Inferring Diversity Dynamics · approaches can’t work: Diversity dynamics: molecular phylogenies need the fossil record Tiago B. Quental and Charles R. Marshall Museum of Paleontology

INTRODUCTION BACKGROUND BUILDING A BETTER LIKELIHOOD CONCLUSIONS

Inferring Diversity Dynamics

Todd L. Parsons

Department of MathematicsUniversity of Toronto

http://www.sas.upenn.edu/˜tparsons/

Mathematical Biology Research SeminarMcMaster University

October 6, 2011


IN COLLABORATION WITH

Helene MorlonEcole Polytechnique

Joshua PlotkinUniversity of Pennsylvania


OUTLINE

INTRODUCTION

BACKGROUND

BUILDING A BETTER LIKELIHOOD

FIRST ATTEMPT

SECOND ATTEMPT

SUCCESS!

CONCLUSIONS


UNDERSTANDING THE DYNAMICS OF

DIVERSIFICATION

There is considerable interest, both theoretical (communityecology) and practical (conservation biology), in understandingthe rates at which new species arise, and existing species perish.


THE FOSSIL RECORD

Diversity dynamics may be inferred using fossil data:

Sepkoski, Paleobiology (1978)

Alroy, Science (2010)

However, many groups lack adequate fossil data (e.g. floweringplants, insects, or birds).


THE FOSSIL RECORD

Diversity dynamics may be inferred using fossil data:

Sepkoski, Paleobiology (1978)


However, many groups lack adequate fossil data (e.g. floweringplants, insects, or birds).


CORRECTING THE FOSSIL RECORD

Moroever, different methods to correct for the incompletenessof the fossil record yield contrasting results:

Marine invertebrates

Benton, Science (2009)

Planktonic diatoms

Rabosky & Sorhannus, Nature (2009)


MOLECULAR APPROACHES

I This suggests an alternate approach, using phylogeniesrecovered from sequence data.

I In this case, rather than having the actual tree of species,we have a reconstructed phylogeny of species that havesurvived to the present:

→

Nee, May & Harvey, Philosophical Transactions: Biological Sciences (1994)

I Together with a branching process model, these may beused to estimate rates of speciation (λ) and extinction (µ).





→







→




UNFORTUNATELY. . .

Extinction rates estimated from molecular phylogenies aretypically too low to be realistic:

Rabosky & Lovette, Evolution (2008)

Estimated speciation (solid) and extinction (dashed) for A. Australian agamid lizards,

B. North American wood-warblers, and C. Australo-Papuan pythons.



Many extant clades should be in decline, but phylogeneticinference does not detect this.


Moreover, direct comparison ofphylogenetic inference with fossildata from the cetaceans (whales,dolphins, and porpoises) showinconsistencies.

Can diversification dynamics be gleaned withoutfossils?As pointed out by Ricklefs [2], a molecular phylogeny, byvirtue of startingwith the living biota, must show a patternof increasing diversity with time regardless of the truediversity dynamics of the clade. This perceptual bias isreflected in the methods developed for inferring diversitydynamics from molecular phylogenies. These typicallyassume that the speciation rate is on average higher than,or occasionally equal to [10,19], the extinction rate. How-ever, it is clear from the fossil record that this is often notthe case. Paleontologically, our primary experience of bio-diversity dynamics is one of the ‘waxing and waning’ ofclades, with many clades showing complex diversity tra-jectories through time. Most clades (including many of theliving clades) have had negative diversification rates atsome point in their history.

The central question that must be answered is: do ourviews of the patterns and rates of diversification changewhenwe have access to the extinct lineages?Herewe arguethat the answer is substantially ‘yes’. If we cannot accessthe extinct lineages, we are blind to their extinction andthe origination dynamics. Thus, not only dowe lack data onthe extinction dynamics of the clade, we also lack data on aportion of their origination dynamics. We support ourconclusion with evidence drawn from computer simu-lations and the fossil record.

‘Time-traveling’ with computer simulationAs discussed above, if there is extinction, computer simu-lations show that diversity-dependent diversification canbe detected in a molecular phylogeny only if the initialspeciation rate is sufficiently high to ameliorate the erosiveeffect of the extinction. A metric that quantifies how highthe initial speciation needs to be is the LiMe ratio (the ratioof the initial speciation rate to the equilibrium extinctionrate) [21]. For example, if the speciation rate is linearlydensity-dependent and the extinction rate density-inde-pendent, then the LiMe ratio must be about !4 to give a50% chance of seeing a decreasing rate of diversification ina molecular phylogeny using the g statistic (with an equi-librium diversity of 100 species and when the computersimulations are stopped the first time the equilibriumdiversity is reached [21]). When the computer simulationsare re-run, keeping track of the g statistic as the diversity-dependent diversification unfolds, we find that, even forlarge LiMe values (ones that should generate negative gvalues), there is only a relatively narrow temporal windowwhere one is likely to see a significantly negative g value(Figure 1). At the initial phases of the radiation, the g value

is not yet significantly negative, reflecting the initial expo-nential phase of the radiation. Strikingly, after the radi-ation is complete, the g value once again becomes non-negative but, rather than reflecting exponential growth, itreflects species turnover at a constant diversity [24]. Thepositive g value seen in molecular phylogenies does notnecessarily mean that the clades are diversifying [24–25].

