# Paleobiology

- The Paleontological Society

## Abstract

Despite substantial advances in plate tectonic modeling in the last three decades, the postulated position of terranes in the Paleozoic has seldom been validated by faunal data. Fewer studies still have attempted a quantitative approach to distance based on explicit data sets. As a test case, we examine the position of Avalonia in the Ordovician (Arenig, Llanvirn, early Caradoc, and Ashgill) to mid-Silurian (Wenlock) with respect to Laurentia, Baltica, and West Gondwana. Using synoptic lists of 623 trilobite genera and 622 brachiopod genera for these four plates, summarized as Venn diagrams, we have devised proportional indices of mean endemism (ME, normalized by individual plate faunas to eliminate area biogeographic effects) and complementarity (C) for objective paleobiogeographic comparisons. These can discriminate the relative position of Avalonia by assessing the optimal arrangement of inter-centroid distances (measured as great circles) between relevant pairs of continental masses. The proportional indices are used to estimate the “goodness-of-fit” of the faunal data to two widely used dynamic plate tectonic models for these time slices, those of Smith and Rush (1998) and Ross and Scotese (1997). Our faunal data are more consistent with the latter model, which we use to suggest relationships between faunal indices for the five time slices and new rescaled inter-centroid distances between all six plate pairs. We have examined linear and exponential models in relation to continental separation for these indices. For our generic data, the linear model fits distinctly better overall. The fits of indices generated by using independent trilobite and brachiopod lists are mostly similar to each other at each time slice and for a given plate, reflecting a common biogeographic signal; however, the indices vary across the time slices. Combining groups into the same matrix in a “total evidence” analysis performs better still as a measure of distance for mean endemism in the “Scotese” plate model. Four-plate mean endemism performs much better than complementarity as an indicator of pairwise distance for either plate model in the test case.

## INTRODUCTION

Faunal data have generally been presented only in a qualitative or semiquantitative way for formulating plate tectonic reconstructions. Faunal provinciality has long been considered to indicate separate biogeographic provinces, yet quantitative scaling of faunal difference in relation to distance is rarely attempted. The advent of continental drift models using the principles of plate tectonics has given fresh impetus to, and provided new hypotheses for, the evaluation of biogeographic signals. Paleomagnetic data have revolutionized our knowledge of the positions of crustal plates in the post-Triassic (e.g., Smith et al. 1973; Torsvik et al. 1991; Trench et al. 1992). However, because the entire ocean floor predating ca. 170 Ma has been subducted or obducted, continental paleomagnetic evidence is often still contradictory for parts of the lower Paleozoic. Paleomagnetic studies do not constrain the relative longitudinal position of plates, and the longitudinal representation of plate positions has been accordingly arbitrary (Mound and Mitrovica 1998). At best, relative longitude is known to ±30° and latitude to ±15° (Scotese and McKerrow 1990). Extensive potential for the use of fossil data in the early Paleozoic thus still exists, not only to discriminate between competing plate models, but also to suggest new relative positions or orientations of plates. We here exploit the abundance of faunal data from radiations of benthic brachiopods and trilobites to devise a more objective method for inferring paleogeography. These data enable us to judge between two published reconstructions that have been incorporated dynamically into computer programs (Fig. 1).

Very few analyses in the modern biogeographic literature examine the mathematical relationship between the distance that separates particular areas and their faunas. One reason may be that, given exact and essentially static geography in the Recent, there is apparently nothing useful to elucidate. This lack of studies is unfortunate for paleogeography, where the response of measures of faunal endemism and similarity (or their counterparts, faunal sharing and complementarity) to distance is of great potential value. Two recent papers are relevant to the problem. Picoli et al. (1991) examined a range of mathematical models to describe the relationship between interbasinal distance and faunal similarity for Cenozoic mollusks with a planktic larval stage, and they found that an exponential model best described the relationship. Similarly, Nekola and White (1999) examined the “distance decay” between floras of modern trees with differing dispersal mechanisms. The exponential relationship they found had different slopes for guilds with different dispersal strategies.

