Body size and diversity exemplified by three trilobite clades
TERZY TRAMMER ANdANDRZEJ KAIM
Trammer, J. & Kaim, A. 1997. Body size and diversity exemplified by three trilobite clades.- Acta Palaeontologica Polonica 42, l, l-12.
Cope's rule concerns only the radiation phase of a clade, overlooking the phase of the clade decline; thus it is incomplete. Changesofbody size during the entire evolutionary history of a clade are exemplified here by three trilobite groups - ftychopariina, Asaphina and Phacopida. Increasing diversity ofthe clade is associated with increase in maximum body size during the radiation phase, and decreasing diversity is generally associated with a decreasein maximum body size. Two basic patterns of the maximum body size changes are observed dwing the decline of the clade. The first one is characterized by a high correlation between diversity and the maximum body size, and indicative of species atfrition that is nonselective with respect to the body size. The second one is characterized by a weak correlation between diversity and maximum body size, and typical of selective species attrition in relation to size.
Jercy Trammer, Instytut Geologii Podstawowej, Ilniwersytet Warszawski, al. Zwirki i Wigury 93, PL-02 -089 Warszaw a, P oland; Andrzej Kaim, Instytut Paleobiologii PAN, ul. Twarda 5l/53, PL-00-l14 Warszawa' Poland.
Introduction
The evolutionaryhistory of an extinct cladeconsists of at leasttwo phases:the phase of diversificationor radiation (when speciesnumber increase)and the phaseof its decline or reduction (when the number of speciesdrop). Cope's rule, an old (Cope 1887)but recentlymodernized (Stanley 1973) generalization, claims that most groups have evolved from small towards larger body size. This rule concernsonly the first, diversification phase of a clade's history (Stanley 1973: figs 5-7). In effect, our knowledge of body size evolution is biased and incomplete as it disregardsthe evolution of body size during the secondphase, when a clade was on the wane.The aim of the presentcontribution is to presentan analysisof changesin body sizeduring the complete evolutionary history of a clade. Body size and diversity: TRAMMER & KAIM
^45 E
I -s z^^ ILI JU J 225 4. zo T
u lc o 10
5
0 40 60 80 loo 12o J:"" ,.J:tt" (';T
Fig' 1. Body length related to cephalonlength in Ptychopariina,Asaphina and Phacopida.Since body length and cephalonlength are strongly correlated,length of cephaloncan be used as a size-index.Data measured from Harrington et al. (1959).
The problem will be exemplified by three trilobite groups:the Early Cambrian- Late Ordovician suborder Ptychopariina Richter, 1933; the Early Cambrian-earliest Silurian suborderAsaphina Salter, 1863; and by the Late Cambrianto Late Devonian .orderPhacopida Salter, 1864. Body size for individual specieswas measuredon all adult specimensof the aforementionedtaxa illustratedinthe Treatiseon Invertebrate Paleontology(Harrington et al. 1959).Such data have been proven by our analysisto be statisticallyrepresentative. Since in mostcases the illustrationsintheTreatise show only cephalons,we measuredcephalons' lengths as a proxy for body size.we found that cephalonlength is well correlatedwith the whole body length (Fig. 1) and thus may be used as a size-index.Finally, we conshuctedsize-frequency histograms for various stagesof the evolutionary history for eachtrilobite group studied. We areaware that since1959, when the trilobite volumeof Treatisewas published, a large number of new trilobite taxa has been described.we decided,however, to conductthis study on the basisof knowledgeof trilobite history recordeduntil 1959. which, incompleteas it is, demonstratestrends in trilobite evolution.The histogram relating nilobite speciesnumber to their body size (Fig, 8) is typical also for other animaltaxa (compareMay 1988:figs 2 and6; Jablonsky1996: fig.10). This validates the representativenessof the Treatisedata.It would be interestingto seeif datafrom a larger sample - such as the future edition of the Treatise - will confirm oul conclusionsbased on the limited subset.