Thus, a non-negative g value might indicate: (i) trueexponential growth (or even increasing rates of diversifica-tion); (ii) logistic growth in its earliest stages (which iseffectively exponential); (iii) a wide range of diversity-de-pendent diversification processes; (iv) a clade that is main-taining a constant diversity and is simply experiencingspecies turnover (Table 1). With more imaginative compu-ter simulationswemay find additional scenarios that couldlead to non-significantly negative g values. We also notethat for real molecular phylogenies (assuming that theirdynamics were governed by diversity-dependent diversifi-cation), we do not have a simple way of determining wherethey are in their diversification trajectory (but see [21]).Interpreting the process of diversification for phylogenieswith non-significantly negative g values is fraught withdifficulties.

Figure 1. Expected behavior of the g statistic, derived from computer simulations,as radiations with density-dependent diversification proceed. Each simulationassumed diversity-dependent speciation and constant extinction (Box 2), with anequilibrium diversity of 50 lineages. For each simulation, the g values werecalculated throughout the diversification process. Note that the time axis ismeasured in units of the expected species duration instead of absolute time. Thesolid line represents the average g value for 1000 simulated trees and the dashedline the 95% confidence interval. The light-blue area represents g values thatsignificantly support a model of decreasing diversification rate (g less than –1.645).When the LiMe ratio (the ratio of the initial speciation rate to the equilibriumextinction rate) is low (left panel), only seldom does the g value indicate thedecreasing diversification rate. When the LiMe ratio is high (right panel), the g

statistic only detects the decline in diversification for a short window of time that is(asymmetrically) centered about the time the clade first achieves the diversityequilibrium. Modified from Liow et al. [24].

Table 1. Range of diversity dynamics consistent with molecular phylogeniesa

g less than –1.645 Ref g equal to or more than –1.645 Ref

Rate of diversification Decreasing Increasing; mildly decreasing; constant

Possible diversity trajectories 1) Logistic growth [17] 1) Exponential growth2) Early phases of logistic growth [24]2) Constant diversity through time

with diversification pulse and heritableextinction within sub-clades

[19]3) Logistic growth with a low LiMe ratio [17,21]

3) Declining diversity? [current article]4) Species turnover at equilibrium diversity [24]

aWe assume here that the g value for a phylogeny reflects the true diversification of the clade, and is not an artifact of the inability to distinguish the most recently divergedspecies [23], or due to errors in the construction of the chronogram [22].

Opinion Trends in Ecology and Evolution Vol.xxx No.x

TREE-1248; No. of Pages 8

3

Quental & Marshall, TREE (2010)


SHOULD WE ABANDON MOLECULAR APPROACHES?These difficulties have led some to suggest molecularapproaches can’t work:

Diversity dynamics: molecularphylogenies need the fossil recordTiago B. Quental and Charles R. Marshall

Museum of Paleontology and Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720, USA

Over the last two decades, new tools in the analysis ofmolecular phylogenies have enabled study of the diver-sification dynamics of living clades in the absence ofinformation about extinct lineages. However, computersimulations and the fossil record show that the inabilityto access extinct lineages severely limits the inferencesthat can be drawn from molecular phylogenies. Itappears that molecular phylogenies can tell us onlywhen there have been changes in diversification rates,but are blind to the true diversity trajectories and rates oforigination and extinction that have led to the speciesthat are alive today. We need to embrace the fossilrecord if we want to fully understand the diversitydynamics of the living biota.

Molecular phylogenies and rates of diversificationUnderstanding the patterns and processes of diversifica-tion has long been of interest to paleontologists as we usethe fossil record to document biodiversity change throughtime [1]. However, interest in diversity dynamics amongbiologists and the greater public has grown, particularly aswe become aware of the impact of humans on the bio-sphere. Among biologists, the study of biodiversitydynamics was invigorated by the proposal that the ratesand processes of diversification could be inferred frommolecular phylogenies. This is particularly importantgiven that many taxonomic groups have poor-to-non-existent fossil records [2]. In particular, the pioneeringcontributions of Nee et al. [3–5] and Harvey et al. [6]provided methods for detecting mass extinction eventsand for estimating speciation and extinction rates frommolecular phylogenies despite the absence of the extinctspecies (but see [7] and [8]). This work was presaged byThompson [9], and there were also parallel efforts by Hey[10] andYang andRannala [11]. In 2000, Pybus andHarvey[12]widened the scope of enquiry by shifting attention to thepatterns and, through them, the processes of diversification.This line of research continues to be valued bybiologists andnew approaches continue to be developed [13,14]. In thiscontribution, we will discuss how our inferences about thepattern and processes of diversification change when wehave direct access to extinct species.

Describing patterns and inferring processes frommolecular phylogeniesTo study patterns of diversification, Pybus andHarvey [12]introduced the g statistic (Box 1). This is a simple tool for

determining if a molecular phylogeny is consistent with aconstant rate of diversification, or if there has been adecrease in the diversification rate (inferred when molecu-lar phylogenies have significantly negative g values).There is now a sizeable literature which indicatesthat many clades have decreasing diversification rates[12,15–17], in fact for about half of the 160 well-sampledphylogenies analyzed [15]. Considerable attention is nowfocused on exploring the implications of these findings forthe evolutionary and ecological mechanisms responsible

Opinion

Corresponding author: Marshall, C.R. ([email protected]).