Although there have apparently been no large-scale studies of Paleozoic biogeography along the lines of these two papers, faunal provincialism for the Ordovician, in particular, has been known and discussed by many authors since the time of Nicholson and Etheridge (1878). Spjeldnaes (1961) qualitatively studied the provinciality of a wide range of groups and attributed distributions to the influence of paleoclimates. Whittington (1966) focused on trilobite families, finding high levels of endemism (even at that taxonomic level) that diminished toward the end of the Ordovician. Williams's (1969) study on coefficients of association between brachiopod faunas and Whittington and Hughes's (1972, 1973) studies on trilobite provinciality were landmarks in quantitative analysis for the Ordovician. Whittington and Hughes (1972) assessed dissimilarities between faunas at generic and family levels, applying Simpson's Index using nonmetric multidimensional scaling. Burrett (1973) used a wider range of organisms and plotted an index of provinciality. Campbell and Valentine (1977) were the first to ask whether modern marine provinces are distinctive at the higher taxonomic resolution characteristic of many fossil data. Valentine et al. (1978) expanded on this work, using a stochastic computer model to evaluate empirical data through the Phanerozoic. However, metric indices of distances that scale as proportions have seldom been used for such studies.

More recent work has expanded the importance of faunal data for the Paleozoic and suggested criteria on which they should depend, stressing the importance of evaluating comparable biofacies (an issue that we do not directly examine here). Cocks and Fortey (1982) qualitatively examined faunal evidence supporting oceanic separations for paleocontinents now comprising the North Atlantic area (Fig. 1), and Cocks and Fortey (1988) described the distribution of peri-Gondwanan marginal faunas. Fortey and Cocks (1992) explicitly examined and compared two contrasting paleogeographic models by using parsimony analysis of endemicity (PAE) to assess quantitatively which model was better supported by the faunal data. Some other studies have integrated faunal and biogeographic evidence in a semiquantitative fashion, notably Harper et al. 1996 in an assessment of the terranes around Iapetus.

As an attempt to resolve the paleogeography of the areas surrounding the present-day North Atlantic, we use faunal data to discriminate between competing plate tectonic models. We investigate indices derived from comprehensive faunal lists of brachiopods and trilobites. In particular, we examine the potential of such biogeographic indices to generate new sets of interplate distances based on specific tectonic models. We consider each plate as represented today by its centroid, the point that approximates the center of mass of a uniformly thin plate projected perpendicularly to the earth's surface. We reconstruct back in time (retro-project) these same points within differing tectonic models. We look at the fit of two of the simplest biogeographic indices that scale positively and metrically in proportion to distance, mean endemism (ME) and complementarity (C), with matrices of inter-centroid distances based on existing plate models. Once relationships to distance at a particular time are found, it is possible to place plates in their optimal positions for metric consistency with a given data set, although we make no attempt to weight plates according to certainty of paleolatitude. Here we examine what relative position for Avalonia—and thus rescaled distance—best reflects the unweighted biogeographic signals from the faunal data for a given time slice.

A number of complicating parameters are not considered here, because our objective is to examine the potential of a simple metric methodology on a large data set for particularly well sampled plates. We do not take into account the effect of ocean currents in relation to a given paleogeography (for a model see Christiansen and Stouge 1999). We also ignore the latitudinal span of a given great-circle distance at a given time—the greater the latitudinal span for a given distance, the greater should be the range of climatic contours traversed and therefore the greater the “effective” biogeographic distance. This may affect the potential of faunal data to discriminate longitude as well as they do latitude, although we are dealing with vast distances. We also ignore the strong possibility of a latitudinal gradient in generic diversity in our faunal data, although such effects should be controlled by our normalization of the data. In this paper, we emphasize the relative magnitude of interplate distances rather than the actual positioning of a network of plates on a global grid.

We take as given that, for a given plate model and pairwise plate matrix, the spatial extent of plates in question (expressed here as the sum of present day inter-centroid distances) remains the same. The position of a given plate clearly depends on its position in a preceding time slice (temporal autocorrelation), and this imposes dynamic constraints on rescaling of distances. We also assume that shallow- to deep-water biofacies, preserved across a number of transgressive and regressive events, have been adequately sampled for the four major plates in consideration. In spite of our oversimplification of the potential influences on biotic distributions, the fact that the results from our objective methods agree well with other paleocontinental distribution models derived from geologically independent evidence implies that our simplifying assumptions are reasonable first approximations.