Results
In the Ptychopariina, the phase of diversification continued from the Early to Late Cambrian, and the phase of reduction from the Late Cambrian to Late Ordovician (Figs 2, 3). In the Asaphina,the phaseof diversificationpersisted from the Middle cambrian to Early ordovician, and the phase of reduction from the EarJy to Late ACTA PALAEONTOLOGICAPOLOMCA (42) (I)
1oT n=2 rara 0 l----.--- -+ OiOovician Ptychopariina
10T n=B Middre n=403 O+-___.@-&-q Ordovician t8 toT n=39 Earty (/) 101& Ordovician tlJ oF6@G*@al--- - ul [L 50 @^ IL o30 t ,2o u rn dl l z 30 20 Middle Cambrian 10 0 30 20 Early '10 Cambrian 0
CEPHALON LENGTH (mm)
Fig. 2. Frequency distribution of cephalon length among speciesof Ftychopariina in various intervals of their evolutionary history. During the Cambrian radiation phase, maximum body size and diversity increased,while during Ordovician reduction phasethe marimum body size and diversity decreased'Data (including stratigraphic range of the clade) from Harrington et aI. (1959).
Ordovician (Figs 4, 5). We found that in both groups, maximum cephalonsize is strongly correlatedwith diversity. It increasedduring the radiation phase,and de- creasedduring the reductionphase (Figs 2-5).To the contrary,we did not find strong correlationbetween maximum cephalonsize and speciesnumber in the Phacopida (Figs6,7).
Discussion
Many moretrilobite specieshad smalland intermediate cephalon sizes than large sizes (Fig. 8). Such a body size pattem is largely result of cladogeneticdiffrrsion (passive trend of McShea 1994) awayfrom an originally small-sizedancestor as shown by Gould (1938)and McKinney (1990a,b). The processof diffirsion and,in addition,the highertotal originationrate of smaller-sizedspecies then large ones(Dial & Marzluff 1988;McKinney 1990b)implies that the origin of a large trilobite specieswas a rare or improbableevent. Thus the correlationbetween maximum body size and diversity may be discussedand explainedin termsof probability. 'seAIEArq ,(rer ezrse8rul pe,lerqo" slerlrrueru leql pelou (AtAil,(epe15 puu sleuruuru ',firsre.nrp ere ,frsre.trp qBFIpuu ezrsdpoq a3.re1ueel11eq uo4eleJros;o seldurexereq16 'se11uourlue q8r,qJo spoued qll^\ eu4l uI peprcuro3,(1pnsn seplder pge spodoqcuJq 'sproplneu'srr€JeJruruero; Suoure srruoJ JI1ImBIS;o ecuermcto eql leql pelou (ZOZ'd 'Eg6I'196I) rellnhtr'erolureqpng '(qOOOt ,{euu5trc11:886I plnog !9361ueppegce6 'sueeplssoqord'sesroq :1-g s8g :g26y f,eyrc1g)sluepor pue smesoc,(1ed'sueecelec 'selruoruure'spodorqcerq 'suereJrmrr€JoJ JoJ os1e ]nq selrqoll4 JoJfluo lou prlel sI eInJ slrlJ '(S-Z s8rg) requrnuselceds Surseercur qlIA sos?eJJulsellqollrl Suouruezrs ,(poq rrmrrrrxeru.{q.t sr sIg.T.'seseeJoul d11 ueq sesueJJepdueq,l luql lceJeql luo4 s/(oIIoJ ',(Iluenbesuoc q 'ut8rro slr ro; ,(msseceusuo4el3eds go d71Joqumu eq1 sr re8rel oql 'prru '(8 '3rg) 'serceds ur8uo s1r;o d,(llpqeqord erp sI Jeilerusoql eIqolLD e Jo ezrs 'uolsnJJrp ,(poq eqt re8rel eq] crleueSopulseql Jo 1ro1rreql q8norql 1IeJo lsrlJ ,(lereueg 'ur8uo '8ID .setcedsyo ,ftr-p.qeqordeql e1eruqseol (8 epp pcutdure eql esn,{eureu6 '00I ro 16'971qenbe Suol ruur 0g sreqrueuqlr,l serceds1o qSpo aql roJ drusseceu 'prreq 'I0'0 r/71suorprceds;o requrnu oqt reglo eql uO ro 000I/0I spnbe Suoltuut 99 '?uo1 selqelueserderqIA sarcads;o ur8uo go d firyqeqord eql ueql urur 0S oJe0I 'serceds 'e8ere,re ,(1uo 9961Suogm;1 uo suorlercedsd71 speeu serceds sr-q1;o ur8uo eq] 'aseqd ueql 'dsr sercedsue,rr8 e;o ur8uo eq];o ,(lqgqeqordeql g - uollelp€r arl;,
'p ta paE',nHreu?el'rs eurr;'z'alduro4 e,nc'(L68'0 = r) uoqelexorSuorts ,fts,r eql etou ,*.,|?rXflr] 'g '8tg 3io e8eluecred e su pepold euruedoqcr(14u1 dlrsrelrp prre ezrs ,(poq umrnrxuru eq1 m se8ueq3 (Byrt)f u\rr creoto:le 6e? 09t 009 099 0L9
ueuquPc _l o o o o o ^uPf q d_< 0t P< = ol, 9c; Fm 5r- :3 -'o_ XE qH +a :9. 56 -.(D a:.<- '!) 6' 9) 0) Fo - 5 f, = :r
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1ltn1T)IT dilhlh111gJ :{ystaup puo ans Kpog ACTA PALAEONTOLOGICA POLONICA (42) (1)
Late Ordovician
a lrJ O tu 8 (L o U) 4 IL o 2 E. 0 I.IJ d]
f 4 z 2 0 t0 20 Asaphina tt=87 Middle Cambrian
CEPHALON LENGTH (mm)