Glossary

Boundary-crosser method: used to estimate the diversity at the boundary ofadjacent geological time intervals by counting the number of taxa that musthave crossed the boundary because they are known before and after it. Thismethod has the desirable property of assuring the co-existence of taxa.Chronogram: a phylogeny with branch lengths adjusted so that they areproportional to absolute time. Also known as a ‘time tree’.Crown group: a monophyletic clade that contains all extant members of theclade in addition to its last common ancestor and all of its descendants, bothliving and extinct.Diversification rate: the rate of origination (speciation, l) minus the rate ofextinction (m): (l – m).Diversity trajectory: a curve that portrays the number of species through time.It permits one to see if a given clade was diversifying or declining.Equilibrium diversity: assuming diversity-dependent diversification, the ex-pected number of taxa when the speciation and extinction rates are balanced,i.e. when the net diversification rate is zero.Frequency ratio (FreqRat): statistical method used to measure the incomplete-ness of the fossil record by estimating the sampling probabilities per unit time(r) using the frequency of taxa that have stratigraphic ranges of one (ƒ1), two(ƒ2) or three (ƒ3) time intervals: r = (ƒ2)2/(ƒ1)(ƒ3).g statistic: a statistic that describes the center of mass for the nodes in achronogram compared with the expected center of mass under a pure birthmodel. Nodes concentrated towards the base of a tree indicate a decrease indiversification rates, yielding negative g values.LiMe ratio: the ratio of the initial speciation rate (Lambda initial) and theextinction rate at equilibrium (Mu equilibrium). It assumes the existence ofdiversity equilibrium and, along with the size of the clade, and where the cladehappens to be in its diversity trajectory, it plays a major part in determining theshape of the phylogeny.Logistic growth: in the context of diversity dynamics describes the accumula-tion of species where the growth is initially exponential but as diversityaccumulates the rate of species accumulation decreases, leading to a diversityplateau (the equilibrium diversity).Net diversification rate: the average diversification rate needed to account forthe diversity of a clade at any point in time. The actual diversification ratesexperienced by the clade might have been very different from this retro-spective average rate.Paleobiology Database: worldwide database of fossil collections. It providestaxonomic, geographic, stratigraphic, taphonomic, environmental, and collect-ing data, as well as providing analysis tools (http://paleodb.org).Sampled in bin (SIB) method: method used to estimate diversity by countingthe number of taxa within each geological time interval. Has the desirableproperty of being amenable to corrections of the incompleteness of the fossilrecord, but will typically overestimate the total diversity unless the timeintervals are short with respect to the longevity of the taxa studied.Stem group: Extinct taxa that lie phylogenetically between the last commonancestor of the living species of a clade (the crown group), and the nearestliving relatives of that clade.


0169-5347/$ – see front matter ! 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tree.2010.05.002 Trends in Ecology and Evolution xx (2010) 1–8 1

ORIGINAL ARTICLE

doi:10.1111/j.1558-5646.2009.00926.x

EXTINCTION RATES SHOULD NOT BEESTIMATED FROM MOLECULAR PHYLOGENIESDaniel L. Rabosky1,2,3,4

1Department of Ecology and Evolutionary Biology, Cornell University, Ithaca, New York 148532Cornell Laboratory of Ornithology, 159 Sapsucker Woods Road, Ithaca, New York 148503Department of Integrative Biology, University of California, Berkeley, California 94720

4E-mail: [email protected]

Received February 17, 2009

Accepted December 3, 2009

Molecular phylogenies contain information about the tempo and mode of species diversification through time. Because extinction

leaves a characteristic signature in the shape of molecular phylogenetic trees, many studies have used data from extant taxa only

to infer extinction rates. This is a promising approach for the large number of taxa for which extinction rates cannot be estimated

from the fossil record. Here, I explore the consequences of violating a common assumption made by studies of extinction from

phylogenetic data. I show that when diversification rates vary among lineages, simple estimators based on the birth–death process

are unable to recover true extinction rates. This is problematic for phylogenetic trees with complete taxon sampling as well as

for the simpler case of clades with known age and species richness. Given the ubiquity of variation in diversification rates among

lineages and clades, these results suggest that extinction rates should not be estimated in the absence of fossil data.

KEY WORDS: Adaptive radiation, extinction, macroevolution, phylogenetics.

Time-calibrated molecular phylogenies contain information aboutboth the timing and rate of species diversification and thus providea complementary window into macroevolutionary processes thatare often obscured by the incompleteness of the fossil record. Nu-merous studies have used molecular phylogenies to characterizethe pattern of species diversification through time (e.g., Nee et al.1992; Harmon et al. 2003; Ruber and Zardoya 2005; McPeek2008) and to quantify differences in diversification rates amonglineages (e.g., Mooers and Heard 1997; Moore et al. 2004; Mooreand Donoghue 2007; Rabosky et al. 2007).

One of the most intriguing applications of molecular phy-logenies involves the inference of extinction rates from data onextant taxa only (Nee et al. 1994a; Paradis 2003; Maddison et al.2007; Ricklefs 2007). It may seem counterintuitive that livingspecies could provide any information on historical extinctionrates, but this is indeed possible, because the shape of phyloge-netic trees is influenced both by the net rate of lineage diversifi-cation through time as well as the ratio of the extinction rate µ to

the speciation rate !. This parameter (µ/!), denoted by ", is alsoknown as the relative extinction rate and is critically importantin determining the distribution of speciation times that occur ina molecular phylogeny. Differences in the relative extinction ratecan result in different phylogenetic tree shapes, even for cladesdiversifying under precisely the same net diversification rate (de-noted by r, where r = ! ! µ). This phenomenon occurs becausehigh relative extinction rates lead to high-lineage turnover throughtime, which changes the “age structure” of nodes in a phyloge-netic tree. When " is low, many species will be relatively old, butwhen " is high, most species will be young, simply as a func-tion of high-lineage turnover. This leads to the appearance of atemporal acceleration in the rate of diversification through time,a phenomenon that has been explored by numerous prior studies(Nee et al. 1994b; Rabosky 2006b).

