JOURNAL OF MORPHOLOGY 269:840 864 (2008) Patterns of Morphospace Occupation and Mechanical Performance in Extant Crocodilian Skulls: A Combined Geometric Morphometric and Finite Element Modeling Approach Stephanie E. Pierce, 1 * Kenneth D. Angielczyk, 2 and Emily J. Rayfield 1 1 Department of Earth Sciences, University of Bristol, Bristol BS8 1RJ, UK 2 Department of Geology, Field Museum of Natural History, Chicago, Illinois 60605 ABSTRACT Extant and fossil crocodilians have long been divided into taxonomic and/or ecological groups based on broad patterns of skull shape, particularly the relative length and width of the snout. However, these patterns have not been quantitatively analyzed in detail, and their biomechanical and functional implications are similarly understudied. Here, we use geometric morphometrics and finite element analysis to explore the patterns of variation in crocodilian skull morphology and the functional implications of those patterns. Our results indicate that skull shape variation in extant crocodiles is much more complex than previously recognized. Differences in snout length and width are the main components of shape variation, but these differences are correlated with changes in other regions of the skull. Additionally, there is considerable disparity within general classes such as longirostrine and brevirostrine forms. For example, Gavialis and Tomistoma occupy different parts of morphospace implying a significant difference in skull shape, despite the fact that both are traditionally considered longirostrine. Skull length and width also strongly influence the mechanical performance of the skull; long and narrow morphotypes (e.g., Tomistoma) experience the highest amount of stress during biting, whereas short and broad morphotypes (e.g., Caiman latirostris) experience the least amount of stress. Biomechanical stress and the hydrodynamic properties of the skull show a strong relationship with the distribution of crocodilians in skull morphospace, whereas phylogeny and biogeography show weak or no correlation. Therefore, ecological specializations related to feeding and foraging likely have the greatest influence on crocodilian skull shape. J. Morphol. 269:840 864, 2008. Ó 2008 Wiley-Liss, Inc. KEY WORDS: Crocodylia; morphology; disparity; function; phylogeny; biogeography; correlation Analysis of shape variation, in both extant and fossil organisms, and the evaluation of causal factors underlying morphological disparity offer insights into large-scale macroevolutionary patterns (Foote, 1997). A variety of biological processes, such as long-term evolutionary diversification, ontogenetic development, and even disease or injury, produce differences in shape between species, individuals, or their parts. Differences in shape may also signal different functional roles or responses to environmental or geographical selective pressures (Zelditch et al., 2004). Studies of disparity in extant animals are increasingly common, and have been used to address basic issues in evolutionary biology, including whether shape variation is related to phylogenetic structure (Stayton, 2005), ecological correlates (Claude et al., 2004), developmental constraints (Zelditch et al., 2003), or biomechanical performance (Hulsey and Wainwright, 2002). Nonetheless, only a few studies have adopted a fully integrated, holistic approach to the study of disparity by synthesizing data from multiple fields of inquiry (Milne and O Higgins, 2002). This article explores patterns of morphological variation in modern crocodilians through an analysis of skull shape and assesses factors that are potentially responsible for those patterns. The crocodile skull is a unique and complex structure that has been extensively used in phylogenetic (e.g., Clark, 1994; Brochu, 1997) and functional studies (e.g., Busbey, 1995; Daniel and McHenry, 2000; Erickson et al., 2003; Metzger et al., 2005; McHenry et al., 2006), and it has served as a model for interpreting the paleobiology of extinct species (e.g., Massare, 1987; Taylor, 1987; Taylor and Cruickshank, 1993; Sereno et al., 1998, 2001; Meers, 2002). Despite this great interest, however, there has been little quantitative documentation of Contract grant sponsors: UK Overseas Scholarship, University of Bristol Postgraduate Scholarship, Natural Sciences and Engineering Doctoral Scholarship, Sir James Lougheed Award of Distinction, University of Bristol Alumni Association. *Correspondence to: Stephanie E. Pierce, The Royal Veterinary College, Hawkshead Lane, North Mymms, Hatfield, Hertfordshire, AL9 7TA, United Kingdom. E-mail: spierce@rvc.ac.uk Published online 21 May 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/jmor.10627 Ó 2008 WILEY-LISS, INC.
