Global occurrence trajectories of microfossils: environmental volatility …folk.uio.no/tores/Publications_files/Liow et al. 2010.pdf · 2010-04-09 · Global occurrence trajectories

Global occurrence trajectories of microfossils:environmental volatility and the rise and fall of individual species

Lee Hsiang Liow, Hans Julius Skaug, Torbjørn Ergon, and Tore Schweder

Abstract.—Species arise and establish themselves over the geologic time scale. This process ismanifested as a change in the relative frequency of occurrences of a given species in the global pool ofspecies. Our main goal here is to model this rise and the eventual decline of microfossil species using amixed-effects model where groups each have a characteristic occurrence trajectory (main effects) andeach species belonging to those groups is allowed to deviate from the given group trajectory (randomeffects). Our model can be described as a ‘‘hat’’ with logistic forms in the periods of increase anddecline. Using the estimated timings of rises and falls, we find that the lengths of the periods of riseare about as long as the lengths of the periods when species are above 50% of their estimated maximaloccurrences. These latter periods are here termed periods of dominance, which are in turn about thesame length as the species’ periods of fall. The peak rates of the rises of microfossils are in generalfaster than their peak rates of falls. These quantified observations may have broad macroevolutionaryand macroecological implications. Further, we hypothesize that species that have experienced andsurvived high levels of environmental volatility (specifically, periods of greater than average variationin temperature and productivity) during their formative periods should have longer periods ofdominance. This is because subsequent environmental variations should not drive them to declinewith ease. We find that higher estimated environmental volatility early in the life of a speciespositively correlates with lengths of periods of dominance, given that a species survives the initialstress of the environmental fluctuations. However, we find no evidence that the steepness of the riseof a species is affected by environmental volatility in the early phases of its life.

Lee Hsiang Liow. Center for Ecological and Evolutionary Synthesis, Department of Biology, University ofOslo, Blindern Post Office Box 1066, Oslo N-0316, Norway. E-mail: [email protected]

Hans Julius Skaug. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen N-5008, Norway

Torbjørn Ergon. Program for Integrative Biology, Department of Biology, University of Oslo, Blindern PostOffice Box 1066, Oslo N-0316, Norway

Tore Schweder. Department of Economics, University of Oslo, Post Office Box 1095 Blindern, Oslo N-0317,Norway

Accepted: 10 October 2009

Introduction

The study of origination rates, extinctionrates, and diversity in the fossil recordrequires that we be able to estimate whenindividual lineages originated and becameextinct. However, as with most other types ofbiological sampling (Clark and Bjørnstad2004; Hortal et al. 2007; Royle et al. 2007),sampling in the fossil record is incomplete formany reasons. This incompleteness affectsour estimates of times of originations andextinctions. For a species to be recorded in adatabase along with information on bothgeologic time and geographic location, notonly must the species have been extant, butalso individuals of that species must havebeen fossilized despite taphonomic and sedi-

mentary vagaries, survived weathering, beensampled by a paleontologist, and been cor-rectly identified and aged (Holland 2000;Kidwell and Holland 2002; Peters 2006).Various methods have been proposed tocorrect for sampling incompleteness or to fillin temporal gaps in sampling. They aredeveloped variously to reduce biases inspecific ways as demanded by the questionsbeing posed. These methods include sam-pling standardization in diversity studies(Alroy et al. 2001, 2008; Bush et al. 2004;Kiessling and Aberhan 2007), excluding sin-gletons in rate calculations (Foote 2000),setting confidence intervals to stratigraphicranges (Marshall 1990, 1994, 1997; Solow 2003;Weiss and Marshall 1999), exploiting phylo-genetic approaches (Angielczyk and Fox 2006;

Paleobiology, 36(2), 2010, pp. 224–252

’ 2010 The Paleontological Society. All rights reserved. 0094-8373/10/3602–0004/$1.00

Pol and Norell 2006; Wills 2007), embracingsampling or preservation parameters in hy-potheses (Foote 2003; Wagner 2000), andincluding the use of capture-recapture ap-proaches to account for detection probabilitiesthat are less than one (Nichols and Pollock1983; Nichols et al. 1986; Connolly and Miller2001; Liow et al. 2008).

A complicating factor that has been em-phasized recently is the inevitably lowerprobability of sampling a lineage at both thebeginning and end of its existence: lineagestend to be more frequently sampled in themiddle of their life spans, probably owing totrue increases in abundance and geographicextent (Foote et al. 2007; Liow and Stenseth2007). Occurrence probabilities must taper offto zero both at the beginning and at the end ofa species’ life. Hence inferences regarding thetiming of origination and extinction mustnecessarily be highly dependent on how thistapering is modeled. This temporally chang-ing probability of sampling a lineage isespecially problematic when considering fac-tors that contribute to lineage persistence(Viranta 2003; Liow 2007a,b) because theassumption that rank order durations oflineages are not affected by sampling maynot hold despite sampling standardizationsand corrections.

In this paper, we develop a model (the‘‘hat’’) to estimate the changing probabilitythat individual microfossil species are de-tected given that they are globally extant,using their observed occurrences in geologictime. Our model is designed to estimate thetimes of the rises and declines of speciesinstead of the less tractable times of speciationand extinction. As a natural consequence ofthe model, we can also obtain estimates of therates of rise and fall of species. We propose‘‘period of dominance’’ as a new and morerobust measure of species persistence. Todemonstrate our model, we use data fromthe public microfossil database, NEPTUNE(Lazarus et al. 2007) to estimate the temporaloccurrence trajectories of diatoms, nanno-plankton, planktic foraminifers, and radiolar-ians. We find that peak rise rates are ingeneral quicker than peak fall rates for all thefour microfossil groups and that the lengths

of periods of rise, dominance, and fall are ofthe same order of magnitude.

Using estimates from our ‘‘hat’’ model, wealso explore a new idea, namely, that thevolatility in the environment experienced andsurvived by a young or establishing speciesinfluences the length of its period of dom-inance. We reasoned that species that areinitially conditioned to survive changingconditions have better chances of survivingsubsequent environmental changes. Hencewe predict that species that had enduredvolatile beginnings should have a longerperiod of dominance. This prediction isdistinct from how their individual popula-tions respond to shorter-term, more localenvironmental fluctuations. There is someindication that higher estimated environmen-tal volatility early in the life of a planktonspecies correlates positively with its lengthsof periods of dominance.

Materials and Methods

Species Occurrence Data.—NEPTUNE (Laz-arus 1994; Spencer-Cervato 1999; Leckie et al.2004; Lazarus et al. 2007) is an online speciesoccurrence database based on the Deep SeaDrilling Project and its descendant, the OceanDrilling Project, maintained and hosted byCHRONOS (http://chronos.org). It comprisesdata from species of planktic microfossils,diatoms and nannoplankton (phytoplankton)and radiolarians and planktic foraminifers(zooplankton), groups commonly surveyedby biostratigraphers. Our raw data are henceoccurrence data of populations with aconfirmed identity (‘‘resolved_species_id’’ inNEPTUNE) identified by a unique sampleidentification number (‘‘sample_id’’) in thedatabase, as we have previously used (Liowand Stenseth 2007). Only valid taxa are used inour study, and specimens identified with ‘‘sp.,’’‘‘aff.,’’ and ‘‘cf.’’ are excluded. This filteringgives us occurrence data from a total of 732nannoplankton species, 688 diatom species,608 planktic foraminifer species, and 634radiolarian species. The raw data are eachassociated with present-day spatial coordi-nates, based on where the deep-sea core wasdrilled, and an absolute geological ageestimated by age models specific for given

OCCURRENCE TRAJECTORY OF SPECIES 225

cores (Spencer-Cervato 1999). We limit the datato occurrence data after the Cretaceous(65.5 Ma) because of possible differences intaxonomic practice among workers collectingMesozoic and Cenozoic data (Smith 2007),because occurrence data are much sparserbefore the Cenozoic, and because thepaleoenvironmental proxy data are better forthe Cenozoic. We use only species that arefound in at least ten unique samples (i.e.,associated with unique ‘‘sample-ids’’). Thesubset of the data we ultimately used henceconsists of 312 nannoplankton species, 328diatom species, 271 planktic foraminiferspecies, and 342 radiolarian species.