Fig. 4. Frequency distribution of cephalon length among speciesof Asaphina in vmious intervals of their evolutionary history. During the Cambrian-Early Ordovician diversification phase,the maximum body size and diversity increased, while during the Ordovician phase of the reduction, maximum body size and diversity decreased.Data (including stratigraphic range ofthe clade) from Harrington et aI. (1959). rapidly during only in about 10 to 15 Ma of radiation.Bivalves, in contrast,needed much more time to develop large-sizedforms. This conforms with our explanationof the phenomenonsince mammals'diversification rates were much higherthan diversi- fication rates of bivalves (Stanley 1979) so that mammalsachieved high diversity much faster. The reduction phase.- To explain why maximum body size decreasesalong with decreasingdiversity during the reductionphase, two hypothesescan be suggested: a deterministicone and a stochasticnull hypothesis.The deterministichypothesis assumeselevated extinction and/or reduced origination rates of large-bodiedtrilobite speciesin comparisonwith small-sizedones, as postulated by Stanley(1979) and Vrba Body size and diversitv: TRAMMER & KAIM
vo1o0 Asaphina 90 80 m diversity
70 a m€uimum bodysize 60 r = 0.840 50 40 \\
30 20 10 \
c c (s (sc o.S >,'6 :c)nt.S 9'F 9'< e.9 =F ,:Y O .=o qB 9E -p >E -E >E c L i's o c o o o 550 500 450 439 GEOLOGICTIME (Ma)
Fig. 5. Changesof the maximum body size comparedto changesof the diversity in Asaphina, plotted as a percentageof maximum. Note the strong correlation between size and diversity (r = 0.840). Data from Fig. 4. Time scaleafter Harland et al. (1990).
(1983) for all large species.But the assumptionmentioned above is not necessaryto explainthe patternvisible in Figs 2-5. The null hypothesis,on the otherhand, assumes that the taxonomicattrition of speciesduring the reductionphase is nonselectivewith respectto species'bodysize. For successivetime periods,it assumesequal extinction probability q for all speciespresent, no matter if small, intermediateor large (for the positive test seeAppendix). To make the problem simpler, the hypothesis omits the influenceof speciesorigination because during the reductionphase the rate of extinc- tion greatly exceedsthe rate of speciation(compare Harrington et al. 1959; Foote 1988).An equalextinction probability for all the speciesdoes not meanan equaltotal probabilityofextinction for all body sizespresent, because individual body sizeclasses are not equally numerousat the beginningof the reductionphase (see e.g., size-fre- quencyhistogram for the Late Cambrianin Fig. 2).Large-bodiedtrilobite speciesare rare; in general,the smallerare trilobite species,the more frequentthey are (Figs 2,4 and 8). For example,only one Late Cambrian Ptychopariinaspecies falls into the cephalonsize class of 118-120 mm, but over 40 speciesfall into the size class of 24 mm (Fig. 2). If one speciesfrom the size class 118-120mm becameextinct, the ACTA PALAEONTOLOGTCAPOLONTCA (42) (t)
oI wq t1=7 Late 6l @ m r mt Devonian 10 20 30 4- n=15 Middle' 'l uevonlan 6l ffi@ WHq*' ,ry 10 20 30 40
I Early fi= 42 Devonian 4 0 a ul - ;8 o-4 a rLo o E H8
=4 z0
8 Middle 4 Ordovician 0
Phacopida 12 n-192 8 4 Early 0 Ordovician 10 20 CEPHALON LENGTH (mm)
Fig. 6. Frequency distribution of cephalon length of Phacopida species in various intervals of their evolutionary history. Data (including stratigraphic range of the clade) from Harringlon et. al. (1959). Body size and diversitv: TRAMMER & KAIM
o/o1oo
90 80 70 60
50 Phacopida 40 w diversity 30 q,\ maximum t 20 bodysize 10 r =0.242 0
o .g o.g (5 o'S F:O lg'6 o.9 o >.9 o'F i-o (s< :.S 6> 'itr rrr .YO -! : oerd i >p a fit >o O o o o o o o
517 363 GEOLOGIC TIME (Ma)
Fig. 7. Changesof maximum body size compared to changesof diversity in Phacopida.Note the weak correlationbetween size and diversity (r = 0.242).DatafromFig. 6. Time scaleafter Harland et al. (1990). whole size class would vanish.But even if 35 speciesfrom the size class 24 mm becameextinct, this classwould continue. In accordancewith the laws of probability concerningcompound events, when extinction probability of each species is 4, then the probability of simultaneous extinction of two speciesis f , the probability of simultaneous extinction of three speciesequals q3 andthe probability of extinction of ruspecies equals { . The more species-richa groupof species,the smalleris the probabilityof its completeextinction. Sinceclasses of smallbody sizeare species-rich and classes oflarge body sizeare very poor in species,classes