CROCODILE SKULL SHAPE AND BIOMECHANICS 841 the broad-scale patterns of morphological variation present in extant crocodiles. The absence of a quantified framework makes it difficult to extrapolate studies of particular species to crocodiles in general and to substantiate hypotheses about the paleobiology of fossil forms. SHAPE VARIATION Extant crocodilians include 23 species of alligators, caimans, crocodiles, and gharials found throughout the world s warm temperate and tropical regions. All species are semiaquatic and do not venture far from estuaries, swamps, lakes, streams, or rivers (Cogger and Zweifel, 2003). Despite their low modern diversity, crown group crocodilians have a rich fossil record and long evolutionary history extending back to at least the Campanian (Upper Cretaceous) (Brochu, 2003). One of the most notable features of crocodilian evolution is the multiple instances of convergence/parallelism in cranial morphology (Densmore and Owen, 1989; Cleuren and Vree, 2000). A classic example of this pattern is the acquisition of a longirostrine snout morphology specialized for a piscivorous diet (e.g., Gavialis gangeticus, Tomistoma schlegelii, and Crocodylus cataphractus). Because crocodiles interact with their surroundings using their snouts, the observed morphological similarity may indicate that a limited number of anatomical solutions exist for any given ecological problem, that crocodilians have only utilized a very limited portion of their possible ecomorphological space, or the clade has been presented with only a limited set of ecological opportunities during its history. The frequency of convergent and/or parallel evolution in skull morphology within the Crocodylia was not recognized initially, and early taxonomists frequently classified crocodiles based on overall similarity in skull shape. Species were placed into two broad categories based on rostral shape: longirostrine, those crocodiles with elongated, tubular snouts, and brevirostrine, all other remaining species (e.g., Lydekker, 1888; von Zittel, 1890). However, these categories not only do not reflect the phylogeny relationships of crocodiles, they are also overly simplistic descriptions of skull shape. In recent years, various attempts have been made to subdivide the classical short/long snouted morphotypes into finer categories using linear measurements and qualitative assessments. Busbey (1995) subdivided crocodile snouts based on cross-sectional dimensions and the ratio of rostral lengthto-skull length: long (>70%), normal (<70 55%>), and short (<55%). Brochu (2001) slightly modified Busbey s categories into slender-snouted, generalized, and blunt-snouted (along with two other categories only seen in fossil forms), and reorganized the distribution of species within each group. Most recently, McHenry et al. (2006) classified crocodilians based on ecomorphotype as follows: 1) longirostrine crocodiles specialized for a piscivorous diet; 2) tall and narrow mesorostrine crocodiles suited to feed on large, less agile prey; 3) flat and broad mesorostrine crocodiles suited to feed on large, more agile prey; and 4) brevirostrine crocodiles which feed on small aquatic invertebrates and terrestrial vertebrates. Although these categories are useful descriptors of some aspects of crocodilian cranial shape, they do not describe the entire range of shape variation present. Questions concerning the extent and nature of morphological disparity within and between groups are best addressed in terms of the distribution of taxa in some form of morphospace (Foote, 1997). Morphospaces are theoretical or empirical constructs defined with reference to any number of quantifiable elements of form, which together describe aspects of morphological variation within a group of interest (Wills and Fortey, 2000). Traditionally, morphometric data consist of measurements such as length, depth, and width. However, such data sets are not ideal because they usually capture relatively little information about shape (e.g., many of the measurements are primarily made along few dimensions or are redundant), and their measurements frequently are highly correlated with size (Zelditch et al., 2004). Alternatively, landmark-based geometric morphometrics offer a means to evaluate variation in shape independent of size, and is a powerful tool for creating quantitative morphospaces. Geometric morphometric techniques are particularly useful when investigating covariation between shape and other characteristics of interest, permitting the exploration of macroevolutionary patterns such as convergent evolution (Stayton, 2006), character displacement (Adams and Rohlf, 2000), and the link between form, function, and phylogeny (Monteiro and Abe, 1999). Factors Affecting Shape In his phylogenetic review of crocodilian cranial shape, Brochu (2001) compared both living and fossil forms in the context of their taxonomic, geographic, and temporal distributions. By mapping a discrete-state character that qualitatively described snout shape over phylogeny, he found that a few basic morphotypes have arisen multiple times in distantly related lineages. Some clades were found to be morphologically uniform, but geographically widespread, whereas sympatric taxa tended to be morphologically disparate and distantly related. For example, the three crocodilians (Osteolaemus tetraspis, C. niloticus, C. cataphractus) living in western Africa today are morphologically very disparate and phylogenetically distinct. Brochu s study substantiated the suspicion of nearly all previous workers (e.g., Langston, 1973;
842 S.E. PIERCE ET AL. Russell and Wu, 1997), that snout shape is very labile within crocodilians and that similar snout morphologies may not be phylogenetically or biogeographically restricted. As a result, Brochu hypothesized that the morphological divergences within crocodilians reflected ecological separations, but neither he nor any previous authors assessed these observations in a quantitative fashion. As many of the previous studies on crocodilian skull shape concluded that morphological diversity among taxa is driven by ecological specializations, the link between skull shape and function has been studied using a variety of techniques. Busbey (1995) investigated the structural consequences of rostral shape in various extant crocodile species using beam theory (Young and Budynas, 2001). By calculating the second moment of area (a property of a shape that is used to predict its resistance to bending) along the length of the snout, Busbey concluded as follows: 1) second moments of area increased posteriorly along the snout in all taxa, indicating an increase in the ability to withstand bending stress along the rostrum in an anteroposterior