Paleoenvironmental Proxies.—The paleoen-vironmental proxies used in this study, d18Oand d13C, are based on an updated versionsupplied by Mark Pagani (personal commu-nication 2007) of previously published data(Zachos et al. 2001). Briefly, the oxygenisotope fractionation between ambient sea-water and foraminiferal calcite during calcifi-cation is temperature dependent such that ahigher d18O reflects lower temperature (Fair-banks et al. 1980, 1982). It also reflects thevolume of the ice sheets present, because icesheets sequester 16O, such that the d18O valuesalso represent a time-averaged record of high-latitude sea-surface temperature (Zachos et al.2001). Here we simply interpret d18O as aproxy for paleotemperature, where a higherd18O (%) implies a lower temperature. On theother hand, plants and marine algae prefer-entially assimilate 12C during photosynthesis.Thus the carbon isotope effect of organiccarbon versus inorganic carbon is measurablein the dissolved inorganic carbon (DIC)reservoir, incorporated by marine microor-ganisms (e.g., benthic foraminifera) such thatd13C can be interpreted as a proxy forpaleoproductivity. We recognize that carboncycle and global climate are closely linked butthat cause, effect and feedback loops arecomplex (Sundquist and Visser 2005).

Modeling.—In order to estimate the timesand peak rates of global rise and fall ofspecies, we assume (1) that detection isprimarily a function of global abundance, (2)that global occurrence is a reflection of bothdetection and local site occupancy probabil-

ities, and (3) that local preservation is onlyinfluenced by global abundance (see ‘‘Discus-sion’’). For clarity, we explicitly avoid using theterm ‘‘occupancy’’ as commonly used inpaleobiology (Jernvall and Fortelius 2004; Raiaet al. 2006; Foote et al. 2007). Instead we followthe usage in ecology (MacKenzie et al. 2006),whereby occupancy is the true presence of aspecies at a space and time. Occurrence, on theother hand, is detection given occupancy. A nulloccurrence can be either true absence orpresence but with non-detection. We do notdeal with partitioning non-detection and ab-sence in this paper, because the estimation ofspecies-specific parameters already contributesto a high computational burden. However, wewant to remain explicit about our assumptions,especially regarding detection probabilities.Further, we interpret the changes in occurrenceover the life span of each species as causallylinked to the ecological processes of establish-ment, population growth, dispersal, local ex-tinction, and global extinction, as well as theevolutionary process of speciation.

We define the occurrence of a species at asite and in a time bin as its presence in theNEPTUNE database within that site and timebin. An occurrence thus results from either (1)the species being present at the time andspace in non-negligible abundance and thesubsequent fossilization of the same speciesand its later detection and correct identifica-tion; or (2) fossils of a different species havingbeen misidentified as the focal species andthought to be present at the given time andspace. We allow for the second possibility byestimating p0, the probability of a co-occur-ring species in the same sample beingclassified as the focal species (see below).

The occurrence trajectory of any givenspecies is assumed to be hat-shaped (Fig. 1),with an initial rise in occurrence following alogistic curve, followed by a period ofdominance, and finally tapering off followinganother (independent) logistic curve. Thiscontinuous hat-shaped trajectory is modeledthrough an occurrence probability function:

h(t)~L:l lS t{Sð Þð Þ:l lT t{Tð Þð Þ ð1Þwhere l(z)~1=(1z exp ({z)) is the logisticfunction and t is time in millions of years.

226 LEE HSIANG LIOW ET AL.

The parameters are constrained such that0vLƒ1, lSw0, lTv0, SwT. The componentl lS t{Sð Þð Þ thus increases toward an asymp-totic level of 1 while l lT t{Tð Þð Þ decreasesfrom 1. The height of the ‘‘hat’’ is controlledby L, the estimated theoretical maximumoccurrence. A species is characterized by thefive parameters L, S, T, lS, and lT . S and T arethe absolute time point of inflection ofoccurrence increase and the absolute time ofthe inflection of occurrence decline, respec-tively. S and T are also interpretable as whenthe species has reached half of its estimatedtheoretical maximum occurrence during itsrise and its decline respectively. The twoparameters lS and lT determine the max-imum rates of increase toward and decreasefrom L, respectively. In summary, the prob-ability of a species occurring at a randomlychosen site in a given time bin in the database

is modeled as the ‘‘hat’’ value h(t) for thatspecies.

We grouped the data into 32 time bins of2 Myr each. These time bins are much coarserthan the finest temporal resolution of thedatabase, which is 330 Kyr (Spencer-Cervato1999). The relatively wide time bins of 2 Myrwere used in order to reduce the computa-tional burden involved in optimizing thelikelihood function numerically. A species isrecorded as present in a sample as long as atleast one specimen of that species is found;i.e., we do not use information regardingabundance of specimens within samples,sometimes recorded in NEPTUNE.

Let yij be the number of records of species ifrom a total of kj samples in time bin j(samples within time bins identified byunique ‘‘sample_id’’ values in the database).We assume that yij follows a binomialdistribution with the parameters (kj, hij),where hij is the probability that species i isrecorded (i.e., extant and detected) in time binj and where kj is the number of independentsamples (trials) within the time bin. Theprobability of a species being recorded intime bin j with midpoint tj, i.e., its occurrenceprobability, is hij~hi tj

� �.

Hence, the ‘‘hat’’ model for discrete timebins for a given species i is

hij~Li:l lS,i(tj{Si)� �

:l lT,i(tj{Ti)� �

ð2ÞTo allow the basic parameters of the modelL, lS, S, lT, T (Fig. 1) to be species-specific,we use the following mixed-effects model:

Li~l(m1zu1,i), u1,i*N(0, s12) ð3Þ

lS,i~ exp (m2zu2,i), u2,i*N(0, s22) ð4Þ

Si~q25,izm3zu3,i, u3,i*N(0, s32) ð5Þ

lT,i~{ exp (m4zu4,i), u4,i*N(0, s42) ð6Þ

Ti~Siz exp m5zu5,ið Þ, u5,i*N(0, s52) ð7Þ

where N(0, s2) denotes a zero-mean normal

distribution with variance s2, and l(z) is thelogistic function as before. Here, the m’s arethe main effects shared by all species in eachof the four groups (diatoms, nannoplankton,radiolarians, and planktic foraminifers). Theu’s are ‘‘random effects’’ or individual devia-