of large-bodiedspecies are characterizedby a higher total probability of extinction in comparisonwith the classesof small-bodiedspecies, despitethe fact that the loss of speciesdiversity is nonselectivewith respectto body size.During the waning of a clade,large species become extinct faster than small ones (Figs 2, 4), not becausethey are largerbut becausethey are lessnumerous (compare McKinney 1990b:p.107,frg.4.I0;Jablonsky1996:p.277). Attrrtionof acladeduring its decline will tend to causegradual extinction of size classesin order of their increasingspecies richness (that means classes including smallerand smaller species). In effect, the maximum body size decreasesalong with decreasingdiversity during the reductionphase of a clade'shistory Gigs 2*5). ACTA PALAEONTOLOGICAPOLONICA (42) (I)
a 2O0 TU Trilobita o t! 160 (L a I rzo E H80 z40
CEPHALON LENGTH (mm)
Fig. 8. Distribution of body size in speciesof Trilobita. Length of the cephalon is used as the size-index. The peak on the side of small sizes results from much higher speciationrate of the smaller species.Data measuredfrom Harrington et al. (1959).
We prefer this stochastichypothesis, instead of the deterministicone, as it is the simpleEnull hypothesis.Another reasonis that it seemsto be supportedby investiga- tions concerningcranidial shape outline which suggestnonselective species extinction during the decline of the Ptychopariinaand Asaphina (Foote 1993). The lack of evident dependenceof maximum body size on diversity in the reduction phase. - As mentioned earlier, the maximum body size and diversity are weakly correlatedin Phacopida,so it may happenthat althoughdiversity decreases, the maximumbody sizeincreases (Figs 6, 7). This stronglysuggests a taxonomicattrition that is selectivewith regard to body size during the reduction phase.In general, elevatedorigination and/or reduced extinction in regionsofmorphospace occupied by large-bodiedspecies had to exist, otherwiselarge-bodied species, which belong to species-poorsize classes,should have vanishedfirst when diversity decreased.In particular,the origin of new large-sizespecies in the faceof decreaseof total diversity resulting from the extinction of smaller speciesin the Late Ordovician (Fig. 6) has causedthe lack of evidentdependence of maximum body size on diversity.The data of Foote( 1993) alsosuggest selective species reduction during the declineof Phacopi- da and seemto supportour conclusionsin this respect. Test for presenceor absenceof selectivity.- As shown above,there are two basicpatterns of body size evolution during the reductionphase of a clade.The first one with a high correlationbetween the diversity andthe maximumbody size(Figs 3, 5), and the secondone that is characterizedbyweak correlationbetween these two variables(Fig. 7). The fust patternmay serveas an indicatorthat the cladedecline is nonselectivewith regard to body size, while the secondone may indicate a clade reductionselective with respectto size. Size-diversityrelationship ofrecenttaxa. - The analysisofthe interdependence of body size and diversity in the fossil recordmay add a historicalperspective to the studyof body sizepatterns of speciesdiversity of Recenttaxa. Size-frequency plot for speciesof a Recenttaxon may be comparedto a single film frame while time-series 10 Body size and diversity: TRAMMER & KAIM plots for fossil speciesto a whole film sequence.Rosenzweig (1995: pp.73-:77) presenteda sufilmary of knowledge of the body size patternsof the speciesdiversity amongRecent groups. Most extant organismsdisplay more numerousintermediate- size speciesthan either very large or very small ones.But thereare alsogroups where small body sizesare most common.Rosenzweig (1995) admits that ecologistscannot explainthe size-diversitypatterns of Recenttaxa. The messagefrom the fossil record (Stanley 1973 and this paper) is that the unimodal size-diversityrelationship may appear at the beginning of the diversification phase, and at the termination of the reductionphase of thehistory of a cladewhen diversity is relativelylow. The positively skewedbody size histogramsare typical of periods of the acme of a clade when diversity is relatively high. Thus the shapeof the body sizehistogram of a particular Recenttaxon may dependon the cunentphaseof the evolutionaryhistory of the clade.