direction; 2) taxa with more tubular snout morphologies tended to have smaller section moments and, therefore experienced greater bending stresses under comparable loads; 3) an oreinirostral (tall) skull is less susceptible to dorsoventral bending stress than a platyrostral (low) skull; and 4) a platyrostral skull is less susceptible to mediolateral bending stress than an oreinirostral skull. These mechanical features of the crocodilian skull were proposed to reflect adaptive responses to varying habitat and trophic demands. For example, the broad, flat snout of Alligator mississippiensis was hypothesized to be an adaptation for withstanding compression and axial torsion during rolling maneuvers that are used to destabilize and dismember prey. More recently, McHenry et al. (2006) used finite element modeling (FEM) to investigate the relationship between rostral shape and biomechanical performance in crocodilians. Six specimens were chosen to reflect, as far as possible, the full range of rostral shape in living crocodilians: Caiman crocodylus (juvenile and adult), A. mississippiensis, C. johnstoni, Melanosuchus niger, and Paleosuchus palpebrosus. Comparison of the mechanical performance of the six FEMs indicated that tall skulls performed best under vertical loading, whereas tall and wide skulls performed best under torsional loading (e.g., M. niger and P. palpebrosus). As such, Busbey s (1995) hypothesis that a dorsally flattened skull (e.g., A. mississippiensis) is optimized for torsional loading was not supported. Instead, McHenry et al. suggested that the crocodilian skull represents a mechanical compromise between strength for subduing and processing food and hydrodynamic efficiency for catching agile aquatic prey. Accordingly, both the longirostrine (G. gangeticus, T. schlegelii, C. cataphractus) and platyrostral (as exemplified by A. mississippiensis) conditions were portrayed as two possible means to increase speed of attack while at the same time minimizing drag. As there appears to be a definite relationship between skull shape and biomechanical performance, it is reasonable to hypothesize that environmental selection acting on mechanical properties would result in variation in cranial morphology. STUDY OBJECTIVES The primary objective of this study is to analyze the interplay between shape variation, biomechanics, evolutionary, and environmental factors. To accomplish this goal, we adopt an integrative and interdisciplinary approach to the study of crocodilian skull shape, combining for the first time geometric morphometric techniques, FEM, and correlation statistics to address the following questions: 1) What is the main quantifiable pattern of shape variation in the skull roof of modern crocodilian species? 2) How does each taxonomic group contribute to total morphological disparity? 3) Is there an association between morphology, phylogeny, and biogeography? 4) How does skull shape affect mechanical performance? 5) Which constraints phylogeny, function, and/or biogeographic distribution are responsible for the observed morphological variation? This study tests and extends the results of previous studies, particularly those of McHenry et al. (2006), by first generating theoretical skull shapes to develop a heuristic method for investigating problems of evolutionary biomechanics, and then analyzing skulls from all 23 living crocodilian species within that context. Our ultimate objective is to cast new light on the patterns and processes responsible for morphological variation in modern crocodilians and to provide a framework for documenting the morphological diversity of this clade throughout its evolutionary history. MATERIALS AND METHODS Phylogeny The phylogenetic relationships of crocodilians are comparatively well understood and are based on diverse sources of information, including both morphological and molecular data (Densmore and Owen, 1989; Brochu, 1997). However, some areas of the tree still remain unresolved. Most notably, the placement of G. gangeticus is unclear. Although morphological evidence points toward this species being the basal sister taxon of all other extant crocodilians, molecular datasets argue that Gavialis is a derived crocodilid that is closely related to T. schlegelii (Norell, 1989; Willis et al., 2007). Furthermore, the relationships among species belonging to Crocodylus (the true crocodiles) and Caiman remain poorly resolved (see Brochu, 2003 for a review). In this study, we used a modified version of Gatesy et al. s (2004) morphology-based topology (see Fig. 1)
CROCODILE SKULL SHAPE AND BIOMECHANICS 843 Fig. 1. Phylogenetic relationships within extant crocodilians. The phylogeny is a modified version of Gatesy et al. s (2004) morphology-based typology, with Isisfordia duncani used to root the tree. Numbers correspond to the nodes used to draw the phylogeny onto morphospace (see Fig. 5). and their supermatrix topology, which was based on the assessment of both morphological and molecular data, to assess the relationship between skull shape, morphospace occupation, and phylogeny. In the morphology tree, G. gangeticus is reconstructed as the most basal extant crocodilian, whereas T. schlegelii is nested within the Crocodylidae. Two major polytomies were included within the Crocodylidae because no consensus currently exists with respect to relationships within this group; and C. latirostris was considered to be more closely related to M. niger than to other Caiman species. The supermatrix topology differed from the morphology-only tree in its placement of G. gangeticus as the sister group of T. schlegelii within the crocodilids, the greater resolution among crocodilid species, the sister group relationship between C. cataphractus and O. tetraspis, and the poorer resolution of the clade including Caiman and Melanosuchus. Geometric Morphometrics Specimens, landmarks, and superimposition. A total of 126 specimens (see Appendix A), representing all 23 extant crocodilian species and one fossil outgroup species (Isisfordia duncani; Salisbury et al., 2006), were used in this study. The majority of individuals were adults, but in a few cases juvenile specimens were sampled as well (Appendix A). Sample sizes for different species varied (1 19 specimens per species), reflecting specimen availability in museum collections visited. The majority of specimens in the data set were skeletonized osteological specimens; however, seven individuals were adapted from illustrations made by Iordansky (1973), and the outgroup was an idealized reconstruction (Salisbury et al., 2006). We quantified skull shape variation using two-dimensional (2D) landmark-based geometric morphometrics. Crocodile skulls are relatively flat