Methods for analyzing species diversification rates oftenmake assumptions about the constancy of diversification ratesthrough time (Magallon and Sanderson 2001) or among lineages

1 8 1 6C" 2010 The Author(s). Journal compilation C" 2010 The Society for the Study of Evolution.Evolution 64-6: 1816–1824









Opinion


Glossary




ORIGINAL ARTICLE

doi:10.1111/j.1558-5646.2009.00926.x




























Opinion


Glossary




ORIGINAL ARTICLE

doi:10.1111/j.1558-5646.2009.00926.x





















OUR QUESTION

Can we reconcile molecular phylogenies with the fossil record?

35 30 25 20 15 10 5 0

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ber o

f spe

cies

Myrs

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f spe

cies

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f spe

cies

Myrs

Myrs

a b

dc

35 30 25 20 15 10 5 0

050

100

150

200

BalaenopteridaeDelphinidaePhocoenidaeZiphiidaeother cetaceans

35 30 25 20 15 10 5 0

020

4060

8010

012

0

BalaenopteridaeDelphinidaePhocoenidaeZiphiidaeother mysticetesother odontocetes

Myrs35 30 25 20 15 10 5 0

Fossil data

Phylogenetic inference

In particular, can we successfully use molecular phylogenies tounderstand cetacean diversity dynamics?


OUR QUESTION

Can we reconcile molecular phylogenies with the fossil record?

35 30 25 20 15 10 5 0

Num

ber o

f spe

cies

Myrs

250

50

100

150

200

0

Num

ber o

f spe

cies

Num

ber o

f spe

cies

Myrs

Myrs

a b

dc

35 30 25 20 15 10 5 0

050

100

150

200


35 30 25 20 15 10 5 0

020

4060

8010

012

0


Myrs35 30 25 20 15 10 5 0

Fossil data


In particular, can we successfully use molecular phylogenies tounderstand cetacean diversity dynamics?



I Use the branch lengths as datum for a likelihood,measuring time backwards from the present:

t1

t2

t3

t4

1 2 3 4

t1

t2

t3

1

t4

2 3 4

t1

t2

t4

1 2

t3

3

t1

t2

cl

cl

4

−→ {(t2, t1), (t3, t2), (0, t2), (t4, t3),

(0, t3), (0, t4), (0, t4)}

I ti is the time at which the ith observed species differentiated.

I N.B. Each divergence time occurs as RHS of a pair twice.LHS is the next divergence time (or the present, t = 0) forthe daughter species. We’ll write the pairs as(si,1, ti), (si,2, ti).



I Use the branch lengths as datum for a likelihood,measuring time backwards from the present:

t1

t2

t3

t4

1 2 3 4

t1

t2

t3

1

t4

2 3 4

t1

t2

t4

1 2

t3

3

t1

t2

cl

cl

4

−→ {(t2, t1), (t3, t2), (0, t2), (t4, t3),

(0, t3), (0, t4), (0, t4)}

I ti is the time at which the ith observed species differentiated.I N.B. Each divergence time occurs as RHS of a pair twice.

LHS is the next divergence time (or the present, t = 0) forthe daughter species. We’ll write the pairs as(si,1, ti), (si,2, ti).


FIRST ATTEMPT

From this data, build a likelihood function:I Model speciation/extinction as a time-inhomogeneous

branching process with rates λ(t) and µ(t).

I Assume extant species are sampled with probability fI Let

Φ(t) = P {a lineage is not in the sample|it was alive at time t}

Ψ(s, t) = P{

a given lineage produced no descendants in our sampleduring the time interval (s, t), and it survived from t to s

}.

I L(t1, . . . , tn) = f nΨ(t2, t1)∏n

i=2 λ(ti)Ψ(si,1, ti)Ψ(si,2, ti)


FIRST ATTEMPT


branching process with rates λ(t) and µ(t).I Assume extant species are sampled with probability f

I Let


Ψ(s, t) = P{


}.

I L(t1, . . . , tn) = f nΨ(t2, t1)∏n



FIRST ATTEMPT


branching process with rates λ(t) and µ(t).I Assume extant species are sampled with probability fI Let


Ψ(s, t) = P{


}.

I L(t1, . . . , tn) = f nΨ(t2, t1)∏n



FIRST ATTEMPT


branching process with rates λ(t) and µ(t).I Assume extant species are sampled with probability fI Let


Ψ(s, t) = P{


}.

I L(t1, . . . , tn) = f nΨ(t2, t1)∏n



Φ(t)

I First,

Φ(t+∆t) = P{

lineage goes extinctin (t, t + ∆t)

}+ P

{no extinction and speciation, but

neither lineage is observed at present

}+ P

{no extinction or speciation in (t, t + ∆t),

but lineage is not observed at present

}

= µ(t)∆t + (1− µ(t)∆t)λ(t)Φ2(t)

+ (1− µ(t)∆t)(1− λ(t)∆t)Φ(t) + o(∆t).

I Take ∆t→ 0: dΦdt = µ(t)− (λ(t) + µ(t))Φ(t) + λ(t)Φ2(t)

I Φ(0) = P {a lineage is not in the sample|it was alive at time 0} = 1− f .