Conclusions
1. Cope'srule did not concerna completeevolutionary history of a cladebut only its radiationphase. 2. Cope'srule may be redefinedas follows: 'During the radiationphase of a clade, the maximum body size is diversity dependentin such a way that increasein diversity is associatedwith increasingmaximum body size'. 3. The radiationphase is selectivewith respectto body sizebecause small-bodied specieshave much higher total speciationrates than large-bodied ones. 4. During the decline of a clade, the maximum body size is in principle diversity dependentsuch that decreasein diversity is associatedwith a decreasingmaximum body size. 5. There are two basic patterns of maximum body size modifications during the decline of a clade. The first pattern occurs when strong correlation between the diversity and the maximumbody size exists,and is indicativeof nonselectivespecies attrition. The secondpattern occurs when the correlation between diversity and the maximum body size is wealg and is indicative of the attrition which is selective in respectto size.
Acknowledgments
We thank an aronymous reviewer, M.J. Benton, M. Foote, K. Malkowski, and M.L. McKinney for help and discussion during the preparation ofthis paper.
References
Cope,E.D. 1887.The Origin of the Fittest, 1426. D. Appleton & Co, New York. Dial, K.P. & Marzluff, J. 1988.Are the smallestorganisms the most diyerse?- Ecology 69,1620-1624. Foote,M. 1988.Survivorship analysis of Cambrianand Ordoviciantrilobites. - Paleobiologyl4,258J1l. Foote, M. 1993. Discordanceand concordancebetween morphological and taxonomic diversity. - Paleobiolo gy 19, 185204. ACTA PALAEONTOLOGICAPOLONICA (42) (I) 11
- Gould, S.J.1988. Trends as changes in variance:a new slanton progressand directionality in evolution. Journal of Paleontology 62, 319-329. Harlan4 w.B., Armstrong,R.L., Cox, A.V., Craig, L.8., Smith, A.G., & Smith, D.G. 1990.A Geologic TimeScale 1989,1J63. CambridgeUniversity Press,New York. Harrington, H.J., Henningsmoen,G., Howell, B.F., Jaanusson,V', Lochman-Balk, Ch', Moore R'C', PoulsenCh., Rasetti,F., Richter, E., Richter, R., Schmidt, H., Sdzuy, K., Struve,W', Stormer,L'' Stubblefield,C.J., Tripp, R., Weller, I.M., and Whittington, H.B. 1959.Systematic descriptions. 1n.' R.C. Moore (ed.), Treatiseon Invertebra.tePaleontology, Part O, (Arthropoda 1),01704539. Geo- logical Society of America and University of KansasPress, Lawrance, Kansas. Jabloniky, D. 1996.Body size and macroevolution.1n.' D. Jablonsky,D.H. Erwin & J.H. Lipps (eds)' EvolutionaryPaleobiology, 256-289. University of ChicagoPress, Chicago' 'Eohippus'(Hyracotherium) MacFadden,B.J. 1986.Fossil horses from to Eqaus:scaling, Cope's Law, and the evolutionof body size.- Pal eobiolo gy 12, 355-3 69. May, R.M. 1988.How many speciesare there on Earth'!- Science24l'1441-1449' McKinney, M.L. 1990a.Classifying and analysingevolutionary trends. 