structures; therefore, little distortion is caused by projecting a specimen s 3D shape into a 2D plane. Skulls were positioned with their parietal table level with the horizontal plane and photographed in dorsal aspect by positioning the camera perpendicular to the specimens long-axis, ensuring comparability between the images and dorsal-view drawings. A scale was included with the images to record the size of each specimen. A total of 65 landmarks were digitized on the images using TpsDig 2.04 (Rohlf, 2005). Sixty of the landmarks are bilaterally symmetrical and five are located along the midline of the skull. To avoid inflating degrees of freedom in the statistical analyses, symmetric landmarks from one side of each specimen were reflected onto the other and the average position for each pair of landmarks was calculated using BigFix 6 (Sheets, 2001a). Subsequent analyses were carried out on these half specimens (see Fig. 2). All landmarks are either Type 1 (e.g., intersection point of three bones) or Type 2 (e.g., extreme end point of a bone) in the classification of Bookstein (1991). Landmark coordinates for all specimens were superimposed using the generalized least squares or Procrustes method (Rohlf, 1990) in the program CoordGen 6f (Sheets, 2001b) to remove the effects of position, orientation, and scale from the data set. To determine whether the amount of shape variation in the data set was small enough to permit statistical analyses, a correlation of the Euclidean distance between all pairs of aligned and scaled specimens and Procrustes distances between all pairs of specimens was conducted using TpsSmall 1.20 (Rohlf, 2003a). The correlation was extremely high (r > 0.99999) indicating that the area of shape space inhabited by the specimens is small enough that projection into the tangent plane does not introduce distortion. Shape variation, phylogeny, and biogeography. To assess morphospace occupation and shape variation, the data set was converted into shape variables (i.e., partial warp and uniform component scores), and these were subjected to a principle components analysis (PCA) using PCAGen 6n (Sheets, 2001c). The PCA computed by PCAGen is equivalent to a relative warps analysis (RWA) in which the scaling factor (a) is equal to zero. Axes that represent most of the useful information on shape variation were selected by plotting eigenvalues, or percentages of total variance, against ordinal number of principle components (i.e., a scree plot) and finding the inflection point. This was further confirmed by performing a v 2 statistic based on the likelihood-ratio criterion (Zelditch et al., 2004; also see Anderson, 1958 and Morrison, 1990). By focusing on only the axes that were significant descriptors of shape variation as measured by these criteria, we were able to summarize main patterns of variation while reducing the dimensionality of the data and filtering out noise. As noted earlier, our sample primarily consists of adult specimens. However, adult specimens within each species display minor size variation, and in a few cases we also sampled juveniles of particular species. Because our samples for each species include differently sized individuals, it is possible that allometric shape differences exist among the specimens of each species. Intraspecific allometric differences are not the primary focus of this study, but the presence of such variation in the data could make comparisons between species more difficult. Therefore, it was necessary to investigate the relationship between size and shape in our data set to ensure that differences in size were not the primary factor controlling shape difference among specimens. To assess the entire data set, we ran a multivariate regression of the partial warp and uniform component scores against centroid size, using the consensus form of all specimens as the reference specimen. In addition to this, the scores of the specimens along the significant PC axes were regressed on centroid size to check whether shape differences described by the PC axes were correlated with size. The regressions were run in TpsRegr 1.28 (Rohlf, 2003b) and Systat version 12.00.09. Although the PCA allows us to define a morphospace by quantifying and summarizing variation within our data set,
844 S.E. PIERCE ET AL. Fig. 2. Landmarks used in this analysis. 1: Anterior tip of premaxillae contact. 2: Posterior tip of premaxillae contact at the narial opening. 3: Anterior tip of nasal bones (or contact of nasal-premaxilla). 4: Nasal-nasal-frontal contact. 5: Midline of supraoccipital. 6: Premaxilla-maxilla contact along lateral margin. 7: Premaxilla-nasal-maxilla contact. 8: Nasal-maxillaprefrontal contact. 9: Nasal-frontal-prefrontal contact. 10: Maxilla-prefrontal-lachrymal contact. 11: Maxilla-lachrymaljugal contact. 12: Maxilla-jugal contact along lateral margin. 13: Jugal-lachrymal-orbit contact. 14: Lachrymal-prefrontalorbit contact. 15: Prefrontal-frontal-orbit contact. 16: Frontalpostorbital-orbit contact. 17: Anterodorsal tip of postorbital bar. 18: Anteroventral tip of postorbital bar. 19: Frontal-parietal-postorbital contact. 20: Parietal-postorbital-supratemporal fenestra contact. 21: Posterodorsal tip of postorbital bar. 22: Posteroventral tip of postorbital bar. 23: Jugal-quadratojugalinfratemporal fenestra contact. 24: Postorbital-squamosalquadratojugal contact. 25: Postorbital-squamosal-supratemporal fenestra contact. 26: Midpoint of supratemporal fenestra along parietal. 27: Parietal-squamosal-supratemporal fenestra contact. 28: Jugal-quadratojugal contact along lateral margin. 29: Quadratojugal-quadrate contact along lateral margin. 30: Medial condyle of quadrate. 31: Posterolateral tip of squamosal (wing). 32: Parietal-squamosal contact along posterior margin. 33: Lateral contact of parietal-supraocciptal. 34: Midlateral margin of external narial opening. 