X. Modeling Stratophenetic Series

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1 GEOS 33001/EVOL October 2007 Page 1 of 12 X. Modeling Stratophenetic Series 1 Differences between evolutionary and stratophenetic series 1.1 Sequence in thickness vs. time 1.2 Completeness of sampling Problem with statistical power As directional pattern sampled with decreasing completeness, ability to reject symmetric random walk diminishes. 2 Principal approaches 2.1 Symmetric random walk as null hypothesis See topic 4. Random walk with p=q= E(S n ) + 2 4pqn 20 E(S n) + 4pqn S n 0 E(S n ) 20 E(S n ) 4pqn 40 E(S n ) 2 4pqn Time steps (n)

2 GEOS 33001/EVOL October 2007 Page 2 of Generalized random walk Maximum likelihood estimates of mean step size and variance in step size (Hunt 2006). 580 GENE HUNT FIGURE 1. Three example step distributions (top) used to generate corresponding evolutionary sequences of 100 steps (bottom). When the mean of the step distribution is zero, increases and decreases are equally likely and the overall dynamics are nondirectional (A, C). Step distribution B has a positive mean, and therefore will tend to produce positively trended evolutionary sequences. With increasing step variance, evolutionary sequences are more volatile, with larger positive and negative excursions (compare C with A). Generalized Random Walk (Hunt 2006) depends only on step (see below), it is a natural measure of directionality. Before proceeding, it is necessary to clarify some terminology. What I refer to here as a general random walk includes the whole class of models that are characterized by evolutionary transitions that (1) are independent from each other and (2) are homogenous over time; i.e., they are drawn from the same step distribution through the interval of interest. When unqualified in the paleontological literature, the term random walk generally implies an unbiased random walk, and some would restrict the term only to this subset of models. Throughout this paper, I use general random walk to refer to all models that meet the above two criteria and unbiased random walk to denote the special case of nondirectional random walks ( step 0). It is important to note that modeling phyletic evolution as a random walk does not imply any particular evolutionary process; many different microevolutionary scenarios produce evolutionary sequences that can be described as random walks (Hansen and Martins 1996; Roopnarine et al. 1999). Although phenotypic transitions are modeled as random draws from a step distribution, this should not be understood to imply that the evolutionary changes themselves are unrelat- 2.3 Log-rate vs. Log-interval comparisons (Gingerich 2001; Gingerich and Clyde 1994)

3 GEOS 33001/EVOL October 2007 Page 3 of 12 Rate (slope) versus interval length for a simple sequence. Morphology Time Ideal log(rate)-log(interval) comparisons A B Directional Morphology log Evolutionary rate slope=0 Time log Interval length C D Static Morphology log Evolutionary rate slope= 1 Time log Interval length E F Random Morphology log Evolutionary rate slope = 1 2 Time log Interval length

4 GEOS 33001/EVOL October 2007 Page 4 of Comparing within- and between-lineage changes(charlesworth 1984, Cheetham 1986) High ratio of between- to within-lineage variance as operational test for punctuation.

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7 GEOS 33001/EVOL October 2007 Page 7 of 12 B. 2.5 Inversion of stratophenetic series via model of stratigraphic variation (Hannisdal 2006, 2007) Assume, or empirically constrain, model of basin filling, depth, grain size Assume or empirically constrain habitat preferences (preferred depth and grain size, tolerance about preferred value) Forward model: Assume pattern of phenotypic evolution, predict stratophenetic series Inverse model: Observe stratophenetic series, including morphology and abundance, and infer evolutionary pattern H A N N I S D A L esses be exin modern ry phenom- 982; Schluare the preion (Erwin am 1999)? ons depend sure evolutterns from rphological ion are govecological, s. The relns among e scale and mic) of the e of organstanding of parameters nalysis and model of pic evolut-sediment, ed from de- The model vestigating sitional ar- It is shown stratopheple yet unortant immodes of e nature of Figure 1. Outline of model components and their input/output relations. The basin fill model (SedFlux) takes a series of input files (defining basin dimensions, sediment properties, and process parameters) and outputs various seafloor properties, including water depth d and sediment grain size g, as well as depositional characteristics (e.g., sedimentation rate q). These output variables are used to drive models of (1) abundance, predicting population size N (the sum of population sizes per spatial bin n) based on a species habitat preferences; (2) evolution, predicting population changes in phenotype f in response to selection and drift; and (3) preservation, predicting the number of preserved fossils K. substrate properties, and sedimentation rate along an onshore-offshore profile, according to userdefined sediment input, process parameters, and sea level change; a model of abundance predicting the distribution of individuals according to the species habitat preferences and peak abundance; a

