GEOS 36501/EVOL 33001 10 February 2012 Page 1 of 26XI. Sampling Models, 3: Introduction totime-varying models (forward problems)1 Basic framework(see Foote 2000, Paleobiology Supplement to 26(4):74-102, Foote 2001, Paleobiology 27:796[erratum], and Foote 2003, Journal of Geology 111:125-148, 752-753 [erratum])1.1 Let there be n time int ervals, each characterized by a set oforigination, extinction, and sampling rates: pi, qi, and ri.1.2 Use time series of p and q to predict true time series of NbL,NF t, NF L, and Nbt(and thus Nband Nt).1.2.1 These quantities all scale to Nbi.1.2.2 Nbfor continuous-turnover mo del (q.v. ) in which origination andextinction occur at constant per-capita rate within an interval:1. Let the age of the bottom boundary be at time t = x.GEOS 36501/EVOL 33001 10 February 2012 Page 2 of 262. Let N0= 1 at t = 0.3. Let ptand qtbe time-specific rates.4. ThenNx= expZx0(pt− qt) dt .5. Or, if we divide time into intervals and assume constant p and q within an interval(while still varying among intervals):Nbi= expi−1Xj=1(pj− qj) ,if the rates piand qiare expressed per lineage per interval. If instead they areexpressed per lineage-million-years, and interval durations are given by ∆ti, then wehave:Nbi= expi−1Xj=1(pj− qj)∆tj .1.2.3 Nbfor pulsed-turnover model (q.v.) in which originations are all at startof interval and extinctions at end of interval (so all lineages extendthroughout the interval):1. In this model, P is the number of new lineages produced per lineage extant at thestart of the interval. (So number of originations is equal to Nb· P and total intervaldiversity is Nb[1 + P ].)2. In this model, Q is the extinction probability. (So number of extinctions is equal toNb[1 + P ]Q).3. ThusNbi=i−1Yj=1(1 + Pj)(1 − Qj).1.3 Use time series of p, q, and r to determine samplingprobabilities for given time intervals.Probability of being sampled in a given interval of time (either before a reference point,after a reference p oint, or between two reference points) is obtained as the integral, over allpossible durations, of the probability of having a certain duration multiplied by theprobability of being sampled given that duration.GEOS 36501/EVOL 33001 10 February 2012 Page 3 of 26Journal of Geology E r ratum 753Table 1. Expressions Used to Calculate Survivorship ProbabilitiesQuantity/model Expression:Nbt•C⫺qNeb•P N (1 ⫺ q)b:NbL•C⫺qN (1 ⫺ e )b•P Nqb:NFtCCp⫺q ⫺pNe (1 ⫺ e )bPP Np(1 ⫺ q)bCPp ⫺pNe(1 ⫺ q)(1 ⫺ e )bPC⫺qNpeb:NFLCCaif ,⫺qN (e ⫹ p ⫺ 1) p p qbif(p⫺q) ⫺qqe ⫹ (p ⫺ q)e ⫺ pNp( qbp ⫺ qPP NpqbCPpNq(e ⫺ 1)bPC⫺qNp(1 ⫺ e )bPA(i):•Cnk⫺1 k⫺1 nn⫺S q ⫺q ⫺S r ⫺S q ⫺S rmk m k kmpi⫹1 mpi⫹1kpi⫹1 kpi⫹1冘 e (1 ⫺ e )1⫺ e [1 ⫺ P (k)] ⫹ e 1 ⫺ e [1 ⫺ P (n)][( ) ]{( ) }( ){( ) }()DFbL Akpi⫹1•Pnk⫺1 nk⫺1 n⫺S r ⫺S rm kmpi⫹1kpi⫹1冘写1 ⫺ q (q )1⫺ e [1 ⫺ P (k)] ⫹ 写 (1 ⫺ q )1⫺ e [1 ⫺ P (n)]{( ) } {( ) }()mk DFbL k A()[]kpi⫹1 mpi⫹1 kpi⫹1PB(i):C•i⫺1i⫺1 i⫺1 i⫺1 i⫺1⫺S p ⫺p ⫺ S r ⫺S p ⫺ S rmk m k kmpk⫹1 mpk⫹1kp1 kp1冘 e (1 ⫺ e )1⫺ e [1 ⫺ P (k)] ⫹ e 1 ⫺ e [1 ⫺ P (1)][( ) ]{ ( ) } ( ){ ( ) }()DFFt Bkp1P•i⫺1 i⫺1 