We acknowledge that the spatial entities used in the analysis are of different size; different plate areas and asymmetric shapes are not ideal scenarios for a test case and may have some effect on our results. However, we have controlled for the most important potential effect, an area-biogeographic one. The number of endemics for a plate should, and indeed does, depend on the size of the faunal list, which itself should depend on area, but we have normalized this number by faunal totals. The position of the centroid is constrained by the size of the plate. The area and asymmetry of a plate also constrain the range of feasible positions for that plate. We have ignored these parameters in order to simplify the analysis, but they are incorporated implicitly in plate tectonic modeling programs.

We consider briefly the complication that the levels of cosmopolitanism may change with time in a manner that is not directly related to the absolute distances between plates. For example, a proportion of the shared fauna could accumulate over a long period of time merely by chance, and geographic range size for long-lived taxa should expand with time (“age and area” of Willis 1922), if only because range size must start very small (see Miller 1997). Range size during the Ordovician has also been shown to increase across the onshore-offshore framework (Miller 1997), and this will increase the probability of sampling a particular taxon in later time slices in cases where availability of suitable biofacies is limiting. If the range-size frequency distributions of particular indicator taxa change over time, so will the relationship between biogeographic indices and distance. Thus, the parameters for a relationship existing at one time slice are not a priori applicable to another time or taxonomic group. More specifically, during the end-Ordovician glaciation event, cosmopolitanism was at a selective premium for climatic reasons alone. For example, relatively few genera made up the widespread *Hirnantia* brachiopod fauna broadly associated with glacial events, in contrast to the different and more diverse mid-Ashgill faunas that preceded it (Sheehan and Coorough 1990). For different faunal groups sharing a common geography and environment (here, marine and continental shelf), we would expect a broad similarity in their biogeographic signal. However, differences in ecology and dispersal mechanism between groups, and evolutionary history with respect to terranes, should also affect potential to indicate relative distance. We examine the issue of differential performance at the generic level for both trilobites and brachiopods; these are some of the best-represented groups in the Paleozoic fossil record.

The approach we take is to analyze brachiopods and trilobites separately and together (independent data sets versus “total evidence”). If there is a relationship between a biogeographic index and a particular plate model, we should expect additional data to strengthen that relationship if the model really does approach the “real” distances. We examine the differing potential and robustness of two different biogeographic indices across five Ordovician–Silurian time slices for these two major taxonomic groups.

Our method uses faunal lists as the basis for reconstructing paleogeography. Another method that may claim to be objective has recently been developed (e.g., Ebach and Edgecombe 2001), one that uses cladistic analyses of critical groups and relates these to paleogeography by way of area cladograms. This method requires intimate knowledge of the species-level phylogenetics of component groups before paleogeography can be deduced. For the very large number of taxa used in our paper such an approach is not yet feasible. However cladistic area methods should provide a complementary and independent test of the kinds of summary methods described in this paper, and, we believe, vice versa.

## METHODS

#### Faunal Data and Time Slices

We extracted faunal data at the generic level from a newly compiled global-scale database (deposited in the Natural History Museum archive and in that for IGCP Project 410 “The great Ordovician biodiversification event”), originally derived from a large list of references selected by L.R.M.C. and R.A.F. We also drew extensively from databases developed by Rex Doescher (for brachiopods), Jonathan Adrain (for trilobites; see Adrain et al. 1998) and Arnold Miller (for both groups). The individual records are generic occurrences of brachiopods and trilobites recorded for named localities and stratigraphic horizons. Before input, the taxonomy was standardized for monophyletic genera by J. M. Adrain and R.A.F. (trilobites) and L.R.M.C. (brachiopods), including updates from the revised Brachiopoda volumes of the *Treatise on Invertebrate Paleontology* (Williams et al. 2000). Our database contains over 30,000 records of brachiopods and trilobites for five divisions of the Paleozoic of no greater than 15 Myr. These time slices are lower and upper Arenig, lower and upper Llanvirn, lower Caradoc, and Ashgill (not including Hirnantian), representing the Ordovician; and Wenlock, representing the Silurian, by which time the oceans between three of the four plates are considered in several models to have largely closed. Using these data, we have compared two different models of reconstruction. These models, both computer programs, are those of Ross and Scotese (1997) (hereafter, referred to as “Scotese”), and Smith and Rush (1998) (hereafter, referred to as “Smith”). These two programs make an interesting and worthwhile comparison, because that of Smith, AtlasWinPro, is based on paleomagnetic data, whereas that of Scotese, PaleoGIS for ArcView, is based on a wide but undeclared range of paleomagnetic, geological, chemical and biogeographic data. The latter model is an update from Scotese and McKerrow (1990). Crucially, both programs can automatically recover the paleocoordinates of retro-projected points (e.g., Fig. 2) that have first been attached to specific fragments, although that for Smith is cursor based and requires manual input of each centroid.