I Thus, Φ(t) = 1− e∫ t

0 λ(u)−µ(u) du

1f +∫ t

0 e∫ s

0 λ(u)−µ(u) duλ(s) ds

.


Φ(t)

I First,

Φ(t+∆t) = P{


}+ P



}+ P



}= µ(t)∆t + (1− µ(t)∆t)λ(t)Φ2(t)

+ (1− µ(t)∆t)(1− λ(t)∆t)Φ(t) + o(∆t).




0 λ(u)−µ(u) du

1f +∫ t

0 e∫ s


.


Φ(t)

I First,

Φ(t+∆t) = P{


}+ P



}+ P



}= µ(t)∆t + (1− µ(t)∆t)λ(t)Φ2(t)

+ (1− µ(t)∆t)(1− λ(t)∆t)Φ(t) + o(∆t).




0 λ(u)−µ(u) du

1f +∫ t

0 e∫ s


.


Φ(t)

I First,

Φ(t+∆t) = P{


}+ P



}+ P



}= µ(t)∆t + (1− µ(t)∆t)λ(t)Φ2(t)

+ (1− µ(t)∆t)(1− λ(t)∆t)Φ(t) + o(∆t).




0 λ(u)−µ(u) du

1f +∫ t

0 e∫ s


.


Φ(t)

I First,

Φ(t+∆t) = P{


}+ P



}+ P



}= µ(t)∆t + (1− µ(t)∆t)λ(t)Φ2(t)

+ (1− µ(t)∆t)(1− λ(t)∆t)Φ(t) + o(∆t).




0 λ(u)−µ(u) du

1f +∫ t

0 e∫ s


.


Ψ(s, t)

I Proceed as before:

Ψ(s, t + ∆t) = P{

no extinctionin (t, t + ∆t)

}×(P{

no speciationin (t, t + ∆t)

}

+ P{

speciation in (t, t + ∆t), but one ofthe two lineages is not in the sample

})

× P{

the lineage survives from t to swithout any observed daughters

}

= (1− µ(t)∆t) ((1− λ(t)∆t) + 2λ(t)∆tΦ(t)) Ψ(s, t).

I Get dΨ(s,t)dt = ((2Φ(t)− 1)λ(t)− µ(t)) Ψ(s, t)

I Ψ(s, s) = 1.

I Therefore, Ψ(s, t) = e∫ t

s λ(u)−µ(u) du

[1 +

∫ ts e∫ τ

0 λ(σ)−µ(σ) dσλ(τ) dτ

1f +∫ s

0 e∫ τ


]−2

.


Ψ(s, t)


Ψ(s, t + ∆t) = P{


}×(P{


}

+ P{


})

× P{


}= (1− µ(t)∆t) ((1− λ(t)∆t) + 2λ(t)∆tΦ(t)) Ψ(s, t).


I Ψ(s, s) = 1.


s λ(u)−µ(u) du

[1 +

∫ ts e∫ τ


1f +∫ s

0 e∫ τ


]−2

.


Ψ(s, t)


Ψ(s, t + ∆t) = P{


}×(P{


}

+ P{


})

× P{


}= (1− µ(t)∆t) ((1− λ(t)∆t) + 2λ(t)∆tΦ(t)) Ψ(s, t).


I Ψ(s, s) = 1.


s λ(u)−µ(u) du

[1 +

∫ ts e∫ τ


1f +∫ s

0 e∫ τ


]−2

.


Ψ(s, t)


Ψ(s, t + ∆t) = P{


}×(P{


}

+ P{


})

× P{


}= (1− µ(t)∆t) ((1− λ(t)∆t) + 2λ(t)∆tΦ(t)) Ψ(s, t).


I Ψ(s, s) = 1.


s λ(u)−µ(u) du

[1 +

∫ ts e∫ τ


1f +∫ s

0 e∫ τ


]−2

.


Ψ(s, t)


Ψ(s, t + ∆t) = P{


}×(P{


}

+ P{


})

× P{


}= (1− µ(t)∆t) ((1− λ(t)∆t) + 2λ(t)∆tΦ(t)) Ψ(s, t).


I Ψ(s, s) = 1.


s λ(u)−µ(u) du

[1 +

∫ ts e∫ τ


1f +∫ s

0 e∫ τ


]−2

.


UNFORTUNATELY. . .

Still inferring µ(t) ≡ 0.


SECOND ATTEMPT

I If extinction rates are high enough, the probability ofanything surviving to the present is infinitetesimal.

I But, we know that something did survive, so condition onthis event.

I New likelihood function:

L(t1, . . . , tn) =f nΨ(t2, t1)

∏ni=2 λ(ti)Ψ(si,1, ti)Ψ(si,2, ti)

1− Φ(t1)


SECOND ATTEMPT




L(t1, . . . , tn) =f nΨ(t2, t1)


1− Φ(t1)


SECOND ATTEMPT




L(t1, . . . , tn) =f nΨ(t2, t1)


1− Φ(t1)


VICTORY!

The improved likelihood successfully infers rates fromsimulated phlogenies, even when net diversity is decreasing:

!

!

!

!

!

1.0 0.8 0.6 0.4 0.2

0.0

0.5

1.0

1.5

simulated speciation rate at present

estim

ated

spec

iation

rate

at p

rese

nt

!

!

!

!

!

!!

!!

!

0.05 0.10 0.15

!0.4

!0.2

0.0

0.2

simulated rate of decay in speciation rate

estim

ated

rate

of d

ecay

in sp

eciat

ion ra

te

!!