12.' K.J. McNamara (ed.), EvoI- utionary Trends,23-58. The University of Arizona Press,Tucson. McKinney, M.L. 1990b.Trends in body-sizeevolution. In: K.I. McNamara (ed.),Evolutionary Trends, 75-l 18.The University ofArizona Press,Tucson. McShea,D.W.lgg4.Mechanismoflarge-scaleevolutionarytrends.-Evolution48,l747-1763. Miiller, A.H. 1961.Grossabliiufe der Stammesgeschichte,l-116.G' Fischer,Jena. Miiller, A.H. 1963.Lehrbuch der Paleiozoologie.Allgemeine Grundlagen,l-387. G. Fischer,Jena. Rosenzweig,M.L. 1995. SpeciesDiversity in Space and Time, 1-436. Cambridge University Press, Cambridge. Stanley,S.M. 1973.An explanationfor Cope's ntle. - Evolution21,l-26. Stanley,S.M.lg7g.Macroevolution: PattemandProcess,l-332. W.H. Freeman,San Francisco. Vrba,E.S. 1983. Macroevolutionarytrends: new perspectives ontheroles ofadaptationandincidentaleffect. - Science22l,387-389.
ZaleLnofi0 migdzy maksymalnq wielko6ciq ciala a zt62nicowaiieil gatunkowym na przykladzie trzech grup trylobit6w
JERZYTRAMMER i ANDRZEJKAIM
Streszczenie
Ewolucyjnahistoria wymarlej grupy skladasig przynajmniej z dwtfaz: zfazy radiacji - gdyliczba gatunk6wsig zwigksza , orazfazy redukcji- kiedy liczba gatunk6wspada. W fazieradiacji, zgodnie zrcgul4Cope'a,wrazzrosn4c4 liczb4 gatunk6w,roSnie tez na drodzekladogenerycznej dyfu4i (Gould 1988;McKinney 1990a,b) maksymalna wielko6i ciala w grupie (Fig. 2--7),tzn. pojawiaj4 sig gatunki obejmuj4ceosobniki o corazto wigkszych rozmiarachciala. Natomiast w fazie redukcji zachodziodwrotny proces:zmntejszq4cej sig liczbie gatunk6whowatzyszy r6wnoczesny spadek maksy- malnej wielkoSciciala w grupie @ig. 2a). Silna korelacja migdzy liczb4 gatunk6w a maksymalnawielkoscia ciala w czasie redukcji grupy (Fig. 3, 5) wskazuje, ze powoduj4ceredukcjg wymieranie odbywalo sig bez selekcjina wielko6i. Slabakore- lacjamigdzyobiema wielkoSciami (Fig. 7) jest natomiastprzejawem istnienia doboru zaleinegood rozmiar6wciala w czasiewymierania' 12 Body size and diversity: TRAMMER & KAIM
Appendix
Table 1 and Fig. 1. Expected and found speciesnumber of Early Ordovician Ptychopariina to test the null hypothesis.
size class (mm) o2 1/ +o G8 8-10 10,t2 I2-14 14-16 r6-18 Late Cambrian (found) 12 43 30 3I 24 l3 ll 1l 5 specles number Early found 5 i0 9 3 2 I 1 Ordovician expected zXl 813 naL 6!2 5jl2 2X1 zXl 2+l l+l
a1 a^ size class (mm) t8t0 24-26 2628 28-i0 30 32 J Z--)+ Late Cambrian (found) l 2 4 2 l 4 0 I specles number Early found 0 0 0 0 0 0 Ordovician expected l+t 0d-1 l+t 0t1 1+t 1+l 0 0
all I ().^ ul rv IL a o + tr8 l+ expected tu m >7 :) i z + o +rl+llllr I-LI -+TT rT r++
24 26 28 30 32 34 36 CEPHALON LENGTH
Expected species number for each size class was calculated according to the equation: Nz=Nr pt{ttr 'p.tt-p) where: Nz - expected Early Ordovician species number in a given size class, Nr - Late Cambrian speciesnumber found in a given size class, p - probability of speciessurvival (p = Ntz/Nrr), Ntr - total found Late Cambrian species number, Nz - total found Early Ordovician species number.