35: Point on lateral margin of premaxilla corresponding to the mid-lateral margin of the external narial opening. and to visualize the positions of particular species within the space, additional analyses are required to understand how patterns of morphospace occupation correlate with possible causes. To assess the correspondence between skull morphology and our current understanding of taxonomy and biogeography, we mapped the areas of morphospace occupied by taxonomic groups and groups of sympatric species onto the PC plots using convex polygons. Division of species into taxonomic groups and geographic groups was straightforward in most cases. However, we did split the Asian species into those occurring in Asia (i.e., mainland India and China) and Southeast Asia (i.e., the Southeast Asian islands and Australia) based on similarities in species composition (see Appendix A). In addition, we assessed the correspondence between morphology and phylogeny by superimposing a species-level cladogram onto the PC plots using the following protocol. First, the PC1 and PC2 coordinates of the average specimen for each species were calculated and plotted in morphospace. Next, the PC1 and PC2 coordinates of each internal node within the preferred phylogeny were determined using the squared-change parsimony reconstruction method in Mesquite v 1.06 (Maddison and Maddison, 2006), and the corresponding point was then plotted onto the same graph. The values of the internal nodes represent the maximum-likelihood estimate of the ancestral states and provide a standardized method for drawing the phylogeny onto the morphospace plot. Finally, the internal nodes were connected to each other and their corresponding terminal points using straight lines (for an alternative plotting method, see O Keefe, 2002; Stayton, 2005; Stayton and Ruta, 2006). We can obtain a general sense of correlation from inspecting the phylogeny superimposed on the PC plots, but this procedure only gives a qualitative estimate and can be misleading if the data sets include multiple axes of correlation (Stayton and Ruta, 2006). To statistically address the issue of association between skull shape and phylogeny, we compared morphological and phylogenetic distances among taxa at the species level. Specifically, we wanted to establish if closely related species tended to cluster more closely in morphospace than more distantly related species. The Mantel test (Mantel, 1967) was used because it allows one to examine the correlation between two distance matrices. The morphological distance matrix consisted of partial Procrustes distances (i.e., Euclidean distances between landmark configurations in the plane tangent to shape space) measured between the average shapes of the species of interest. The phylogenetic distances were calculated in two ways: 1) the equal distance method, where the number of nodes that separate all pairs of species in the phylogeny are tallied (Stayton and Ruta, 2006), essentially giving each branch on the tree an equal length; and 2) the character distance method, where the distances between pairs of species are assessed through an analysis of a character matrix. The latter procedure scales branch lengths to the amount of evolutionary change that occurs along each branch. The branch lengths used were based on the morphology-only and supermatrix topologies and the character matrices of Gatesy et al. (2004). To assess significance, the rows and columns of one of the matrices was subjected to 1,000 random permutations, with the correlation being recalculated after each permutation. Partial Procrustes distances were calculated using TwoGroup 6h (Sheets, 2000), and the Mantel tests were computed with XLSTAT version 2006.5. Disparity. The preceding analyses can provide insight into whether skull shape is related to phylogeny and biogeography, but in addition to this question we were also interested in whether different crocodilian groups, and different geographical regions, display similar amounts of shape variation, or disparity. For example, do the Crocodylidae display significantly more or less disparity than the Alligatoridae? Similarly, but at a narrower taxonomic level, does Crocodylus display significantly more or less shape disparity than Alligator? In addition, does Africa contain more or less shape disparity than South America? To answer these questions, we adopted Foote s (1993) definition of disparity, under which the disparity of the subgroup of interest is a function of the distance between the mean shape of
CROCODILE SKULL SHAPE AND BIOMECHANICS 845 the subgroup and the mean shape of all of the subgroups (i.e., the grand mean shape). We then grouped the 23 species into families (all of which are monophyletic; see Fig. 1), genera (all of which but one are monophyletic; see Fig. 1), and geographic regions, and examined how the different subgroups contributed to the total amount of disparity exhibited by the data set. Disparity was measured by using a version of Foote s (1993) metric modified for application to geometric morphometric data (Zelditch et al., 2003, 2004). The method calculates the partial disparity of each subgroup by summing the squared Procrustes distances between the mean shape of each subgroup and the grand mean shape of all the subgroups in the data set. The total disparity of the data set is determined by adding up the partial disparities of each subgroup. To assess whether the partial disparities of each subgroup were significantly different, we made a series of pairwise comparisons using two-sample t-tests (Zelditch et al., 2004). It is important to stress that this technique focuses on how disparate the mean shape of each subgroup is from the grand mean shape of morphospace rather than the degree of disparity or variation within a particular subgroup (represented by an area or volume in morphospace occupied by members of the group). Disparity calculations were carried out in DisparityBox 6h (Sheets, 2006) and significance tests in TBox (Sheets, 2003). Finite Element Modeling Model construction and properties. We examined a total of 27 2D FEMs of crocodile skulls. Four of these represent the extreme shapes along PC1 and PC2 which were created to develop a theoretical framework for examining the relationship between mechanical performance and skull shape variation described by the morphospace. In addition, 2D models of all 23 species were constructed to test the viability of using PC end points as a proxy for interpreting the relationship between the pattern of species distribution in morphospace and skull strength, as well as to gain insight into the mechanical properties of real morphologies. Although a 2D model is not entirely reflective of the morphology of the crocodilian skull, it can be used as a first approximation for biomechanical investigations and can be tested further through the use of more detailed models in the future (Rayfield, 2004). The models were based on bilaterally symmetric landmark configurations that were generated by back-reflecting the 35 landmarks used in the geometric morphometric analyses (see Fig. 2). This new data set was loaded into the program Morphologika (O Higgins and Jones, 2006), where skull shapes representing the extreme points along PC axes 1 and 2 and the average shape of all 23 extant species were digitally captured. This new sample was then redigitized in TpsDig 2.04 (Rohlf, 2005) to capture coordinate data, imported into CoordGen 6f (Sheets, 2001b), and again superimposed with centroid size set equal to one to ensure that all models were created at the same scale. The