9 Now, suppose we have a diversity path over m time steps: n1, n2, ..., nm.Then, for some candidate set of rates p0, p1, and p2, we would calculatethe likelihood as L(p0, p1, p2|n1, ..., nm) =

∏mi=1 Pr(ni−1 → ni|p0, p1, p2) [or the

support as S(p0, p1, p2|n1, ..., nm) =∑m

i=1 ln Pr(ni−1 → ni|p0, p1, p2)]. We wouldthen find, numerically, the values of p0, p1, and p2 that maximize thelikelihood.

5.6.10 Comments: Method is very promising because it rests on readilyavailable diversity data, but it has barely been applied. Its sensitivityto general incompleteness should be explored (setting aside thequestion of temporal variation in sampling). For example, suppose halfof all taxa are sampled in each interval of time. The probability ofgoing from 2 to 3 is not the same as the probability of going from 4 to6. How does this affect the robustness of the method?


6 Inverse Methods, 2: Interval-by-interval rates

6.1 All taxa in an interval fall into one of four exclusive groupsdepending on whether they first appear before or during theinterval and last appear during or after the interval.

6.2 Using the subscripts b and t to refer to taxa that cross thebottom and top boundaries of the time interval, and using Fand L to refer to first and last appearance, we can denotethe numbers of taxa in the four categories as NbL, NFt, NFL,and Nbt. The number of taxa making a first appearance inthe interval is equal to NF = NFt + NFL and the numbermaking a last appearance is equal to NL = NbL + NFL. Thenumber alive at the beginning of the interval is equal toNb = NbL + Nbt, and the number alive at the end of theinterval is equal to Nt = NFt + Nbt.


6.3 Exponential survivorship implies that Nbt = Nbe−q∆t, where ∆t

is the interval length. Thus we can estimate extinction rateas q = − ln(Nbt/Nb)/∆t (see the following table).

6.3.1 If extinction rate varies through the interval, then Nbt = Nbe−

R ∆t0 q(T ) dT

and − ln(Nbt/Nb)/∆t yields the mean rate.

6.4 Likewise, for origination rate: p = − ln(Nbt/Nt)/∆t.

6.5 Note that rate measures that normalize by interval lengthtacitly assume that the originations and extinctions aredispersed throughout the interval. If we have reason tobelieve that this is not the case—i.e. that the number ofevents does not really depend on interval length—then itmay be best not to divide out by ∆t.

Taxonomic counts and taxonomic rate measures.Quantity Symbol EquivalenceTaxa at beginning of interval Nb NbL + Nbt

Taxa at end of interval Nt NFt + Nbt

Total diversity in interval Ntot NbL + NFt + NFL + Nbt

Number of first appearances NF NFt + NFL

Number of last appearances NL NbL + NFL

Proportional origination PO NF /Ntot

Proportional extinction PE NL/Ntot

Proportional origination per m.y. POm.y. PO/∆tProportional extinction per m.y. PEm.y. PE/∆tVan Valen’s origination rate VO NF /[(Nb + Nt)/2]/∆tVan Valen’s extinction rate VE NL/[(Nb + Nt)/2]/∆tPer-capita origination rate per Lmy p −ln(Nbt/Nt)/∆tPer-capita extinction rate per Lmy q −ln(Nbt/Nb)/∆t


Effect of interval length (∆t) on two common measures of extinction

Number of extinctions ∼∫ ∆t

0qe(p−q)tdt

Total diversity ∼ 1 +∫ ∆t

0pe(p−q)tdt

0 5 10 15 20

0.0

0.2

0.4

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Interval length (m.y.)

Pro

port

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extin

ct

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Interval length (m.y.)

Pro

port

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extin

ct p

er m

.y.


7 Inverse Methods, 3: An alternative (untested)

approach to estimating interval-by-interval rates.

7.1 Let there be m intervals with extinction rates q1, ..., qm, where1 is the oldest and m is the youngest.

For simplicity, consider rates per interval rather than per m.y. (this has no effect onultimate result).

7.2 For each interval, the Li be the Lyellian proportion—theproportion of taxa found in that interval that are still extanttoday.

Treat taxa as ranging through entire interval, i.e. if they are found in an interval they areassumed to be extant at the start of the interval.

7.3 Then:

• L1 = e−q1e−q2e−q3 ...e−qm−1e−qm ,

• L2 = e−q2e−q3 ...e−qm−1e−qm ,

...

• Lm−1 = e−qm−1e−qm ,

and

• Lm = e−qm

7.4 Taking logarithms, we have ln Li =∑m

j=i−qj.

7.5 In matrix form we have:−1 −1 −1 · · · −10 −1 −1 · · · −10 0 −1 · · · −1...

. . .

0 0 0 · · · −1

q1

q2

q3...

qm

=

ln L1

ln L2

ln L3...

ln Lm

or

X~q = ~L

Thus~q = X−1~L

Documents

VI. Introduction to Branching Modelsfoote/MODEL/2007/notes2007...for birth-death models. Clades with higher origination and extinction rates are expected to have greater volatility