The two models are similar in their distance relationships between the six different plate pairs across the five major time slices, three of which are shown in Figure 1 (see also Fig. 6A). The greatest differences are in the position of the microcontinent Avalonia, which in the Scotese model is much more distant from the centroid of West Gondwana from early Caradoc times and even more so up to the Wenlock, and considerably nearer Laurentia by the Ashgill. The distance between Baltica and West Gondwana in the Scotese model in the Wenlock is also considerably greater than in the Smith model (Fig. 1, “BW” in Fig. 6A). As the position of Avalonia is the one most in dispute between the models, we focus on this fragment for our analysis.

The Smith and Scotese models used different time slices. The Scotese scale, using the DNAG timescale option, is here calibrated as (base to top; representative radiometric date in Ma, here not necessarily the most recent): L–U Arenig (488–478; 483); L–U Llanvirn (478–458; 473), L Caradoc (458–451; 454.5); Ashgill (448–438; 443); and Wenlock (428–421; 424.5). The Smith scale is Arenig (485–470; 473.75); L–U Llanvirn (470–458; 463), L Caradoc (458–451.8; 454.5); Ashgill (449–443.7; 443.5); and Wenlock (428–423; 425.5). So as to align the scales we used and to minimize the between-scale dating error of ±1–3 Ma, we picked the appropriate mean time for reconstruction for the respective scale as representing the likely mean for the faunal records for each time slice. Any records that were imprecise to a magnitude of more than 15 Ma were excluded from the data set. This value was chosen because the maximum series time slice we used was 15 Ma (the Arenig). The synoptic list among such records for each of these five time series (each of which was taxonomically checked by us for synonymies and other errors) comprised 623 trilobite genera and 622 brachiopod genera. Thus 1245 genera were used for the total evidence analysis (an estimated 80% of the total genera known from these four plates over this time). A subsidiary list was extracted for each plate at each time slice, and the number of genera endemic at each time to one, two, three, or all four plates determined. Figures 3 and 4 show all these endemism figures for four-plate Venn diagrams for the five time slices.

For each trilobite or brachiopod faunal list for every time slice, we calculated two indices—complementarity (C) and mean endemism (ME)—for each of the six plate pairs; both indices scale as proportions in a positive fashion with distance. Complementarity, also known as Marczewski-Steinhaus distance, has a considerable history of use in the ecological literature as a measure of faunal turnover or beta diversity (Colwell and Coddington 1994). It is a true metric that is equivalent to 1 minus the Jaccard index of similarity (equally, it is also a measure of dissimilarity or uncommonality between faunas): where *t*_{1}, *t*_{2} are the total richnesses of the fauna for plates 1 and 2 respectively, *S* is the richness of the mutually shared fauna, and *T* is the total richness of the synoptic fauna (*t*_{1} + *t*_{2} …), for plates 1 and 2 (i.e., _{1,2}). When all genera are shared, *S* = *T* and the index equals 0, whereas when none are shared, *S* = 0 and the index = 1. In this study, values ranged between 0.45 to 0.95.

Here we introduce mean endemism (ME) as a measure of the proportion of endemics in each fauna, designed to capture the strongest signature of endemism (unique endemics for all four plates considered, i.e., ME_{4}), averaged across each plate pair: (2) where *e*_{1}, *e*_{2} are the taxa uniquely endemic (for all the plates under consideration, here 4), to plates 1 and 2 respectively, and where *t*_{1}, *t*_{2} are the total richnesses of faunas on plates 1 and 2 respectively. Endemism is thus normalized to eliminate potential area-biogeographic effects, and averaged to create a measure for each plate pair. Again, this index scales from 0, where there are no unique endemics and the faunas would be expected to show minimal spatial separation, to 1, where all of each fauna are uniquely endemic, and would be most likely to be farthest apart. In this study values ranged between 0.1 and 0.65.