new landmark coordinates were exported as ASCII files, and each coordinate was multiplied by 100 to construct images large enough to view in the FE software. The amplified coordinates of each skull shape were imported into the Geostar geometry creator component of the COSMOSM FEA package (v. 2.8 for Unix; SRAC, CA, and CenitDesktop, UK). The external perimeter of the skull and the boundaries of internal cavities such as nasal opening, orbits, supratemporal and infratemporal fenestrae, were generated by linking the appropriate landmarks with straight lines, thereby creating a generalized skull shape without any sutural contacts. The inner region was meshed to produce an interconnected grid of three-noded triangular FEs representing only the dorsal surface of the cranium. The models were fixed along the posterior edge of the left and right quadrate during loading; as such, stress patterns at these points will be artificially magnified. Elastic isotropic properties were assumed for the models with the following values: E (Young s modulus) 5 6.65 GPa and v (Poisson s ratio) 5 0.35. These values are slightly different from those used in recent FE studies on crocodilian skulls (Daniel and McHenry, 2000; Metzger et al., 2005; McHenry et al., 2006), which employ material properties similar to bovine haversian bone (E 5 10 GPa and v 5 0.4). The estimated Young s modulus in this study is based on Currey s (1987) analysis of the material properties of Crocodylus frontal and prefrontal bones (frontal 5 5.6 GPa; prefrontal 5 7.7 GPa; average 5 6.65 GPa), and a reasonable average value of Poisson s ratio for bone was assumed (Reilly and Burstein, 1974). Potential orthotropic properties of the crocodilian cranium were not included in this study because data are currently not available, but evidence suggests that modeling orthotropic properties results in higher stress values, which are likely to be found concentrated along major axes of stiffness (Daniel and McHenry, 2000). Loading conditions. To measure stress distribution, we used a standard bite force of 5,000 N, which is the expected bite force of a 2.5-m long individual of A. mississippiensis (based on Erickson et al., 2003). This estimate is conservative, because the body size of modern adult crocodiles ranges from 1 to 7 m (Cogger and Zweifel, 2003). Furthermore, the scale of force is of little consequence, as we are not trying to determine the absolute value of stress in the models, only the relative differences in stress between various skull shapes when size is held constant. Bite force was applied at the premaxilla-maxilla contact, instead of a specific tooth or set of teeth, as this is an easily identifiable point on each model. Muscular forces were excluded. Extant crocodilians use several feeding behaviors to catch, subdue, and process prey (Busbey, 1995; McHenry et al., 2006). These behaviors include the following: 1) simple biting, or jaw adduction, which loads the skull in a dorsoventral plane; 2) head shaking, or pitching and yawing, which loads the skull in a mediolateral and dorsoventral plane; and 3) rolling, or twist feeding, which generates torsional loads within the skull. Therefore, FEAs were carried out on three separate load cases for each of the four extreme PC models and the models of all 23 species to quantify stress response in relation to these behavioral loading conditions: Load Case 1 (LC 1): a bilateral bite at the left and right premaxilla-maxilla suture with 2,500 N applied in the z-direction to each side to bend the skull dorsally; Load Case 2 (LC 2): a unilateral bite at the left premaxilla-maxilla suture with 5,000 N applied in the z-direction to induce bending and superimposed torsional loading; and Load Case 3 (LC 3): lateral loading at the left premaxilla-maxilla suture with 5,000 N applied in the y-direction to generate a withinplane lateral bend to the snout. These three load cases are designed to reflect dorsoventral, torsional, and mediolateral loads, respectively. Distribution and magnitude of Von Mises stress were recorded for each model and load case. Von Mises stress is a good predictor of failure for materials such as bone (see Dumont et al., 2005). It is a scalar function of principal stresses 1 3, which is directly proportional to the strain energy of distortion and also mathematically related to the maximum shear stress. Hence it was chosen as an encompassing metric of skull strength (here considered a measure of mechanical performance) in our FE-models. To avoid artificial noise created by fixing the quadrates, stress values along the snout (midpoint of snout; midpoint of snout from bite; and anterior to orbit) of each model were also calculated. Quantifying the Relationship Between Shape and Causal Variables Nonphylogenetic. The degree of concordance between shape variation and the causal variables of interest was assessed first by using raw data, i.e., without taking the possible effects of phylogeny into account. Linear regressions of skull strength and hydrodynamic efficiency versus significant principle component scores were conducted to ascertain whether function and shape were related. In addition, we ran a MANOVA between the factor geography and significant principle component scores, as well as univariate ANOVAs with post hoc Tukey
846 S.E. PIERCE ET AL. Fig. 3. Plot of principle components 1 and 2 showing the location of all 126 specimens. Isisfordia duncani is an extinct taxon used as an outgroup in subsequent analyses. Extreme shapes along each axis are shown at the end points of the axes. Species falling into one of the four quadrants of morphospace were classified as members of a specified morphotype used for further statistical tests (see text for details). Deformation grids indicate the shape change necessary to transform the mean specimen in the data set into the extreme shape. HSD tests on the factor morphotype (a categorical variable with four states created by analyzing the distribution of species within quadrants of morphospace: long/broad [quadrant one] short/broad [quadrant two], short/narrow [quadrant three], long/narrow [quadrant four]) and our functional variables to determine if there was a relationship between shape and geographic occurrence, or between morphotype and function. We chose to use this pattern of subdivision because it is easily repeatable and obvious from an inspection of the PC plots. Skull strength was determined by conducting a finite element analysis as described earlier. Hydrodynamic efficiency was determined as the drag moment, which is a measure of the resistance encountered when rotating an object about a fulcrum through a fluid medium (in this case, when a crocodile sweeps its snout sideways through water