8 ( [ ]) (d d) (g g) f(d, g) p exp. (1) 2 2 2ps s 2 s s d g d g GEOS 33001/EVOL October 2007 Page 8 of 12 The parameters d, g, s 2, and 2 d sg are thus used to control the environmental sensitivity and abundance distribution of a simulated benthic species. Figure 3A shows an example of a density defined by equation (1) for a particular set of parameter values. The number of individuals n in each spatial bin x is found by scaling the peak of the density function (fig. 3A) to the maximal per-bin [ ] x max f(d x, g x) N p n. (2) f(d, g) When applied to the basin fill history from SedFlux, the above model allows a simulated benthic species to track its preferred habitat in space and time. Although the underlying probability density is a symmetric Gaussian, the realized abundance distributions along the seafloor can look drastically different at different times as a result of variability in substrate properties and in the bathymetric profile in response to sea level changes (fig. 3B 3F). This matches the real- Figure 3. Model of habitat preference and abundance. A, Gaussian density representing the probability of occurrence of a benthic species, here controlled by two environmental variables: water depth ( d p 100 m, sd p 80 m) and grain size ( g p 300 mm, sg p 200 mm). The peak of the density function is scaled to the maximum per-bin abundance. B F, Realized abundance distributions across the basin profile at various times throughout a model run, using the basin fill history from figure 2 and the Gaussian model in A. Each bar represents the mean abundance of 200 spatial bins (10 km) along the seafloor. Because of variations in the bathymetric gradient and sediment grain size distribution in response to sea level changes, the abundance distributions look different at different times, ranging from relatively symmetric (B) to highly skewed (E), with a mode that is either variable in both magnitude and position or polymodal (C). Even for a species with broad depth tolerance, the actual area occupied at any given time can be a limited portion of the basin (D).

9 GEOS 33001/EVOL October 2007 Page 9 of B. H A N N I S D A L Figure 4. Model of phenotypic evolution based on the Price equation. Each panel shows the change over time (1000 steps) in the mean phenotypic value (solid line, starting at 0) and the phenotypic variance (dotted line, starting at 1) of the population. The top row of panels depicts different scenarios based on the values of the two regression parameters, from left to right: no selection ( bw,f p 0, bw, (f f) 2 p 0), directional selection ( bw, f 1 0, bw, (f f) 2 p 0), directional selection with stabilizing selection ( bw, f 1 0, b W, (f f) 2! 0), and directional selection with disruptive selection ( bw, f 1 0, bw, (f f) 1 0). The two middle rows are for the same scenarios but with decreasing population size, resulting in 2 stronger drift. The bottom row shows the effect of introducing a population bottleneck (reducing the population 10- fold over a period of time indicated by dashed lines), which can generate a rapid shift in the mean value and a spike in and/or depletion of variance. test operates on the sequence of positive and negative increments in a series, comparing the number of runs (consecutive steps with equal sign) in a series with that expected from a random walk. The scaled maximum test and the Hurst exponent comexpected from a random walk (see The Runs Test, The Scaled Maximum Test, and The Hurst Exponent in the appendix for details). The tests are applied to the stratophenetic series from each location as well as to the microevolutionary

10 GEOS 33001/EVOL October 2007 Page 10 of 12 Figure 6. Numerical experiment showing the effect of depositional architecture. A, Phenotypic evolution as random drift, based on parameters bw, f p 0, bw, (f!f)2 p 0, and N from C. Solid line is for phenotypic mean; dotted line is for variance. B, Gaussian density based on preferred depth p 120 m, depth tolerance p 100 m, preferred grain size p 400 mm, and grain size tolerance p 100 mm. C, Population size through time based on the model in B. Solid line is for total population size; stippled line is for effective population size, which in this case is held constant and small. D, Poisson density function used to determine preservation based on l as a function of the abundance in each model bin. E, Stratophenetic series plotted against synthetic basin fill from figure 2. Sample means (red) and error bars (cyan) are scaled to make the stratophenetic patterns easily visible and have to be projected back onto the blue

11 GEOS 33001/EVOL October 2007 Page 11 of 12 INFERRING EVOLUTION BY INVERSION 109 FIGURE 6. 2-D marginal posterior distributions. Axes correspond to parameter ranges, except for pres (F, ordinate), where the parameter range is truncated as in Figure 5. Solid lines represent 0.1 (inner), 0.5, and 0.95 (outer) confidence contours. Shading is scaled to the peak of each plot to enhance shapes, so shading levels are not comparable between plots. True model indicated by. preserved individuals in a particular sample, an increase in the peak abundance parameter will require a corresponding decrease in the preservation probability parameter, and we would therefore expect the parameters Nmax and prob to be negatively correlated in the posterior. This relationship is also suggested by the slope of their ensemble distribution (Fig. 4E). Note that these are not empirical correlations and do not suggest that more abundant organisms have lower per-capita presmodel correlation matrix, calculated by standardizing each element by the posterior standard errors (Fig. 7). This matrix shows the expected negative posterior correlation between maximum abundance and preservation probability. As mentioned above, the product of these two parameters is treated as a single preservation parameter (pres), and this transformed parameter is consequently positively correlated with the original parameters. Another distinct feature is the positive correla-

12 of the four principal component axes. The solresult of variable unce prior information and through the stratigra because the evolution m homogeneous and con the ensemble of pred properly reflect the dence. Future work w ed inversion approac time-varying posterior GEOS 33001/EVOL October 2007 Page 12 of 12 INFERRING EVOLUTION BY INVERSION FIGURE 9. Ensemble-based inferred pattern of phenotypic evolution. Plots A D correspond to the temporal pattern of population mean shape change along the first four PC axes (cf. Fig. 3). Solid line represents the true pattern (Fig. 1H), dotted line represents the ensemble mean trajectory, and dashed lines are an approximate error envelope representing uncertainty in both age (abscissa) and phenotype (ordinate) calculated as two times the standard error in the ensemble of predicted trajectories. Discu Despite sparse foss little information bey relevant temporal pat albeit with considerab that we can resolve d rameters from the ser is encouraging, and u tion between tempora terns. From the marg tions and the resolutio although the selection are less well resolved parameters, the tend ability implies an evo sistent directionality. does not mean constan represents a statistical the scale of millions o ference is all the mor the stratophenetic seri inated by large fluctu