FIGURE 1. An illustration showing the parameters we areestimating from our ‘‘hat’’ model. The dotted curveshows a ‘‘hat’’ trajectory fitting the fossil occurrence data(vertical solid lines) in 2-Myr bins for Coccolithusmiopelagicus. S and T are the first and second inflectionpoints of the ‘‘hat’’ model and the absolute amount oftime between them is the period of dominance (S, T). Thetheoretical maximum estimated proportion of sites, i.e.,height of the ‘‘hat,’’ is L, shown with a two-headed arrow.The first logistic curve is illustrated with a gray dashedcurve and its inflection point represents the rate of rise ofC. miopelagicus at time S. The time elapsed between 5%and 10% of the height of the C. miopelagicus ‘‘hat,’’ the‘‘initial establishment phase,’’ is labeled as a black bar;the time elapsed between 10% and 50% of the height ofthe ‘‘hat,’’ the ‘‘expansion phase,’’ is labeled as a gray bar.The empirical quantile q25, used for computationalpurposes, is also shown.


tions from the m’s. The term q25,i is used forcomputational convenience, as explained be-low. In other words, each of the four micro-fossil groups has a general ‘‘hat’’ trajectory,but each species within those groups isallowed to have its own characteristic variantdrawn from the group-specific distribution of‘‘hat’’ trajectories. We estimated the m’s forthe four groups separately because of theirdifferences in trophic level and evolutionaryhistory, and because of taxonomic practices,each of which likely contributes to differenttrue values for group-specific main effects.Note that although S occurs in equation (8), T-S is independent of S.

The term q25,i (eq. 5) is the 25% empiricalquantile among time bins for which the givenspecies has been observed (Fig. 1). We param-eterize Si in terms of q25,i to condition the‘‘hat’’ to include the time during which thereare actually occurrence data for the species i.In addition, q25,i eases the computationalburden and makes the assumption of anormally distributed random effect (u3,i)appropriate.

Errors in assigning taxa are not uncommon,both in neontology and paleontology (Sepkoski1993; Foissner 2006). To make the model robustagainst such errors, we introduced a parameterp0, defined as the probability of a co-occurringspecies in the same sample being classified asthe focal species i. The probability of anoccurrence of species i in time bin j is then

p0z(1{p0)hij: ð8Þ

We assume that p0 is the same in every timebin and for every species in a given group. Inpractice, it is difficult to distinguish betweenthe true misclassification of specimens (i.e.,assignment to the wrong species) and a givenspecies not being preserved at a sampledlocation (core); thus p0 accounts for the lack offit of the ‘‘hat’’ trajectory due to both of thesecircumstances. The ‘‘hat’’ model was fittedusing the AD Model Builder (freely availableat http://admb-project.org/).

Periods of Dominance, Rises and Falls, and Per-Species Frequencies of Rises and Falls.—The timeinterval between S and T is termed the periodof dominance (S, T). Of primary interest is the

length of the period of dominance, T-S. Notethat h(t) is peaked for species having shortperiods of dominance and flat for thosehaving longer periods of dominance, relativeto the steepness of their rise and decline. Thelength of period of rise (PR) is the time it takesfor the occurrence probability of a givengroup or species to go from 5% to 95% ofthe estimated theoretical maximum occur-rence (L) whereas the length of period ofdecline (PD) is the time it takes for the same togo from 95% to 5% of L during the fall. Wetabulated NSj

, the number of species that havetheir S values in a given 2 Myr interval, j; NTj

,the number of species with T values in j; andN(S,T)j

, the number of species with periods ofdominance (S, T) overlapping j. Per-speciesfrequency of rise per 2 Myr interval over thetime intervals j (where j 5 1…35) is thenobserved as NSj

=N(S,T)jwhereas the per-

species frequency of fall is NTj=N(S,T)j

. Differ-

enced per species frequency of rises are henceobserved as (NSjz1

=N(S,T)jz1{NSj

=N(S,T)j), and

differenced per species frequency of fall is(NTjz1

=N(S,T)jz1{NTj

=N(S,T)j).

Exploration of Preservation Trends.—Thequality of preservation for a given speciesthrough its lifetime may affect the estimatesof its rise and fall. For example, speciesthat become part of the geologic recordearlier might be less available for samplingand/or the specimens may be relativelyless identifiable. To investigate these poten-tial inherent temporal biases in the data wedo two empirical checks and one analyticalcheck. First, we plotted the qualitative pres-ervation codes (Good, Moderate, Poor) as-signed to samples in NEPTUNE over time foreach species. We also summarized these pres-ervation codes for each microfossil groupaccording to whether they are associated withsamples before or after our estimated S and Tusing box plots. Second, we plotted residualsfrom the lack of fit of the data to the model foreach of the four microfossil groups over time tocheck if there are unusual trends that may bedue to general abnormalities in preservation.These residuals (for each species and each time

bin) are calculated as h{hh=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihh(1{hh)=n

qwhere

h is an observed frequency of occurrence, hh its


value fitted by the ‘‘hat’’ model, and n thenumber of sites at a given time bin. Last, wesupply a brief theoretical analysis of the typesof biases that are generated if global preserva-tion gradually improves over time.

Environmental Volatility.—We denote thetwo paleoenvironmental proxies in generalby denv,t, that is, either d18O or d13C values attime t, and define their lag-1 differenced timeseries as Dt~denv,t{denv,t{1. We use a stochas-tic volatility (SV) model (Harvey et al. 1994)motivated by the fact that the standarddeviation of Dt changes over time (Fig. 2).The time series Dt is mean-subtracted, and weassume that Dt*N(0, tt

2)where tt is theenvironmental volatility process we want tomodel (Fig. 2). As a model for tt we usethe stochastic process: log (tt)~mzXt, whereXt ~ aXt{1zet is a zero-mean stationary first-order autoregressive process, with indepen-dent innovation terms et*N(0,v2) (Brockwelland Davis 1987). This model yields theproperties we seek: tt~ exp (Xt)w0 and posi-tive serial correlation (cor(tt{1, tt)w0) whenaw0. In other words, we use the smoothedestimate of the standard deviation of thedifferenced series of the paleo-proxies(Fig. 2). In doing so, for time intervals withlow values of estimated volatility, tt, thetemporal differences in the environmentalproxies, Dt, will be small in absolute value,and vice versa. Estimates of the parametersm, a,v are obtained using maximum like-lihood and empirical Bayes estimates areused for the volatility series tt (Skaug andFournier 2006).

Our time bins t for calculating average denv,t

are 0.1 Myr in length. This choice is acompromise between two constraints. Weneed to make the bins wide enough to haveenough observations of the proxy data in eachbin. However, making time bins for estimat-ing volatility too large would reduce therelevance of the scale of volatility for theevolutionary and ecological processes we areconcerned about. Moreover, 0.1 Myr bins docapture the time scale of some of the longerso-called climate aberrations including thePaleocene-Eocene Thermal Maximum (PETM)at 55 Ma, the early Oligocene glaciation at

34 Ma, and early Miocene glaciation at 23 Ma(Zachos et al. 2001). The age estimates for thesepaleoenvironmental proxy data are accurate onorbital time scales or at the resolution of 20 Kyr(M. Pagani personal communication 2008).