by flexing the head about the cranial-cervical joint), and was taken from McHenry et al. (2006, Fig. 8). Drag data were only available for a 14-species subset and the PC scores were adjusted accordingly. Phylogenetic. In addition to the analyses run on the raw data, we also carried out tests on a data set that had been adjusted to account for the possible effects of phylogeny to ensure that any apparent patterns were not artifacts caused by phylogenetic relationship among the species of interest. Phylogenetically independent contrasts (PIC) (Felsenstein, 1985) were calculated for two sets of phylogenetic trees, one including all 23 crocodilian species and another including the 14-species subset used to analyze drag. Each set of trees consisted of two topologies with associated branch lengths based on the character matrices of Gatesy et al. (2004): 1) topology and branch lengths based on the morphology-only character matrix, and 2) topology and branch lengths based on the supermatrix (morphology and molecular) character matrix. Branch lengths for both trees in the 14 and 23 taxa data sets were calculated using parsimony and ACCTRAN optimization in PAUP* 4.10b10 (Swofford, 2001). Unresolved branches were given extremely small, non-zero branch lengths, which is functionally equivalent to making the polytomies hard. For the 23-species tree, characters were optimized on the tree directly, whereas for the 14- species subset tree, taxa were pruned and excluded, and the characters were optimized on the reduced tree. PICs were subsequently calculated for significant principle component scores, average Von Mises stress along the snout during a bilateral bite and lateral force, and drag moment (14 species subset only) using Compare version 4.6b (Martins, 2005). An investigation into the correspondence between total shape, as described by the complete set of partial warp and uniform component scores, and function was not possible because of low sample size (n 5 23) as compared with the number of landmark coordinates. Regressions through the origin were performed on the PICs using Systat version 12.00.09. RESULTS Geometric Morphometrics Principal components analysis. The PCA carried out on the partial warp and uniform component scores of all 23 modern crocodilians shows that much of the total shape variance is captured by the first two axes, with over 76% of the variance explained (see Fig. 3). However, there is a
CROCODILE SKULL SHAPE AND BIOMECHANICS 847 Fig. 4. Principle components 1 and 2 with convex polygons encircling (A) the three major extant crocodilian families; and (B) the eight major extant crocodilian genera. These taxonomic groupings correspond closely to monophyletic clades of crocodilians (see Fig. 1). A, Alligatoridae; a, Alligator; C, Crocodylidae; c, Caiman; cr, Crocodylus; G, Gavialidae; g, Gavialis; i, Isisfordia; m, Melanosuchus; o,osteolaemus; p,paleosuchus; t,tomistoma. dominant trend, PC1, which explains 66.2% of the total variance. PC1 and PC2 were the only PC axes judged significant using the procedure described earlier, and we will focus on these for the remainder of the article. The multivariate regression reveals that shape is significantly correlated with centroid size (Wilk s L 5 0.117, P < 0.0001; Goodalls F 5 24.75, P < 0.0001), although the regression model only explains 16.5% of the variance in the data set. Regression of the first two significant PCs on centroid size reveals that the shape variation described by these axes is also significantly related to size (P 5 0.002). Taken together, the regression results indicate that some allometric differences exist among crocodile species, but that allometry is not the only, or main, source of shape variation in the data set. Further analysis of the allometric differences, particularly in the context of ontogenetic shape changes within individual species, likely represents a fruitful area
848 S.E. PIERCE ET AL. TABLE 1. Partial disparity and percentage of total disparity for each extant crocodilian family and genus within morphospace TABLE 2. Pairwise comparisons of partial disparities of family and genus using two-sample t-tests Taxon Partial disparity Percentage Taxonomic pair P-value Alligatoridae 0.00443 23 Crocodylidae 0.0057 30 Gavialidae 0.00889 47 Total 0.01912 100 Alligator 0.00093 8.24 Caiman 0.00131 11.61 Crocodylus 0.00187 16.58 Gavialis 0.004 35.46 Melanosuchus 0.00066 5.85 Osteolaemus 0.00067 5.94 Paleosuchus 0.00022 1.95 Tomistoma 0.00162 14.36 Total 0.01128 100 Alligatoridae/Crocodylidae 0.089 Alligatoridae/Gavialidae <0.001 Crocodylidae/Gavialidae 0.003 Alligator/Caiman 0.002 Alligator/Crocodylus 0.001 Alligator/Gavialis <0.001 Alligator/Melanosuchus 0.335 Alligator/Osteolaemus 0.462 Alligator/Paleosuchus <0.001 Alligator/Tomistoma <0.001 Caiman/Crocodylus 0.072 Caiman/Gavialis <0.001 Caiman/Melanosuchus 0.001 Caiman/Osteolaemus 0.002 Caiman/Paleosuchus <0.001 Caiman/Tomistoma 0.196 Crocodylus/Gavialis <0.001 Crocodylus/Melanosuchus 0.002 Crocodylus/Osteolaemus 0.002 Crocodylus/Paleosuchus 0.002 Crocodylus/Tomistoma 0.643 Gavialis/Melanosuchus <0.001 Gavialis/Osteolaemus <0.001 Gavialis/Paleosuchus <0.001 Gavialis/Tomistoma <0.001 Melanosuchus/Osteolaemus 0.983 Melanosuchus/Paleosuchus <0.001 Melanosuchus/Tomistoma <0.001 Osteolaemus/Paleosuchus 0.002 Osteolaemus/Tomistoma <0.001 Paleosuchus/Tomistoma <0.001 Significance assessed using a Bonferroni-corrected a level of 0.017 for family and 0.002 for genus. Significant differences are highlighted in bold. for further research, although such analyses are beyond the scope of this study. The first PC axis describes variation in the length of the snout (especially the length of the maxilla), size of the supratemporal fenestrae, and orientation of the skull lateral to the orbits (see Fig. 3). In contrast, the second PC describes variation in width of the snout, length of the nasal bones, and size of the orbits (see Fig. 3). As such, PC1 mainly discriminates short- and long-snouted morphotypes, whereas PC2 discriminates broad- and narrow-snouted morphotypes. The scatter plot demonstrates that modern crocodilians fall within all four quadrants of morphospace, but that the morphospace is not uniformly occupied. Instead, the specimens group into two clusters: one large primary cluster that incorporates the long/narrow, short/narrow, and short/broad morphotypes, and one smaller secondary cluster at the extreme of the long/broad morphotype (see Fig. 3). These clusters are roughly parallel to one another, but are separated by a distinct area of unoccupied morphospace. The four skull morphotypes corresponding to quadrants of morphospace (long/narrow, short/narrow, long/broad, and short/broad) will be used to categorize crocodilians