We also examined mean endemism with respect to two plates only: (3) where *u*_{1}, *u*_{2} are the taxa unique to plates 1 and 2 respectively, and where *t*_{1}, *t*_{2} are the total richnesses of faunas on plates 1 and 2 respectively. Because by definition *u* and *S* are complementary (where *S* is the mutually shared fauna; see eq. 1), ME_{2} is also equivalent to the average complementarity value for either of each plate pair, normalized by the richness of each plate. We also analyzed the cosmopolitanism of the faunas, not to use as a metric but simply as a comparative measure against which we could interpret our two indices, by calculating the number of genera in each fauna for one, two, three, and four plates (Fig. 5) (see also Miller 1997). This is also a measure of the range-size frequency distribution of the faunas at each time slice.

## CALCULATING DISTANCES BETWEEN PLATES

#### Defining Centroids

We first created a pairwise distance matrix between the centroids of four different major plates. Then, using each program, we retro-projected the present-day centroid for each plate (coordinates given in Table 1) so that we could compare the differing reconstructions. Thus, in each reconstruction exactly the same points are being compared relative to the coastline files (see below). Errors in the calculated inter-centroid distances may still occur, owing to differing rotations of Baltica between programs for example, but are minor. Our data were all geo-referenced in decimal latitude and longitude and plotted geographically to eliminate any obvious errors. Figure 2 gives an impression at two spatial scales of the sampling density of our data across the terranes under consideration, each dot representing an individual fossil locality for either trilobites or brachiopods. This sampling density compares favorably with many studies of Recent faunal distribution.

Our original data were first integrated with the Scotese model by using a GIS “cookie cutter” that assigns a plate code to each record (equivalence between plate codes in the two tectonic models is given by Table 2 as well as below). This technique ensures that any records that should fall in irrelevant plate fragments are disregarded. Our four major plates are defined as follows:

1. Baltica: as in Cocks and Fortey (1998)

2. Laurentia: because of their modern dispersal, major parts of greater Laurentia (Greenland, Irish island arcs, Grampian [= Scotland], Midland Valley, Spitsbergen, and Caledonides) were disregarded for calculation of the centroid

3. Avalonia: western sites in Avalonia were ignored, because only in the Scotese model are they placed adjacent to the eastern part of Avalonia (even though eastern and western Avalonia were very likely a united terrane [Cocks et al. 1997]).

4. West Gondwana: here taken to include Armorica plus Perunica [Bohemia] plus Montagne Noire, Iberia [including Portugal], Sardinia, Carnic Alps, northwest Africa (Morocco), northeast Africa, and South America including Paraná Basin but excluding the Precordillera of Argentina, a Laurentian fragment (Astini 1999).

Next, we calculated the great-circle distance in kilometers (assuming the present-day mean Earth radius of 6371 km for the purposes of spherical trigonometry) between paleolatitudinal and paleolongitudinal pairs of points representing the retro-projected centroid of each plate at each time zone, for the two plate models.

We calculated the final present-day centroids by using the coastline files (i.e., the *x*,*y* polygon vertices for coastline boundaries from the Scotese model) for a particular fragment or group of fragments. We detail our method here because calculating a centroid on a globe on the basis of coastline vertices in latitude and longitude, although not simple computationally, is highly relevant to reassigning distances accurately in paleogeographic reconstruction, and there is more than one possible solution. Each latitude and longitude was converted to Cartesian *x*, *y*, and *z* values, then the average of each of these values was calculated over all vertices of the coastline. Next, the resulting mean point was projected back to a longitude and latitude point on the surface of the sphere, with necessary adjustments to the algorithm in cases where a polygon crossed −180, +180, that was conditioned on the trigonometric values. The sole disadvantage of this technique is that complexity of the coastline can bias the centroid, so an additional slight adjustment had to be made in a few cases, such as northern North America, where the northern part of the coastline is exceptionally convolute. However, the method successfully allows computation of a centroid in tricky cases such as a polar-straddling polygon (e.g., West Gondwana in the Ashgill—not feasible using the standard centroid calculation algorithm implemented in the program ArcView 3.1). The algorithm also allows calculation of a centroid for a set of centroids that need to be combined (e.g., the multiple plates that were combined to make up West Gondwana; see Table 2). Because we calculated the centroids using only the Scotese coastline files, identical centroids were retro-projected from the present day using the two plate programs. Thus, the fact that some fragments are merged in the Smith model (e.g., Africa; see Table 2) was of no consequence when comparing the models.