Early Phases of the ‘‘Hat’’: Initial Establish-ment and Expansion Phases.—We calculated themean estimated volatility t of each of the twopaleoenvironmental proxies for each speciesover two time periods during the course ofindividual species ‘‘hats’’ (Fig. 1). The first,‘‘initial establishment,’’ is the estimated abso-lute time that elapses between the time aspecies i first attains 5% of the estimatedtheoretical maximum estimated height of the‘‘hat’’ (Li) and when it attains 10% of Li, i.e.,from Sz(1=lS)loge

595

� �to Sz(1=lS)loge

19

� �.

The second, ‘‘expansion phase,’’ is the esti-mated absolute time that elapses betweenwhen a species first attains 10% of theestimated theoretical maximum estimatedheight of the ‘‘hat’’ and when it attains 50%of Li., equivalent to the time point Si.

A Note on the Explorative Nature of the‘‘Hat’’ Model.—In the ‘‘hat’’ model, we mod-eled species as statistically independent ran-dom deviations from the main effects (the m’sin equations 3–7). The ‘‘hat’’ model hasallowed us to explore the slopes of speciesrises and falls, as well as the times of rises andfalls and the durations of the periods ofdominance, among other measures. Theseare explorations of the data not possiblewithout the ‘‘hat’’ model. However, it is notstraightforward to formally test hypothesesregarding species patterns because of thestructure of the model. Hence throughoutthis paper, we refrain from confirmatoryanalyses but adhere to descriptive statistics,avoiding p-values and the like. Work is inprogress to develop methods to deal with thistechnical difficulty.

Computational Details.—Our mixed-effects‘‘hat’’ model is fitted by the maximum like-lihood method. It contains five random effects(the u’s) per species, yielding about 1500unknown values for each of the four groupswe are considering. For instance, there are 312species of nannoplankton included in ouranalyses. There are thus five group param-


eters, 5 3 312 5 1560 species parameters, oneparameter for the model misfit (p0), and hence1566 unknown parameters to be estimated.The problem of maximizing the likelihoodfunction is challenging because of the largenumber of parameters, in combination with thenonlinear structure of the model. The ADModel Builder (http://admb-project.org/) isspecially designed for these types of problems:

it allows efficient maximization of thelikelihood as well as automatic calculationof parameter uncertainties. Originally, weattempted to estimate the variance param-eters s1

2, . . . ,s52 by using the Laplace

approximation (Skaug and Fournier 2006). Itwas difficult to estimate the s’s from the data,so we manipulated s1

2,s22,s3

2,s42,s5

2 tofind values that were as large as possible (to

FIGURE 2. Volatility in paleoenvironmental proxies over time. The upper two panels show time-series plots of firstdifferences of the average temperature proxy from time bin to time bin. The differences are calculated from binneddata, where each bin spans 0.1 Myr, and are plotted at the second time interval from which the differences arecalculated. The dotted lines are two times the standard deviation (t) of the differences (see ‘‘Methods’’ for details),representing the estimated volatility we use in subsequent sections of this paper. The bottom panel shows the numberof available observations in each time bin contributing to the calculation of the differences for d18O. The equivalent plotfor d13C has somewhat fewer observations relative to d18O but is essentially the same and hence not presented.


constrain the individual species random valuesas little as possible) without compromisingsmooth convergence. The estimates reported inour parameter estimation procedure wereconstrained with s1

2,s22,s3

2,s42,s5

2 at 2.7,1.0, 54.6, 1.0, and 7.4, respectively. This allowsboth model flexibility and the numericalstability of the likelihood procedure.

Results

Group ‘‘Hats’’ : General Observations.—In allthe four microfossil groups, species tend tohave faster rates of rise than fall (Table 1).Foraminifers and nannoplankton achievegreater proportions of sites occupied (L) thando diatoms and radiolarians. The latter twogroups also rise and fall faster (Table 1).Slower risers have a longer period of dom-inance while also reaching a greater propor-tion of sites occupied (Table 1). The standarderrors for all the estimates are reasonablysmall as are the probabilities of model misfit(p0) (Table 1).

Species rise and fall rather slowly (Table 1).A species may be extant long before and afterits estimated times of rise and fall. Forexample, the average nannoplankton specieshas a period of dominance that lasts 5.09 6

0.86 Myr but the lengths of its periods of riseand decline are comparable (5.88 6 0.39 Myrand 7.03 6 0.48 Myr, respectively; Table 1).

Species ‘‘Hats’’ : Model Fit.—As mentioned,each species has its characteristic trajectory ofoccurrence through time, to which we wereable to fit our model. We show examples ofthe data and the fitted models for selected

nannoplankton species to illustrate somecharacteristic trajectories. In general, for spe-cies that are more commonly recorded, suchas Coccolithus miopelagicus and Cyclicargolithusfloridanus (Fig. 3), our ‘‘hat’’ model fits thedata well: there are clearly higher occurrencesof the given species in the middle portion ofits life span. In some situations it appears thata double peaked trajectory could be fitted. Forexample, in D. deflandrei (Fig. 3), there is a dipin proportion of samples in which it is presentaround 38 Ma and a second rise around 36-34 Ma, but as specified by our model, only asingle rise and fall were fitted. Even for data-poor species, such as D. germanicus (only 12samples), we are still able to estimate species-specific realized values of S and T because ofour use of ‘‘main effects’’ or a grouptrajectory, even though the confidence inter-vals for S and T are much wider for thisspecies. We also point out that the groupestimates (main effects) are similar to theaverages of species estimates (random effects)although they are not the same (e.g., compareTables 1 and 2 for slopes and lengths of (S,T)).This follows because the individual estimatesare fitted as normally distributed deviationsfrom the main effects (see eqs. 3–7). Wepresent fits for all the species used in theSupplementary Material (online at http://dx.doi.org/10.1666/08080.s1), not least toshow the goodness-of-fit of our model,better assessed on a case-by-case basis.

Rises, Falls, and Environmental ‘‘Events’’.—The average lengths of periods of dominancefor the four microfossil groups are quite

TABLE 1. Parameter estimates (m1 through m5) are shown for the main effects as described in methods and materials.Individual effects are set to zero for L, lS, lT, and T-S shown in this table. p0 is the probability of model misfit. T-S is thelength of the period of dominance, PR the length of period of rise, and PD the length of period of decline; these are allin Myr. The second row in each group shows the standard errors for each of the estimates (those for T-S, PR, and PD arecalculated using the delta method).

m1 m2 m3 m4 m5 L lS lT p0 T-S PR PD

Diatoms 21.525 0.457 20.672 0.104 2.048 0.179 1.580 21.110 5.69E-03 7.75 3.73 5.310.102 0.079 0.417 0.084 0.165 0.015 0.125 0.094 2.45E-04 1.28 0.30 0.45

Foraminifers 20.848 0.014 0.107 20.085 1.810 0.300 1.014 20.919 2.43E-03 6.11 5.81 6.410.112 0.070 0.459 0.075 0.176 0.023 0.071 0.069 1.50E-04 1.08 0.41 0.48

Nannoplankton 21.119 0.001 0.094 20.177 1.627 0.246 1.001 20.838 2.16E-03 5.09 5.88 7.030.106 0.066 0.432 0.068 0.170 0.020 0.066 0.057 9.97E-05 0.86 0.39 0.48

Radiolarians 21.665 0.320 0.064 0.241 1.683 0.159 1.377 21.273 6.25E-03 5.38 4.28 4.630.099 0.072 0.408 0.074 0.158 0.013 0.099 0.094 1.70E-04 0.85 0.31 0.34


FIGURE 3. Plots of each of our model fits to six illustrative nannoplankton species: Coccolithus miopelagicus,Cyclicargolithus floridanus, Discoaster brouweri, Discoaster deflandrei, Discoaster germanicus, and Thoracosphaera operculata.The numbers in parentheses are the number of samples available for the given species. Vertical bars show the observedproportion of sites occupied by each species in 2 Myr bins. Dotted curves are our fitted curves. S and T show theinflection points for the rise and fall of species, respectively. The upper limits of the fitted curves are plotted at the timeof first occurrence in NEPTUNE plus 4 Myr and the lower limits at the time of last occurrence in NEPTUNE minus4 Myr.