!!

!

!! ! ! !

1.0 0.8 0.6 0.4 0.2

0.0

0.5

1.0

1.5

simulated speciation rate at present

extin

ction

rate

at p

rese

nt

!! ! ! !

!!

!

!

!

0.4 0.2 0.0 !0.2

!0.5

0.0

0.5

1.0

simulated net diversification rate at present

net d

iversi

ficat

ion ra

te a

t pre

sent

!!

!

!

!

Constant extinction, time-varying speciation.

! ! !!

!

0.1 0.2 0.3 0.4 0.5 0.6 0.7

!10

12

34

56

extinction rate at present (simulated)

extin

ction

rate

at p

rese

nt (e

stima

ted)

! ! !!

!

! ! ! ! !

0.05 0.10 0.15 0.20

!10

12

rate of increase in extinction rate (simulated)

rate

of in

creas

e in

extin

ction

rate

(esti

mate

d)

! ! ! ! !

! ! ! ! !

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.5

1.0

1.5

extinction rate at present (simulated)

spec

iation

rate

at p

rese

nt (e

stima

ted)

! ! ! ! !

! !!

!

!

0.4 0.3 0.2 0.1 0.0 !0.2

!3!2

!10

net diversification rate at present (simulated)

net d

iversi

ficat

ion ra

te a

t pre

sent

(esti

mate

d)

! !!

!

!

Constant speciation, time-varying extinction.


HOWEVER. . .

Table 1. Various diversification models fitted to the cetacean phylogeny as a whole

Model nb Description LogL AICc

B-constant 1 No extinction, constant speciation rate -279.0280 560.0796

BD-constant 2 Constant speciation and extinction rates -279.0280 562.1270

B-variable-E 2 No extinction, exponential variation in speciation rate -278.9887 562.0485

B-variable-L 2 No extinction, linear variation in speciation rate -278.9896 562.0502

B-variable-E, D-constant

3 Exponential variation in speciation rate, constant extinction rate -278.9887 564.1204

B-variable-L, D-constant

3 Linear variation in speciation rate, constant extinction rate -278.9896 564.1221

B-constant, D-variable-E

3 Constant speciation rate, exponential variation in extinction rate -279.0280 564.1989

B-constant, D-variable-L

3 Constant speciation rate, linear variation in extinction rate -279.0280 564.1989

nb denotes the number of parameters of each model. LogL stands for the maximal log-likelihood; AICc stands for the second order Akaike’sinformation criterion.

Footline Author PNAS Issue Date Volume Issue Number 1


BACK TO THE DRAWING BOARD. . .

I Thusfar, we’ve assumed that speciation andextinction rates are uniform across clades.

I What if we allow extinction and speciationrates to vary?

I Consider a multitype branching process withrates as types: xi = (λi(t), µi(t))

I Get a type-dependent likelihood:

L(t1, . . . , tn) =fx1 Ψx1 (t2, t1)

∏ni=1 fxiλpi (ti)Ψxpi

(si,1, ti)Ψxi (si,2, ti)

1− Φx1 (t1)

(pi indicates the parent of individual i)

t1

t2

x1

t3

x1

1

x1

t4

x3

3

x3

4

x4

2

x2







L(t1, . . . , tn) =fx1 Ψx1 (t2, t1)



1− Φx1 (t1)


t1

t2

x1

t3

x1

1

x1

t4

x3

3

x3

4

x4

2

x2







L(t1, . . . , tn) =fx1 Ψx1 (t2, t1)



1− Φx1 (t1)


t1

t2

x1

t3

x1

1

x1

t4

x3

3

x3

4

x4

2

x2







L(t1, . . . , tn) =fx1 Ψx1 (t2, t1)



1− Φx1 (t1)


t1

t2

x1

t3

x1

1

x1

t4

x3

3

x3

4

x4

2

x2


A CRUMP-MODE-JAGERS APPROACH I

Haccou P., Jagers P., Vatutin VA. Branching Processes: Variation, Growth and Extinction of Populations. Cambridge

University Press; Cambridge (2005)


A CRUMP-MODE-JAGERS APPROACH III Consider a parent lineage, known to be alive at time t, of

type xp = (λp, µp).

I Remaining lifespan Lp is exponentially distributed:

P{

Lp > t− s}

= e−∫ t

s µp(u) du.

I Birth times of “offspring” are a Poisson point process on[t, t− Lp]: probability of n offspring born at s1, . . . , sn in [s, t]is

e−∫ t

s λp(u) du

n!λp(s1) · · ·λp(sn) ds1 · · · dsn.

I Given types x1, . . . , xn, the probability that all disappearbefore the present is

Πxp (s, t) = e−∫ t

s λp(u) du∞∑

n=0

1n!

n∏i=1

∫ t

sλp(si)Φxi (si) dsi.



type xp = (λp, µp).I Remaining lifespan Lp is exponentially distributed:

P{

Lp > t− s}

= e−∫ t

s µp(u) du.


e−∫ t

s λp(u) du

n!λp(s1) · · ·λp(sn) ds1 · · · dsn.


Πxp (s, t) = e−∫ t

s λp(u) du∞∑

n=0

1n!

n∏i=1

∫ t





P{

Lp > t− s}

= e−∫ t

s µp(u) du.


e−∫ t

s λp(u) du

n!λp(s1) · · ·λp(sn) ds1 · · · dsn.