#### Translating Faunal Indices into Distances

With the inter-centroid great-circle distances calculated from the retro-projected centroids given in Table 1, we are then able to calculate a new set of six pairwise distances for each model. This set sums to the original sum of distances, but in the exact ratios of each biogeographic index for each taxon and for each plate model (results given in Table 3). This is achieved by implementing the equations for each graphical fit at each time slice. For this reason, in our graphs distance is plotted as the dependent variable. The goal is to find the set of (here, six) distances computed from the equation that minimizes the sum of squares of observed deviations from the best-fit line (this new set of distances will have a perfect fit to the chosen line). We examine both exponentially and linearly fitted models to each index (Table 4). The *r*^{2} value in regressions is obtained purely for comparative purposes. We do not report significance levels for each graph because statistical significances (alphas) overall are inflated by two factors: the strong temporal autocorrelation between equivalent plate pairs in different graphs, and the interdependence between groups of three plate pairs representing each terrane. However, this simple technique enables us to determine which taxa perform best as indicators of distance and which plate tectonic models best represent our generic level data. For example, if the biogeographic indices of mean endemism and/or complementarity at a particular time scale as direct proportions with distance, we would expect that each index plotted against distance would fall on a straight line. If we can accept a linear model of distance decay, this model can be used to optimize the position of a plate with respect to other plates.

## RESULTS

#### Cosmopolitanism

As indicated by Figure 5, the range size frequency distribution of both trilobite genera (A) and brachiopod genera (B) changes profoundly and consistently across the 50–58-Myr period, from early Arenig to Wenlock (the different models have differing timescales, the Scotese model having a wider chronological range). The predominance of single-plate endemics in the early Ordovician (greatest in the Arenig for trilobites and Llanvirn for brachiopods, for each time slice representing a typical faunistic “hollow curve” distribution) diminishes substantially as many genera become increasingly cosmopolitan toward the end-Ordovician (late Ashgill) glaciation and mass-extinction event. Notably, the four-plate class (a more or less cosmopolitan distribution, as the four plates encompass a vast area even by the mid-Silurian) reaches its peak in the Ashgill. This is particularly so for trilobites, when 16% of the 215 genera occur on all plates considered. Much of this widespread fauna apparently did not survive the Hirnantian glaciation and was not fully replaced in the sampling record until some time in the Silurian. This consistency between taxonomic groups is promising but, as already pointed out, only applies at any one time.

#### Behavior of the Indices and Taxonomic Groups Across Time Slices

The two faunal indices we have chosen show some encouraging consistency across the Ordovician and Silurian between the two groups (Fig. 6B,C). Indeed, if there was no relationship between brachiopods and trilobites for any one measure, we might expect that the utility of the indices as proportional distance measures would be susceptible to taxonomic idiosyncrasy. In fact, taking all time slices into account, the correlation between groups is comparable for either measure at an *r*^{2} of about 0.625 (Fig. 6B,C). Although the relationship is fairly linear across the time slices for complementarity, a second-order polynomial better fits the correspondence between groups for mean endemism. This can be explained by the higher levels of endemism that appear to characterize brachiopods, especially during the lower Ordovician (when values were higher: points toward the right of the graph; Fig. 6C). However, a nonlinear intertaxonomic relationship does not rule out the potential utility of the mean endemism index, especially when considered for individual time slices.

#### Independent and Total Evidence Analyses for Indices

Because indices vary in absolute value through time, we examine relationships within each time slice (Figs. 7–9). A comparison of linear and exponential models, and trilobites versus brachiopods, is shown in Table 4. The two groups are similar as indicators of modeled inter-centroid distance in terms of *r*^{2} (Table 4), but for mean endemism (ME_{4}) brachiopods seem to have the edge (Fig. 7). The “total evidence” results (Figs. 8, 9) reflect the difference between indices and the two plate tectonic models. Table 4 gives more details, with total evidence performing better overall and particularly for ME_{4}. Also, the fit to either the Scotese or Smith model distances is distinctly better for the linear than the exponential model in each case (Table 4). The performance of ME2 is the most disappointing of the three indices in indicating distance, and marginally worse than complementarity (Table 4).