TABLE 2. Average lengths of periods of dominance (S,T) and ‘‘face-value’’ stratigraphic durations calculated from firstand last occurrence data (Duration) are shown in Myr. Average slopes of the species ‘‘hat’’ are also shown.

T-S Duration lS lT

Median Mean Median Mean Median Mean Median Mean

Diatoms 6.1 9.1 8.7 13.4 2.03 1.93 21.31 21.36Foraminifers 6.2 7.3 11.9 15.2 1.12 1.22 20.95 21.09Nannoplankton 5.0 7.4 10.9 17.2 1.18 1.25 20.90 21.05Radiolarians 5.2 6.9 10.7 13.6 1.78 1.64 21.52 21.50


similar (Table 2), as opposed to raw strati-graphic durations (see Spencer-Cervato 1999).Both per capita rises and falls are correlated inthe pairwise comparisons (Fig. 4), with notime lag except for radiolarian per capitarises. Per capita declines are more stronglycorrelated than per capita declines. However,in comparisons between foraminifers and theother groups, differenced per capita rises arenot significantly correlated whereas falls aremarginally positively correlated (Fig. 5). Inboth nannoplankton and foraminifers, therewere slight increases in the frequency of risesand per capita rise rates after the PETM(55 Ma), a global warming event (Gibbs et al.2006; Zachos et al. 2001) (Fig. 6). The samecannot be said of diatoms or radiolarians, inthe latter case because there are no radiolarianspecies rising before 55 Ma in our data.Looking at the fall frequency, the onlygenerality we observe is that there is a peakof falls in the 4 Ma to 0 Ma bins for all the fourgroups (Fig. 6). The three highlighted events,namely the PETM and the glaciations of theearly Oligocene and early Miocene, do notcoincide with fall rate changes (Fig. 6). Thechanges in per capita rise rates do not appearparticularly related to the timing of the earlyOligocene and early Miocene glaciations forany of the microfossil groups (Fig. 6).

Preservation Trends.—Inspection of the tem-poral distribution of preservation codes foreach species (‘‘Good,’’ ‘‘Moderate,’’ ‘‘Poor’’)gives no indication of a temporal bias (seeSupplementary Material). Summaries of thesefor each microfossil group diagrammed asbox plots show that regardless of preservationquality, more samples are found during theperiod of dominance (Fig. 7). In other words,poorly preserved and well-preserved samplesare distributed similarly in time. Residualsfrom the lack of fit of the data to the model donot show any striking outliers or trends intime (Fig. 8).

Patterns of Environmental Volatility.—Cli-mate has changed through the Cenozoic asreflected through global average values (Za-chos et al. 2001). Because we are concernedabout how environmental volatility affectsspecies, we have to first ask how paleoenvi-ronmental volatility has varied through the

Cenozoic. We find no temporal trend in thedifferences of the time binned average pa-leoenvironmental proxies, i.e., the valuesfluctuate around zero (Fig. 2). This is regard-less of the size of the time bins used, eventhough we only present results using 0.1 Myrbins. Volatility estimates do decrease dramat-ically after 40 Ma for d13C and after about60 Ma for d18O (Fig. 2; dashed lines bracketingsolid line represent differences), but they donot follow the changes in the number ofobservations available. We note here that thetwo estimated volatility time series for d13Cand d18O are correlated (0.57).

Environmental Volatility, Estimated Period ofDominance, and Rates of Rise.—We presentscatter plots of estimated volatility experi-enced at the two phases of establishment(initial establishment and expansion phase),versus the length of the period of dominance(Figs. 9–12). Because these data are non-normal and non-independent, we choosesimply to discuss the scatter plots. In theseplots, we removed species with estimated Tlater than 0 Ma because of interpolationproblems, i.e., the greater uncertainty ofestimating persistence in the future due tothe lack of empirical data even though resultsremain qualitatively the same.

The average volatility experienced duringthe ‘‘initial establishment’’ phase spans aslightly greater range of values that duringthe ‘‘expansion phase,’’ clearly seen at leastfor the oxygen isotope proxy (Figs. 9, 10). Inall of the plots for volatility in d18O over eitherthe initial establishment or the expansionphase, a triangular or fan-shaped distributioncan be seen (Figs. 9, 10. Because there is asharp decrease in global volatility at about60 Ma, we plotted the species experiencingthe beginnings of either their initial establish-ment (Fig. 9) or expansion phase (Fig. 10)before and after 60 Ma separately. We findthat species establishing or expanding beforeand after 60 Ma show basically the samepatterns. Patterns of volatility in d13C, how-ever, show no such trends (Figs 11, 12).Within each temporal group, there is arandomly distributed scatter of volatilitiesand periods of dominance.


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FIGURE 6. Left, Absolute frequencies of rises and falls for 2 Myr bins plotted in absolute geologic time. Right, Per capitafrequencies. The three global climate events—the Paleocene-Eocene Thermal Maximum (PETM, 55 Ma), earlyOligocene glaciation (34 Ma), and early Miocene (23 Ma) glaciation—are indicated with arrows.


FIGURE 6. Continued.


There is no observed trend in environmen-tal volatility and rate of rise for any of the fourgroups (Figs. 13–16).

Discussion

Pondering the ‘‘Hat’’.—Occurrences of fossiltaxon are commonly denser in the middle ofits observed duration. In other words, speciesand genera are more detectable in the middleof their lifetimes, as seen in mammal genera(Jernvall and Fortelius 2004), marine inverte-brate genera (Foote 2007), Cenozoic mollus-can genera (Foote et al. 2007) and Cenozoiczoo- and phytoplankton species (Liow andStenseth 2007). We have already shown that amodel of occurrence trajectories should behump-shaped (Liow and Stenseth 2007), andthe ‘‘hat’’ model we develop here satisfies thisrequirement and also supplies parametersthat give biological meaning.

Because occurrence trajectories are oftenhatlike and even more commonly haveshallow slopes, raw global stratigraphic dura-tions are clearly poor estimates for taxon

durations. An increase in sampling effortpotentially can change the stratigraphicranges of species drastically. However,changes in sampling effort should have littleinfluence on our estimated lengths of periodsof dominance, a measure of a species’duration of ecological dominance. Althoughglobal distributions of duration are quitedifferent for the four microfossil groups(Spencer-Cervato 1999), the average esti-mated lengths of periods of dominance aresimilar (Table 2).