Πxp (s, t) = e−∫ t

s λp(u) du∞∑

n=0

1n!

n∏i=1

∫ t





P{

Lp > t− s}

= e−∫ t

s µp(u) du.


e−∫ t

s λp(u) du

n!λp(s1) · · ·λp(sn) ds1 · · · dsn.


Πxp (s, t) = e−∫ t

s λp(u) du∞∑

n=0

1n!

n∏i=1

∫ t



A CRUMP-MODE-JAGERS APPROACH IIII There are two ways that the parent lineage may leave no

observed descendants:

(i) the parent lineage survives to the present, but is notobserved, and leaves no descendent lineages. Thus Lp > t,and the probability of this event is

(1− f )e−∫ t

0 µp(u) duΠxp (0, t).

(ii) the parent lineage dies at some 0 < s < t. Thus Lp = t− s,and the probability of this event is∫ t

0e−∫ t

s µp(u) duµp(s)Πxp (s, t) ds.

I Thus,

Φxp (t) = (1− f )e−∫ t

0 µp(u) duΠxp (0, t) +

∫ t

0e−∫ t


I Get a similar expression for Ψxp(s, t).



observed descendants:(i) the parent lineage survives to the present, but is not

observed, and leaves no descendent lineages. Thus Lp > t,and the probability of this event is

(1− f )e−∫ t



0e−∫ t


I Thus,

Φxp (t) = (1− f )e−∫ t


∫ t

0e−∫ t







(1− f )e−∫ t



0e−∫ t


I Thus,

Φxp (t) = (1− f )e−∫ t


∫ t

0e−∫ t







(1− f )e−∫ t



0e−∫ t


I Thus,

Φxp (t) = (1− f )e−∫ t


∫ t

0e−∫ t







(1− f )e−∫ t



0e−∫ t


I Thus,

Φxp (t) = (1− f )e−∫ t


∫ t

0e−∫ t




UNFORTUNATELY. . .

Infinite sums and products aren’t very useful inpractice.

Can deal with this in one of two ways.


UNFORTUNATELY. . .

Infinite sums and products aren’t very useful inpractice.

Can deal with this in one of two ways.


I. THE RIGHT WAY

I Suppose we had a parametric model, νθ(xp, x′), for theprobability a parent lineage of type xp has a descendentlineage of type x′.

I The probability of having descendent lineages of typex1, . . . , xn at times s1, . . . , sn is

e−∫ t

s λp(u) du

n!λp(s1) · · ·λp(sn)νθ(xp, x1) · · · νθ(xp, xn) ds1 · · · dsn dxp · · · dxn

I Πxp(s, t) reduces to

e∫ t

s λp(s)(∫

X Φxp (s)νθ(xp,x) dx−1)−µ(u) du

.

I Get closed form for Φxp , Ψxp , and the likelihood.I However, we don’t have an appropriate νθ(x, x′).


I. THE RIGHT WAY



e−∫ t

s λp(u) du



e∫ t

s λp(s)(∫


.



I. THE RIGHT WAY



e−∫ t

s λp(u) du



e∫ t

s λp(s)(∫


.



I. THE RIGHT WAY



e−∫ t

s λp(u) du



e∫ t

s λp(s)(∫


.

I Get closed form for Φxp , Ψxp , and the likelihood.

I However, we don’t have an appropriate νθ(x, x′).


I. THE RIGHT WAY



e−∫ t

s λp(u) du



e∫ t

s λp(s)(∫


.



II. THE WAY WE ACTUALLY DID IT

I By definition, none of the offspring in the expression forΠxp leave any descendants.

I Assume all are identical to the parent: Φx1 = · · · = Φxn = Φxp .I Then,

Πxp (s, t) = e∫ t

s λ(u)(Φxp (u)−1) du

and

Φxp (t) = (1− f )e∫ t

0 λp(u)(Φxp (u)−1)−µp(u) du

+

∫ t

0e−∫ t

s λp(u)(Φxp (u)−1)−µp(u) duµp(s) ds.

I Differentiating by t shows Φxp(t) satisfies previous ODE.I Types only allowed to change at observed speciation

events.




I Assume all are identical to the parent: Φx1 = · · · = Φxn = Φxp .

I Then,Πxp (s, t) = e

∫ ts λ(u)(Φxp (u)−1) du

and

Φxp (t) = (1− f )e∫ t


+

∫ t

0e−∫ t



events.







and

Φxp (t) = (1− f )e∫ t


+

∫ t

0e−∫ t



events.







and

Φxp (t) = (1− f )e∫ t


+

∫ t

0e−∫ t


I Differentiating by t shows Φxp(t) satisfies previous ODE.

I Types only allowed to change at observed speciationevents.







and

Φxp (t) = (1− f )e∫ t


+

∫ t

0e−∫ t



events.