Complementarity also performs very inconsistently across the time slices (Fig. 8). Although it is a good indicator of distance in the Scotese model in the lower Caradoc and particularly Ashgill (when *r*^{2} = 0.95), no relationship with distance is evident in either the Arenig or Wenlock for either the Scotese or the Smith model. Further, the Scotese model is usually considerably more representative of the complementarity between faunas than is the Smith model, with the possible exception of Llanvirn time. The contrast with mean endemism for this data set is thus important, for mean endemism shows a strong relationship to the sets of distances in either plate tectonic model across all five time slices, and a remarkably close fit, particularly for the Scotese model (*r*^{2} = 0.97), throughout the Caradoc and Ashgill. The Scotese model represents the biogeographic data remarkably well, and considerably better overall for mean endemism (ME_{4}; Fig. 9), than the Smith model, which shows a weak fit to our data in the Wenlock. Distances in the Scotese model also fit the complementarity index for the pooled taxonomic data in the Caradoc and particularly Ashgill (again with a similar pattern of slight deviations of plate pairs from the linear best-fit line). Our generic-level data support the Scotese model well.

For both complementarity and mean endemism, the ranges of both indices (x-axis in Figs. 7–9) broaden substantially by the Ashgill, as each index tends toward lower absolute values, and the range narrows with increasing values again in the Silurian. These changes reflect the broader pandemicity toward the end-Ashgill event (see again Fig. 5).

#### Rescaling the Distances to Suggest a New Model

Table 3 gives rescaled distances for the six plate pairs for our test case, for all three indices and both models for each time slice. It is particularly encouraging that irrespective of the index used, these rescaled distances fall in our example between the minimum (i.e., the sum of radii 1 and 2 for the respective plate pair, simplified as circles) and maximum physically possible distance (i.e., half the earth's circumference less r1 + r2). This is despite the fact that we have not incorporated these basic constraints into our model. In fact, owing to close fit of coastlines, the Scotese and Smith models sometimes show distances marginally below r1 + r2 (see Table 3). There are two possible mirror-image arrays for any network of distances, and in a reconstruction we would naturally pick that which matches best the geography of the computer models. Note also that our rescaled distance arrays are not adjusted to form a closed network on either a plane or a sphere, but they often approximate to one whose vertices actually all join. Distances are interdependent in such a network (imagine the construction of circles around centroids and the feasible errors), a computational problem beyond the scope of this paper. In some cases, a biogeographic index suggests an array implying a more complex, wiggling movement for Avalonia than would be expected from the assumptions of plate tectonics in the context of subduction of a single paleo-ocean. This is the case when the sum of distances for plate pairs {AL, AB, AW} is greater than the sum of {BL, BW, LW}—for example, using complementarity in the Arenig and Wenlock or ME_{2} in the Arenig (Table 3). Once again, this indicates that complementarity is a less accurate descriptor of distance than is ME_{4}.

In the reconstruction in Figure 10, we show for the best performing of our indices, ME_{4}, how these rescaled distances affect the position of Avalonia relative to the other three plates, for three of the five time slices: Arenig, early Caradoc, and Wenlock. Because existing interdependencies embodied in the PaleoGIS model constrain the movement of larger plates, it is only practical to move Avalonia for the purposes of the figure, and so this illustration shows distance adjustments only for the three plate pairs that include the terrane Avalonia (i.e., AL, AB, AW). In the Ashgill, the position of Avalonia in the Scotese model is already close to an optimal position in relation to the biogeographic indices devised from our data. However, the optimal index based on pooled trilobite and brachiopod genera suggests that Avalonia would reach Baltica/Laurentia rather later than suggested by the Scotese model. The mid-Silurian docking of Avalonia with Laurentia/Baltica shown by the Scotese model in Figure 1 would thus be interpreted from this index to be too advanced, with Avalonia also farther from Baltica in early Caradoc times than suggested by either the Scotese or Smith models. Other distances, particularly the Avalonia-West Gondwana distance, are more consistent with the Scotese model than the Smith model (see also Table 3), accounting for the better fit with mean endemism. By using sequence stratigraphy as an independent criterion, such inferred distances can be tested iteratively by field evidence.