Proportionately rare misidentifications ofspecimens assigned to ages rather far outfrom the temporal range of the given speciesalso have little influence on our parameterestimates. Taking Coccolithus miopelagicus asan example (Fig. 3), our estimates are S 5 25.56 0.47 Ma and T 5 8.68 6 0.67 Ma, whereasthe earliest and latest occurrence data inNEPTUNE for this species are 35.32 and0.13 Ma respectively. Although no expertopinion is available to us on the acceptedglobal first occurrence for C. miopelagicus, its

FIGURE 7. Box plots of species’ average proportions of samples with ‘‘Good,’’ ‘‘Moderate,’’ or ‘‘Poor’’ preservationbefore their individually estimated time of rise, S, during their period of dominance (S,T) and after their time of declineT. G-e 5 ‘‘Good’’ preservation before S; G-m 5 ‘‘Good’’ preservation during (S,T); G-l 5 ‘‘Good’’ preservation after T;M-e 5 ‘‘Moderate’’ preservation before S; M-m 5 ‘‘Moderate’’ preservation during (S,T); G-l 5 ‘‘Moderate’’preservation after T; P-e 5 ‘‘Poor’’ preservation before S; P-m 5 ‘‘Poor’’ preservation during (S,T); P-l 5 ‘‘Poor’’preservation after T; N-e 5 Non-assigned preservation before S; N-m 5 Non-assigned preservation during (S,T); N-l 5Non-assigned preservation after T.


‘‘highest occurrence’’ (i.e., most recent occur-rence where it is abundant enough to be auseful index species in biostratigraphy) isreported to be 11.020-10.613 Ma (Raffi et al.2006). The entries for C. miopelagicus inNEPTUNE that are dated much younger than10.6 Ma may involve reworked specimens,erroneous age models, or misidentified speci-mens. But note that T for C. miopelagicus islater than when it is considered abundant.Taking another example, the earliest recordfor Discoaster brouweri in NEPTUNE is

35.47 Ma. However, we estimate that its S 5

9.73 6 0.24 Ma, indicating that the recordfrom 35.47 Ma is an outlier and should bedisregarded. Although our plotting conven-tion allows the left tail to be drawn out for D.brouweri (Fig. 3), data points far left of thegraph have negligible influence on ourestimates. We also are aware that the genusDiscoaster is thought to have gone extinct inthe latest Pliocene (Raffi et al. 2006). T for D.brouweri is 0.5 6 1.0 Ma; thus the estimatedtime interval of probable decline is later than

FIGURE 8. The lack of fit of the hat model to data over time plotted as standardized residuals (unitless).


the time of its extinction that is commonlyaccepted in the microfossil literature. Wetherefore caution that if too many of theidentifications or age models are system-atically biased, our modeling may still beinadequate for specific cases (such as thedecline of D. brouweri). To summarize, ourmodel not only gives confidence intervals forthe times and rates of rises and falls, it alsohelps to identify potential spurious records ofindividual species if these are temporally farremoved from the estimated times of risesand falls. Although we are not able to verifyrecords for each species in the database, we

encourage taxonomic experts to investigatethe fits of species they have studied and toinform those maintaining the database of anyinconsistencies and errors.

Our ‘‘hat’’ model is a general and flexibleone. Even though good fits of such a generalmodel to every single species cannot beexpected, not least because we have notfactored in preservation variations and site-and time-specific stochasticities, the species-specific plots show that the general shape ofthe model is reasonable. Some individualspecies may have their proportional occur-rences far removed from the main ‘‘hat’’

FIGURE 9. Estimated length of period of dominance (S,T) in Myr, versus estimated d18O volatility averaged over theinitial establishment phase of each species (time elapsed between achieving 5% and 10% of the height of the ‘‘hat’’) foreach of the four groups. Species that have risen to 5% of their maximum proportion occupancy after 60 Ma are labeledwith crosses and those that have done the same before 60 Ma are labeled with open circles.


trajectory as indicated by non-zero p0’s, butthese are not an issue for our generalconclusions.

In approaches where observations of com-plete life spans of lineages are required,extant taxa must be excluded, possibly bias-ing the studies (Liow 2007a,b; Viranta 2003).However, Ti can take values close to theRecent or even into the future and Si can takevalues close to 65.5 Ma and older. This allowsus to retain both species that are extant andthose that were established before 65.5 Ma,avoiding the problem of censoring in survivalanalysis (Kalbfleisch and Prentice 1980). Forexample, we used only entries in NEPTUNE

that are younger than 65.5 Ma, but weestimated Thoracosphaera operculata (Fig. 2) tohave S 5 67.1 6 2.5 Ma. However, eventhough we may not obtain precise predictionson the slope and timing of rise of this species,we may still obtain informative estimates andtheir uncertainties by extrapolation. In otherwords, the average length of period ofdominance of the group or clade and thedistribution of the random effects (around T-Sand lT) may inform us on the expected rise ofthis species.

Rises and Falls: Asymmetries and RelativeDurations.—Planktic microfossil species tendto rise faster than they fall. An explanation

FIGURE 10. Length of period of dominance versus volatility in d18O over expansion phases (time between achieving10% and 50% of the height of the ‘‘hat’’). Symbols as in Figure 9.


previously suggested for this quick-rise/slow-fall pattern is the swamping of the databy long-lived, abundant species (Liow andStenseth 2007). These species may havebiological characteristics that allow them toexpand in geographic extent quickly, whiletheir abundance and large geographic rangesprotect them from a rapid decline. Ourestimates (Table 1) show that slower rise ratescorrespond to a greater estimated theoreticalmaximal proportion of sites of occurrence (L),thus failing to support this hypothesis, giventhat L is a good proxy for geographic range.We have also suggested previously that a

species may rise quickly at the beginning ofits observed life span, because its initialproblems of establishment had been solvedduring a period of very low detectability. Thefall is even slower because once the speciesspreads, it needs to experience declines in allthe ‘‘sites’’ in which it had been detected suchthat no one population can ‘‘rescue’’ another,in the sense of metapopulation dynamics,whether the declines in localized sites are dueto climate perturbations, predators, parasites,and/or competitors. This asymmetry in riseand fall may also be due to biased sampling,which we discuss below.

FIGURE 11. Length of period of dominance versus volatilty in d13C over initial establishment phases (see Fig. 9). Speciesthat have risen to 5% of their maximum proportion occupancy after 40 Ma are labeled with crosses and those that havedone the same before 40 Ma are labeled with open circles.


Our modeling exercise has shown thatspecies rises and declines are much slowerthan we previously realized, calling forserious explorations of explanations. Whymight species rise and fall so slowly overgeologic time? First, the final, total disappear-ance of a species happens only when the lastindividual of that lineage dies, but the timefrom decline until the death of the finalindividual can be quite long, with populationabundance declining along the way. Simi-larly, it is difficult to infer how long a specieshas existed before its first observed occur-rence because the probability of detecting itduring the early phase of its history is likely

to be low for several possible reasons: (1) itsmembers are morphologically not well differ-entiated from members of the ancestral line-age; (2) its population size is smaller com-pared with its ancestral lineage (even in thecase of the equal splitting of one lineage intwo, say by a suddenly arising geographicalbarrier); or (3) local populations have not yetspread out geographically so it is much moredifficult to sample them. In other words, thetime spent at maximal occurrence is relativelyshort, compared with the time a species spendsrising and then declining to occurrence valuespresumably low enough for stochastic pro-cesses to ultimately to drive it to extinction.

FIGURE 12. Length of period of dominance versus volatility in d13C over expansion phases (see Fig. 10). Symbols as inFigure 11.


