GEOS 36501/EVOL 33001 25 January 2012 Page 1 of 23VI. Introduction to Branching Models1 Overview1.1 Homogeneous vs. heterogeneous models1.1.1 Homogeneous models: all taxa governed by same rates at all times.1.1.2 Simplicity-reality trade-off1.1.3 Alternative risk models• Temporal variance• Variance among taxa• Dependence on taxon ageGEOS 36501/EVOL 33001 25 January 2012 Page 2 of 231.2 Hierarchical vs. non-hierarchical models1.2.1 Several models are hierarchi cal in the sense that they track theaggregate behavior of lower-level entities (species) within higher-levelentities (clades).1.2.2 Others more explicitly hierarchical in the sense that they include anadditional process that gives rise to new clades.1.2.3 Because we will allow the origin of new clades from existing clades, theexisting clades are rendered paraphyletic—what Raup (1985,Paleobiology 11:42-5 2) referred to as paraclades.GEOS 36501/EVOL 33001 25 January 2012 Page 3 of 231.3 Pure birth vs. birth-death models1.3.1 Pure birth unrealistic, but useful for training intuition1.3.2 For some problems (e.g. dealing with extant clades lacking fossil record)net diversification rate is of greatest interest, and pure birth model is auseful first pass.1.4 Forward vs. inverse problems1.4.1 Forward: predictio n of system behavior, given model assumptions1.4.2 Inver se: estimation of model parameters1.5 Analytical approaches vs. simulation1.5.1 Analytical approach useful when feasible1.5.2 Simulation often necessary when model not analytically tractable2 Pure-birth (Yule) model(Yule, 1924, Phil. Trans. R. Soc. Lond. B 213:21-87.2.1 Historic background2.2 Model structure2.2.1 Discrete time steps2.2.2 p=Probability of branching per unit time; let p0 = (1 − p)2.2.3 Ntdenotes the number of species in the clade at time t; N0= 1 at t = 0.2.3 Model outcome2.3.1 After one time unit:• Pr{N1= 1} = p0• Pr{N1= 2} = p2.3.2 After two time units:• Pr{N2= 1} = p02GEOS 36501/EVOL 33001 25 January 2012 Page 4 of 23• There are two ways to get to N2= 2:P r(N2= 2) = pp02(if N1= 2)+ p0p (if N1= 1)= pp0(p0 + 1)• Pr{N2= 3} = 2p2p0 (if N1= 2)• Pr{N2= 4} = pp2= p3(if N1= 2)2.3.3 We can procee d like this, but it g et s tedious. To simplify, switch tocontinuous time• Let p be the instantaneous rate per lineage-million-years. If one million years issubdivided into n fine increments of 1/n m.y. each, then the probability of multipleevents in a fine increment of time will be negligible. Thus, the probability of noevents in 1 m.y. will be equal to (1 − p/n)n. As n → ∞, this converges to e−p.• In general, the probability that a process acting at a rate p over a time t will yieldzero events is equal to e−pt(Poisson distribution).• Thus, for the Yule proce ss, the proportion of monotypic taxa after time t is expectedto be equal to e−pt.• Applying the same approach to N = 2, 3, ... etc. (and making use of differentialequations), we end up with the probability of having a standing diversity of exactly nspecies after time t (where n ≥ 0, since there is no extinction):Pn,t= e−pt(1 − e−pt)(n−1)• This is one of the classic Hollow Curves.• Note that, for any p and t, the maximal probability is that the clade will bemonotypic. The probability drops off as n increases, and the curve becomes flatter ast increases.• Expectation: E(n, t) = Pn,t= ept, which is intuitively reasonable.• Variance: V (n, t) = ept(ept− 1).• Note that the variance grows exponentially with time. This implies that an enormousrange of seemingly different diversity trajectories may be probabilistically consistentwith a single underlying growth parameter. This same principle holds for birth-deathmodels.GEOS 36501/EVOL 33001 25 January 2012 Page 5 of 23! " # $ %&&'& &'! &'" &'# &'$Yule model (p=0.1)()*+,-./0.12,34,1.45.6,5)17-/+8+494:;:<%:<!:<=:<":<>:<#! " # $ %&%,!&? %,!&@ %,!&> %,!&= %,!&%Yule model (p=0.1)()*+,-./0.12,34,1.45.6,5)17-/+8+494:;.A9/6.1389,B:<%:<!:<=:<":<>:<#GEOS 36501/EVOL 33001 25 January 2012 Page 6 of 23• Note also that the variance increases with the branching rate. Again, this is also truefor birth-death models. Clades with higher origination and extinction rates areexpected to have greater volatility in diversity (and empirically this has beenverified—Gilinsky 1994, Paleobiology 20:445-458). Such clades are also more likely tocrash to total extinction just by chance (ammonoids as a classic example).• Because variance (volatility) increases with the rate of the process, it is also expectedto be higher at lower taxonomic levels.From Gilinsky (1994), who measures volatility as average absoluteproportional change in diversity per m.y. (see his eqn. 1).GEOS 36501/EVOL 33001 25 January 2012 Page 7 of 233 Birth-death model3.1 Model structure3.1.1 p is the instantaneous branching rate per lineage-million-years (Lmy)3.1.2 q is the instantaneous extinction rate per Lmy0246810Number of lineage−million−years (Lmy) within a time intervalTime (m.y.)2462352 15523213.1.3 See Appendix from Raup (1985) (reproduced below) for some of themost important predictio ns of this model.(Note that Raup uses λ and µ where we use p and q.)3.1.4 Of most use to us will be equations for lineage survival (A1-A4),survival of a paraclade (A11-A14), paraclade diversity (A15-A18), andtotal pro ge ny (A28-A30).GEOS 36501/EVOL 33001 25 January 2012 Page 8 of 23GEOS 36501/EVOL 33001 25 January 2012 Page 9 of 23GEOS 36501/EVOL 33001 25 January 2012 Page 10 of 234 Discrete-time (“MBL”) simulation of clade histories4.1 Historically extremely i mportant in consciousness-raisingabout role of stochastic processes in paleontological data.4.2 Because variance depends on magnit ude of rates, it isessential that simulations be empirically scaled (see Stanleyet al. (1 98 1) Paleobiology 7:11 5) .4.3 With discre te-t im e modeling, time steps should be shortenough that one can neglect the probability o f multipleevents resulting from the corresponding continuous-timeprocess within a time step.4.3.1 How would you determine whether, for a given p and q, the time stepsare short enough to yield reasonable approximation to continuousprocess?5 Inverse Methods, 1: Long-term average rates ofspecies within a clade5.1 Analysis of “exact” durations, not binned into age classes5.1.1 Assume exponential survivorship (consequence of time-homogeneousextinction)5.1.2 Assumption ofte n not critical: If q varies over time and q(T ) denotes thetime-specific rate, then