i⫺1i⫺1 i⫺11 p 1k⫺S r ⫺S rm kmpk⫹1kp1冘写 1 ⫺ e [1 ⫺ P (k)] ⫹ 写 1 ⫺ e [1 ⫺ P (1)]{( ) } {( ) }DFFt B()() ()()kp1 mpk⫹1 kp11 ⫹ p 1 ⫹ p 1 ⫹ pmk k:PDFbt••⫺r1 ⫺ e:PDFbL•Cb ⫺(q⫹r) ⫺q[r ⫹ qe ]/(q ⫹ r) ⫺ e⫺q1 ⫺ e•P⫺r1 ⫺ e:PDFFtC•c ⫺(p⫹r) ⫺p[r ⫹ pe ]/(p ⫹ r) ⫺ e⫺p1 ⫺ eP•⫺r1 ⫺ e:PDFFLCCdif ,⫺p ⫺(p⫹r)Np r 1 ⫺ ep[1 ⫺ e ]b⫺⫺ p p q2{}Np⫹ rp (p ⫹ r)FLif(p⫺q) ⫺(q⫹r)(p⫹r)Npr[e ⫺ 1] pqe [e ⫺ 1]b⫺qp⫹⫺e (e ⫺ 1) p ( q{}N (q ⫹ r)(p ⫺ q)(p ⫹ r)(q ⫹ r)FLPP⫺r1 ⫺ eCPe p ⫺rr(e ⫺ 1) ⫹ p(e ⫺ 1)pp(r ⫹ p)(e ⫺ 1)PCf q ⫺rr(e ⫺ 1) ⫹ q(e ⫺ 1)qq(r ⫹ q)(e ⫺ 1)Note. In the two-character code for model, the first character denotes origination and the second extinction; ,C p continuous. A bullet means the expression applies to either model for the corresponding process. Nbis the true standing diversityP p pulsedat the start of the interval; because all relevant numbers scale to Nb, this can be arbitrarily set to unity.aFoote 2000a, eqq. (6b) and (6c).bFoote 2000a, eq. (27b).cFoote 2000a, eq. (28b).dFoote 2000a, eqq. (29b) and (29c).eLet z represent time within an interval of duration t, where and are the beginning and end of the interval, respectively.z p 0 z p tBy assumption, there is no extinction until the end of the interval. Thus, the density of origination at time z is equal topz pte /( e ⫺(cf. Foote 2001a, eq. [3]). Because all lineages originating within the interval extend to the end, the probability of preservation,1)given origin at z and extinction at t, is equal to . It is necessary to integrate the density of origination times the probability⫺r(t⫺z)1 ⫺ eof preservation over all values of z. Thus, , which is equal to the expression in the table once ttpt pz ⫺r(t⫺z)P p [1/(e ⫺ 1)] e [1 ⫺ e ]dz∫0DFFLis set to unity.fDerived as in the foregoing footnote, with origination and extinction reversed.GEOS 36501/EVOL 33001 10 February 2012 Page 4 of 261.3.1 Use time series of p and r to predict probability PBithat a taxon extantat start of an inte rval i will be sampled sometime before the interval.1.3.2 Use time series of q and r to predict probability PAithat a taxon extantat start of an inte rval i will be sampled sometime before the interval.1.3.3 Use values of pi, qi, and rifor a given interval to predict the probabilitythat a taxon will be sampled during an interval.This depends on whether it is, in reality (i.e. prior to sampling), a member of thecategories NbL, NF t, NF L, or Nbt. The corresponding probabilities are denoted PD|bL, PD|F t,PD|F L, and PD|bt, where the subscript i has been omitted for clarity.1.4 Let XbL, XF t, XF L, and Xbtbe the observed numbers of taxa ineach of the four categories.Note that a taxon observed to be in the bt-category must have been so in reality, but, forexample, and F L-taxon could in reality have been in any of the four categories prior tosampling.1.5 Determine the expected numbers of observed taxa X:XbL=NbLPBPD|bL+ NbtPBPD|bt(1 − PA),XF t=NF tPAPD|F t+ Nbt(1 − PB)PD|btPA,XF L=NF LPD|F L+ NbL(1 − PB)PD|bL+ NF tPD|F t(1 − PA)+ Nbt(1 − PB)PD|bt(1 − PA),andXbt=NbtPBPAand of course we have a