Environmental Events and Temporal Patternsof Rise and Fall.—Per capita rise and fall ratesappear to be correlated for pairs of groups ofmicrofossils. Each group tends to declinewhen other groups are declining, and this isespecially prominent for foraminifers. Thismay indicate that the microfossil groups areresponding to the same environmentalchanges. Although some studies have singledout the effects of prominent environmentalevents on some microfossil groups (Gibbs etal. 2006; Kamikuri et al. 2005), we do notobserve such events to affect the frequency ofeither rises or falls. Why may that be? First,

our estimates are global and even thoughsome populations may be affected adverselyby environmental change, other populationsof the same species may mitigate the effects ofthat change. Second, our bins are rather wide(2 Myr); hence, unlike studies restricted tosingle deep-sea cores with very high temporalresolution, we cannot detect events that occuron a finer time scale, even if they are large inmagnitude, if the effects are not long-lasting.We have no reason to believe that time binsize will affect our general conclusions in thispaper. The ‘‘hat’’ trajectory fits most speciesquite well and dividing the data temporally at

FIGURE 13. Estimated rates of rise (lS) versus estimated d18O volatility averaged over the initial establishment phase ofeach species (time elapsed between achieving 5% and 10% of the height of the ‘‘hat’’) for each of the four groups.Species that have risen to 5% of their maximum proportion occupancy after 60 Ma are labeled with crosses and thosethat have done the same before 60 Ma are labeled with open circles.


a finer scale is not justified. A third explana-tion for our findings is that we did notinclude rarer and perhaps more environmen-tally sensitive species. Fourth, our approachestimates the times of a species’ rise and fall,rather than studying the times of its origina-tion and extinction. The time points of riseand fall are not expected to translate in astraight forward manner to origination andextinction time points, not least because theslopes of rises and falls for species and groupsare different. Fifth, environmental distur-bances over the long term may be moreconsequential to occurrence dynamics thanpreviously suspected (De Blasio and De

Blasio 2009); hence, large, abrupt events mayhave affect the rise and fall of species lessthan does the accumulation of lesser environ-mental changes. Last but not least, bioticinteractions may modulate the effects ofenvironmental disturbance.

Pondering Preservation.—As is common forpaleontological data, our data contain fewersampled cores from earlier in geologic time(Spencer-Cervato 1999). A clear benefit of ourmodeling approach is that a reduced fre-quency of samples earlier in geologic timewill not bias the shape of the ‘‘hat,’’ but onlymakes the estimates less precise. Foraminifersand nannoplankton are better-studied groups

FIGURE 14. Estimated rates of rise versus volatility in d18O over expansion phases (time between achieving 10% and50% of the height of the ‘‘hat’’). Symbols as in Figure 13.


in NEPTUNE, where species are more fre-quently identified and entered in the data-base. Our approach does not lead to biasedestimates for different groups, but only leadsto a lower precision of the estimates fordiatoms and radiolarians.

If preservation probability for each speciesthrough time were known, it would be simpleto estimate the true occupancy trajectory of aspecies. However, a species’ preservationprobability is unknown and most likely nota smooth function of time and would have tobe estimated (MacKenzie et al. 2006). Tryingto estimate the preservation explicitly

through our already complex model wouldmake computations more burdensome. Insetting up our model, we made several broadbut explicit assumptions about preservationand detection. We acknowledge that geologi-cal changes such as rise and fall of sea leveland sequence stratigraphic effects can con-tribute to site- or time-specific variations inpreservation (Holland 2000) and detection,whereas biological traits such as size, testrobustness, and mineralogy may contribute tospecies-specific variations. We have, how-ever, no reason to believe that these species-specific and site- and time-specific factors

FIGURE 15. Estimated rates of rise versus in d13C volatility over initial establishment phases. See Figure 13 for details.Species that have risen to 5% of their maximum proportion occupancy after 40 Ma are labeled with crosses and thosethat have done the same before 40 Ma are labeled with open circles.


contribute to the general patterns we havefound here.

On the other hand, it is useful to discuss thefact that the focal parameters of our analysis(T-S, lS and lT) are little affected when thepreservation is slowly increasing with time,whereas T and S are somewhat positivelybiased.

Defining occupancy probability, o, as meanlocal abundance over a time bin, and assum-ing that each individual is preserved withprobability p, then occurrence probability issimply o:p. If preservation probability isconstant, the occupancy trajectory o tð Þ isproportional to h tð Þ.

If we consider the global trend of increasingpreservation probability with geologic time(Spencer-Cervato 1999; Kidwell and Holland2002; Smith 2007), then preservation prob-ability p(t) gradually increases as a functionof time, t. In this case, the true occupancytrajectory o tð Þ is filtered, such that h tð Þ~o tð Þ:p tð Þ. Both occurrence and occupancytrajectories are regarded as hat-shaped, butthe issue remains as to how the initial and theterminal inflection points of the occupancytrajectory are mapped into the correspondingpoints of the occurrence trajectory.

Let Socc be the initial half-points of the firstlogistic part of the occupancy trajectory,

FIGURE 16. Estimated rates of rise versus volatility in d13C over expansion phases (see Figure 14). Symbols as inFigure 15.


corresponding to S. An increasing preserva-tion probability makes h Soccð Þvh Sð Þ and con-sequently SwSocc. There is also a positive biasin T. The derivative of the hat-function at S is(L:lS)=4 and the derivative at T is (L:lT)=4. Bya first-order Taylor expansion of the occur-rence trajectory about its inflection points,we find the bias to be approximatelyS{s&2=lS 1{½p Sð Þ�=½p Yð Þ�ð Þ, where Y is atime at which the rising logistic curve of the‘‘hat’’ is nearly 1 (say 0.99 making Y~Sz

4:6=lS), and T{Tocc&(2=lT) 1{½p Yð Þ�=½p Tð Þ�ð Þwith Y similarly defined. For a species withnearly equal rates of rise and fall, a slowlyincreasing preservation probability will mapthe occupancy trajectory to an occurrencetrajectory closer to the present, but thelength of the period of dominance, T-S,changes minimally. Likewise, the slopes ofrise and fall (ls and lT) are nearly unaffected.Because most of our discussions center onT-S, lS, and lT, we temporarily shelve theissues of how global secular trends and localidiosyncrasies in preservation may affect ourgeneral results.

On the other hand, the quality of preserva-tion for a given species through its lifetimemay affect the estimates of its rise and fall. Forexample, species potentially entering samplesearlier in geologic time are preserved lessfrequently and/or the specimens may berelatively less identifiable. Depending onexactly how these more poorly preservedsamples are distributed temporally, this pres-ervation bias could produce asymmetric rateseither where rises are quicker than falls orvice versa. However, we see no bias in thetemporal distribution of preservation qualityfor any of the four microfossil groups.

Patterns of Environmental Volatility duringFormative Periods.—We predicted that speciesthat had endured volatile beginnings shouldhave a longer period of dominance, all otherthings being equal. If species are able tosurvive environmental fluctuations duringthe earlier phases of their lives, they may beadept at handling unpredictability and fluc-tuations later on. A prominent feature of thephysical environment is temperature, whichin turn may affect another important factor,the productivity of the local environment,

both of which can be estimated in the fossilrecord by using proxies such as d18O and d13C.Temperature changes have been seen as animportant driver of biological change (Thu-nell 1981; Janis 1993; Bown et al. 1994;Schmidt et al. 2003; Gillooly et al. 2005; Huntand Roy 2006; Wright et al. 2006; Currano etal. 2008; Kurschner et al. 2008) and likewise,productivity or energy availability is alsothought to influence diversity (Bonn et al.2004; Gross and Cardinale 2007).