## DISCUSSION

Any new plate tectonic reconstruction should represent existing faunal data better than a previous one. Our reconstructions of the relative position of Avalonia are based on fits to an index derived solely from numbers implicit in faunal lists. We selected the index showing the best and most temporally consistent fit to a pre-existing model (Ross and Scotese 1997) using brachiopod and trilobite generic data. During the lower Paleozoic, the relative distances between Laurentia, Baltica, and Avalonia (and to a lesser extent West Gondwana) are better known than for many major plates elsewhere in the world, so the approach should have application to more problematic terrane distributions. In the future it would be desirable to test our indices against known biogeography of Recent marine organisms with planktonic propagules, whose values might be expected to depend on distance.

Many orientations of arrays of inter-centroid distances on the surface of the earth would be most consistent with particular biogeographic indices in relation to plate sizes and shapes. Here we have demonstrated a simple method for generating pairwise distances based—for linear fits—on a chosen biogeographic index multiplied by the slope plus the intercept of measured distances taken at relevant time slices from existing plate models. The distances suggested by a faunal index are relative, and so at each time a new index is effectively scaled to the sum of the great-circle distances represented in an existing model; but the distances could easily be adjusted to absolute physical constraints such as plate radii and Earth dimensions. We have not attempted to create a model that fine-tunes the reconstructions at each time slice in a dynamic fashion. The constraints of such temporal relationships, although important to take into account, might be expected to decrease the goodness-of-fit of our indices at each time slice.

The arguments used here perhaps appear circular, because ME_{4} fits one of the computer programs that itself incorporates faunal data in an unspecified way. It might perhaps have been preferable to test faunal data against more than one “fauna free” reconstruction program. The way forward is possibly a series of successive approximations in which all sources of data are allowed to contribute toward a “best fit”—this is the process of reciprocal illumination which is common in historical reconstructions.

Bearing all these qualifications in mind, the results are encouraging. The biogeographic signal from four-plate mean endemism (ME_{4}) to distance, in particular, seems remarkably good, and this index fits one model consistently at five time slices from early Ordovician to mid-Silurian times. Interestingly, this index performs considerably better and more consistently than the standard measure of faunal turnover, complementarity (C), which is based simply on the pairwise ratio of shared to total genera. We suggest that there is more signal relating to distance in the highly endemic (or conversely the highly shared) component of each Venn diagram than in faunal turnover. Normalizing by each plate's faunal total should also provide a tighter control for likely differential effects of diversity (e.g., effects based on different paleolatitudes, on different plate areas, and on faunal sampling) than when using the synoptic pairwise total. We are also encouraged that for the best index “total evidence” analysis appears to improve signal in the data. A simple linear model for indices against distance works well for our data sets, but an exponential model does not perform quite as well. Although computationally convenient, this may seem counter-intuitive, since it might be expected that a threshold distance for the dispersal of many taxa would have been reached over the vast spatial scale in which we are working. Previous biogeographic studies, either Tertiary (Picoli et al. 1991) or modern day (Nekola and White 1999), have suggested that an exponential model fits species data better. We can suggest that our good linear fit for mean endemism may be related to the use of only generic data, which though possibly less differentially sensitive to distance, appear to retain substantial biogeographic signal.

Ideally, the question of how faunal similarity scales to distance would have been approached initially from data based on test cases from living shelf faunas. As we have noticed, there are very few such studies. The data-gathering for such a test would entail at least as much effort as has gone into the present compilation, but should be possible for relatively well-sampled groups such as shelled mollusks. It may seem somewhat quixotic to infer continental distributions hundreds of millions of years ago from faunal evidence without the appropriate actualistic models. However, the greater immersion of Ordovician–Silurian continents compared with the Recent, providing ample chances for endemism, may be particularly appropriate for the kinds of methods used here. We await the Recent comparisons to test this.

## Acknowledgments

This work was funded by National Environment Research Council grant GR3/10717. We acknowledge most gratefully the help of T. McCormick, J. Adrain, and A. Miller, who all provided us with additional data used in the present analysis. The manuscript was improved by helpful discussions with C. Moncrieff, R. Colwell, and A. Rushton.

- Accepted 1 March 2002.