In practice, we assumed type-changes only at new clades, andconsidered constant, linear, or exponential functions for λx(t)and µx(t):

Table 2. Statistical support for rate shifts in the cetacean phylogeny


no shift 1 Best-fit model -279.03 560.08

1 shift 5 Best-fit model: shift in the Delphinidae -262.93*** 536.22

2 shifts 6 Best-fit model: shifts in the Delphinidae and Phocoenidae -260.17* 532.85

3 shifts 7 Best-fit model: shifts in the Delphinidae, Phocoenidae and Ziphiidae -256.13** 526.94

4 shifts 8 Best-fit model: shifts in the Delphinidae, Phocoenidae, Ziphiidae and Balaenopteridae -250.13 517.14

nb denotes the number of parameters of each model; this number depends on the assumed number of shifts as well as on the number ofparameters in the best-fit models for the corresponding subclades and pruned phylogeny. The best fit model for the pruned phylogeny was theBD-variable model across the 1-, 2- and 3-shifts models, and the B-constant D-variable model for the 4-shifts model. LogL stands for themaximum log-likelihood of the entire phylogeny; AICc stands for the second-order Akaike’s information criterion [45]. Stars indicate thestatistical support of the corresponding model against the model with exactly one less rate shift, based on the likelihood ratio test: *** standsfor p < 0.001, ** for p < 0.01, and * for p < 0.05. Each additional shift is supported by both the AICc criterion and the likelihood ratiotest. The 4 shifts model could not be compared to the 3 shifts model using the likelihood ratio test because these models were not nested. Inthe most likely model (bold), each of the four largest families have different rates than the remainder of the tree.



Table 3. Diversification models fitted to the cetacean phylogeny after isolating the Balaneopteridae, Delphinidae,Phocoenidae and Ziphiidae families


B-constant 1 No extinction, constant speciation rate -63.17 128.48

BD-constant 2 Constant speciation and extinction rates -63.17 130.77

B-variable 2 No extinction, exponential variation in speciation rate through time -61.49 127.41

B-variable-L 2 No extinction, linear variation in speciation rate -58.57 121.56

B-variable, D-constant

3 Exponential variation in speciation rate, constant extinction rate -56.76 120.40

B-variable-L,D-constant

3 Linear variation in speciation rate, constant extinction rate -58.45 123.79

B-constant,D-variable

3 Constant speciation rate, exponential variation in extinctionrate

-56.33 119.54

B-constant, D-variable-L

3 Constant speciation rate, linear variation in extinction rate -56.60 120.10

BD-variable 4 Exponential variation in speciation and extinction rates -56.22 121.99

BD-variable-L 4 Linear variation in speciation and extinction rates -56.39 122.31

nb denotes the number of parameters of each model. LogL stands for the maximum log-likelihood; AICc stands for the second order Akaike’sinformation criterion. -L stands for linear variation through time



Out of 14 families, only 4 contain more than 5 species

Morlon et al., in revision

balaenopteridae

delphinidae

ziphiidae

phocoenidae

mysticetes

odontocetes

Out of 14 families, only 4 contain more than 5 species.


35 30 25 20 15 10 5 0

Num

ber o

f spe

cies

Myrs

250

50

100

150

200

0

Num

ber o

f spe

cies

Num

ber o

f spe

cies

Myrs

Myrs

a b

dc

35 30 25 20 15 10 5 0

050

100

150

200


35 30 25 20 15 10 5 0

020

4060

8010

012

0


Myrs35 30 25 20 15 10 5 0

Fossil data


When we isolate the 4 richest families, we infer boom-then-bustpatterns of diversity in the remaining mysticetes andodontocetes.

i.e. Expansion in the more recent clades masks thedisappearance of older clades.


35 30 25 20 15 10 5 0

Num

ber o

f spe

cies

Myrs

250

50

100

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0

Num

ber o

f spe

cies

Num

ber o

f spe

cies

Myrs

Myrs

a b

dc

35 30 25 20 15 10 5 0

050

100

150

200


35 30 25 20 15 10 5 0

020

4060

8010

012

0


Myrs35 30 25 20 15 10 5 0

Fossil data


When we isolate the 4 richest families, we infer boom-then-bustpatterns of diversity in the remaining mysticetes andodontocetes.i.e. Expansion in the more recent clades masks thedisappearance of older clades.


35 30 25 20 15 10 5 0

Num

ber o

f spe

cies

Myrs

250

50

100

150

200

0

Num

ber o

f spe

cies

Num

ber o

f spe

cies

Myrs

Myrs

a b

dc

35 30 25 20 15 10 5 0

050

100

150

200


35 30 25 20 15 10 5 0

020

4060

8010

012

0


Myrs35 30 25 20 15 10 5 0

Fossil data


The resulting diversity curve is consistent with the fossil record.


CONCLUSIONS

I Realistic extinction rates, and a diversity trajectoryconsistent with the fossil record, can be inferred from thecetacean phylogeny, but only if we account for rateheterogeneity.

I This gives hope for our understanding of diversitydynamics in the absence of a fossil record.

I Much remains to be done, such as including:

I density-dependent speciation and extinction,I trait-dependent speciation and extinction,I making more use of the tree topology, andI better accounting for data collection biases.


CONCLUSIONS






CONCLUSIONS






CONCLUSIONS



I Much remains to be done, such as including:I density-dependent speciation and extinction,

I trait-dependent speciation and extinction,I making more use of the tree topology, andI better accounting for data collection biases.


CONCLUSIONS



I Much remains to be done, such as including:I density-dependent speciation and extinction,I trait-dependent speciation and extinction,

I making more use of the tree topology, andI better accounting for data collection biases.


CONCLUSIONS



I Much remains to be done, such as including:I density-dependent speciation and extinction,I trait-dependent speciation and extinction,I making more use of the tree topology, and

I better accounting for data collection biases.


CONCLUSIONS



I Much remains to be done, such as including:I density-dependent speciation and extinction,I trait-dependent speciation and extinction,I making more use of the tree topology, andI better accounting for data collection biases.


Thank you for listening!

Documents

Inferring Diversity Dynamics · approaches can’t work: Diversity dynamics: molecular phylogenies need the fossil record Tiago B. Quental and Charles R. Marshall Museum of Paleontology