Currently there is no inherently intuitiveway to determine the length of period duringwhich we should consider the environmentalvolatility at the beginning of a species’ life.This problem is not unique to this study orthis dataset. The period during which specia-tion occurs can be understood in many waysboth theoretically and empirically (Coyne andOrr 2004). For example, speciation may beconsidered the period from when two popula-tions begin to diverge until they attain repro-ductive isolation. Or, it may also be moreloosely considered the time when gene flowceases, regardless of reproductive isolation. Inpractice, however, speciation rates are oftencalculated from diversification rates minusextinction rates (Coyne and Orr 2004). Simi-larly, in the theory of island biogeography andits offspring, invasion biology, the beginning ofestablishment is a difficult concept, because aspecies must first disperse, establish, and thenbuild up populations large enough to make anecological impact and/or for biologists tosample (MacArthur and Wilson 1967).

We emphasize that the exact point of aspecies’ beginning is impossible to establish,either biologically or statistically. Therefore,instead of trying to determine when thespecies began its life, for which the uncer-tainty is very large, we use the period of timefrom when 5% of the maximum height of the‘‘hat’’ is achieved to when 10% of the same isattained as an approximation for the ‘‘initialestablishment’’ phase in a species’ life. As canbe seen from the individual species ‘‘hats’’(Fig. 3 and Supplementary Material), this peri-od, the length of which varies according to boththe slope of increase and the maximum heightof the ‘‘hat,’’ may or may not encompassobserved samples. Similarly, the ‘‘expansion


phase’’ of a species is arbitrarily taken as theperiod of time from when 10% of the maximumheight of the ‘‘hat’’ is achieved to the inflectionpoint, S. This may be interpreted as the periodwhen the species is initially dispersing, increas-ing the number and spread of its populations.Again, we expect that if the species manages todisperse despite environmental volatility, thisearly ‘‘training’’ should prepare it for laterenvironmental perturbations, and hence con-tribute to a long period of dominance.

We found that the average volatility ex-perienced during the ‘‘initial establishment’’phase spans a slightly greater range of valuesthan during the ‘‘expansion phase.’’ This isreasonable because values averaged over alonger period of time are less likely to beswamped by large extremes. In general, thereare few species in any of the four groups thatare likely to persist if they experience anaverage estimated d18O volatility value ofmore than about 0.3. Note that none of theradiolarian species in our data experiencedvery high volatilities—this is in part due totheir arising later.

Given that a species persists after an initialperiod of higher environmental volatility, thelength of its period of dominance (T-S) maybe greater than expected if there is norelationship between T-S and initial environ-mental volatility (Figs. 9, 10). Species thatexperienced lower volatilities at the begin-ning of their lives do not exhibit long periodsof dominance. However, those that experi-ence higher volatilities at the beginning oftheir lives either do not persist much longerthan those that did not experience highervolatilities, or they persist longer than ex-pected. Although volatility in global tempera-ture volatility decreases sharply ca. 60 Ma, wefind that species establishing or expandingbefore and after 60 Ma show basically thesame patterns. Patterns of volatility in d13C, aproxy for productivity, show random scatterswith respect to length of period of dominance.We interpret these as indicating that thevolatility in paleotemperature has an effecton species persistence but not the volatility inpaleo-productivity.

We also expected high environmental vola-tility to dampen rise rates, because attempts to

build populations or disperse may be thwartedby frequent reversals of environmental condi-tions. However, our results show no suchrelationship between environmental volatilityand rate of rise for any of the four groups.

There are, of course, many potential biasesin volatility estimates: for example, theisotope readings serving as our paleoenviron-mental proxies may reflect local events eventhough we assume that the values, averagedover time and space, at least in part reflectglobal tendencies. We emphasize, however,that the mismatch between the size of theenvironmental proxy bins (0.1 Myr) and thatof the raw occurrence data (2 Myr) is notproblematic because we use the estimatesfrom a smooth ‘‘hat’’ to estimate the vola-tilities experienced during the initial andexpansion phases of species. Moreover, theisotope readings come from several differentmicro-organisms, each with their species-specific and habitat-specific effects influenc-ing the readings (Katz et al. 2003). Addition-ally, older species might persist longer simplyby having more time to persist, althoughaccording to our data (Figs. 9–12), the periodof dominance is not strikingly different forolder species (specifically those having Slarger than 60 or 40 Ma). The estimatedperiods of initial establishment and theexpansion phase depend on the form ofspecies ‘‘hats,’’ which in part depend on thelength of the period of dominance. We alsonote that perhaps surviving great environ-mental volatilities even earlier in the lives ofspecies than what we can estimate (i.e.,‘‘initial establishment’’) contributes morestrongly to periods of dominance, but becauseof the uncertainty of estimating the durationsof these very early phases, we are unable fornow to progress further than this attempt.

Conclusions

1. We have shown that it is possible to fit achanging temporal trajectory of occur-rence for individual species, using amodel including both main and randomeffects. Despite needing to estimate largenumbers of parameters, our estimationprocedure converged on reasonable solu-


tions. The parameters estimated, as wellas the trajectory estimated from theparameter values, can be interpretedecologically. A new species has a periodof low detectability, possibly due to lowmorphological divergence, low popula-tion abundance, and/or low global occu-pancy. If this species survives, it can bemodeled as having a sigmoid rise in theproportion of sites occupied, until itreaches a characteristic ‘‘carrying ca-pacity’’ estimated in terms of maximumproportion of sites occupied. As thespecies approaches its final demise, itdeclines as an inverted sigmoid curve inthe proportion of sites occupied, afterwhich it never recovers, although it maylinger on for variable periods of time.

2. In light of this model, it is clear thatestimating the points of the ‘‘true’’ timesof speciation and extinction is close toimpossible. Hence we propose that atleast for certain types of macroevolution-ary and macroecological questions, it maybe more meaningful to use the times ofpeak rise and fall as the primary data.

3. Our model indicates that it takes a verylong time for a species to rise to and fallfrom its maximum frequency of occur-rence: the periods of rise and fall areabout as long as the period of dominance.This observation begs for a biologicalexplanation.

4. The peak rate of rise is generally greaterthan the peak rate of fall.

5. Species with long periods of dominancetend to have been established duringenvironmentally more volatile times. Wepostulate that species that were estab-lished during less volatile times are lesslikely to persist for long periods becauseenvironmental conditions are likely tochange on longer time scales. However,species that were established during morevolatile times do not all have long periodsof dominance; rather, they can be dividedinto those that have a relatively shortperiod of dominance and others with along period of dominance, as compared tospecies established during less volatiletimes.

6. We see little correlation between environ-mental volatility and the frequency of riseof species. Nor do the three well-knownglobal environmental events at 55, 34, and23 Ma affect the frequency of rises in allfour microfossil groups.

7. We close by stressing the explorativenature of this study. The ‘‘hat’’ model isa parametric one that filters raw data inorder to highlight species patterns, allow-ing the exploration of processes of spe-cies’ rises and falls central to evolutionarybiology.

Acknowledgments

We thank M. Pagani for access to the paleo-proxies. Crampton and T. Ezard gave veryuseful and constructive reviews. L. H. Liowthanks D. Lazarus, C. Cervato, and J. Hender-iks for discussions on NEPTUNE.

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