throughout the rest of the article to clarify the evaluation of shape variation and to discuss mechanical performance. Phylogeny and disparity. The distribution of families and genera within morphospace closely reflects our understanding of extant crocodilian relationships and the taxonomy of the clade. With respect to specific familial patterns (Fig. 4A), Gavialidae forms the entire secondary cluster in the extreme of the long/broad morphotype; Crocodylidae encompasses the majority of the primary cluster and includes all three morphotypes encompassed by that cluster; and Alligatoridae extensively overlaps the crocodilids in the primary cluster, but is restricted to the region of morphospace corresponding to the short/narrow and short/broad morphotypes. Results from the disparity tests show (Table 1) that the position of the Gavialidae is responsible for almost half (47%) of the total morphological disparity observed within the modern crocodile families, whereas the positions of the Crocodylidae and Alligatoridae contribute smaller, and almost equal amounts (30 and 23%, respectively). This result is somewhat counterintuitive, given that alligatorids and crocodilids occupy larger areas of morphospace than the gavialids, but it reflects the fact that the relatively small cluster of gavialids is located much farther from the grand mean of morphospace. The t- tests (Table 2) confirm that the partial disparity of Gavialidae is significantly greater than the partial disparities of the Crocodylidae and Alligatoridae. Conversely, the partial disparities of Crocodylidae and Alligatoridae did not differ significantly. In terms of generic patterns (Fig. 4B), Gavialis sits on its own reflecting the unique morphology of the clade; Tomistoma is part of the primary cluster and represents the furthest penetration of this cluster into the long/narrow quadrant; Osteolaemus diverges from crocodylid morphospace and converges on an alligator-like morphology; Crocodylus sits centrally and makes up the majority of the primary cluster reflecting the large variability in skull morphology displayed by its species; Alligator is positioned in an extreme position in the short/narrow quadrant; Paleosuchus overlaps with
CROCODILE SKULL SHAPE AND BIOMECHANICS 849 TABLE 3. Results of the Mantel tests examining the correlation between phylogenetic distance and partial Procrustes distance using the equal distance method and the character distance method Distance method R R 2 Permutations P-value Node 5 1 0.1095 0.012 5,000 0.094 Morphology 0.5158 0.266 5,000 <0.0001 Supermatrix 0.0376 0.001 5,000 0.39 A significant relationship is indicated by high values of R 2 and low values of P. Significant correlations (P < 0.05) are indicated in bold. Fig. 5. A: Plot of the mean skull shape for each of the 23 extant crocodilian species. B: A species-level phylogeny mapped onto morphospace. Numbers correspond to the nodes on Figure 1. Alligator, but lies within the Crocodylus range; Caiman occupies an extreme position in the short/ broad quadrant; and, finally, Melanosuchus extends between Caiman and Alligator signifying a divergence from the typical caiman-like morphotype. Results from the disparity tests show (Table 1) that the position of Gavialis is responsible for over one-third (35.46%) of the total morphological disparity, with Crocodylus, Tomistoma, and Caiman contributing successively lesser amounts (16.58, 14.36, and 11.61%, respectively). The remaining genera contribute very small amounts (<8% each) to the total morphological disparity of modern crocodilian genera. The t-tests (Table 2) confirm that the partial disparity of Gavialis is significantly greater than those of all other genera. They also highlight the fact that the partial disparities of Crocodylus, Tomistoma, and Caiman are not significantly different from each other, but that they are significantly different from all the remaining genera. Projecting a species-level phylogeny onto morphospace (see Fig. 5) results in multiple intersections of branches. This pattern stems from the convergent nature of the skull morphologies of Osteolaemus and alligatorids, and among species of Crocodylus, specifically the long-snouted forms C. johnstoni, C. acutus, andc. intermedius. Osteolaemus is a blunt-snouted crocodilid that sits in the alligatorid realm of morphospace, whereas the longsnouted Crocodylus species lie within two major unresolved polytomies (see Fig. 1) creating star-like patterns when the phylogeny is plotted in morphospace. In addition to these species, the two species of Paleosuchus are somewhat divergent within the morphospace (P. trigonatus is more crocodile-like and P. palpebrosus is more caiman-like ), and M. niger departs from the caiman area of the morphospace, instead sitting closer to Alligator. Finally, the position of the outgroup Isisfordia is noteworthy. It is located within the range of modern crocodilians, specifically in the long/narrow morphotype, and not in a distinct region. If the skull shape of Isisfordia is an accurate representation of shape of the common ancestor of all extant crocodilians, then the long/narrow morphotype would be the basal condition for the clade. Significant matrix correlations, as measured by the Mantel test, indicate a correspondence between partial Procrustes distance and phylogenetic distance (Table 3) when the morphology-based character matrix is used to scale the phylogenetic distance. In other words, the phylogenetic distances between species closely match their proximities in the morphospace. However, when phylogenetic distances are based on the equal distance method or are scaled according to the supermatrix-based character matrix there is no significant relationship between phylogeny and morphospace occupation. Although there is a significant correlation between a morphology-based species level phylogeny and morphospace occupation, phylogeny only accounts for 26% of the variation in skull morphology within modern crocodilians (Table 3). Biogeography. If we project the different geographical areas onto the morphospace (see Fig. 6),
850 S.E. PIERCE ET AL. Fig. 6. Principal components 1 and 2 with convex polygons encircling groups of sympatric crocodilians. Af, Africa; As, Asia; NA, North America; SA, South America; SE As, Southeast Asia. TABLE 4. Partial disparity and percentage of total disparity for each geographic region within morphospace Taxon Partial disparity Percentage Africa 0.00134 13 Asia 0.00418 40 Southeast Asia 0.00176 17 South America 0.00238 23 North America 0.00069 7 Total 0.01035 100 TABLE 5. Pairwise comparisons of partial disparities of geographic regions using two-sample t-tests Geographic pair P-value Taxonomic pair P-value Africa/Asia <0.001 Asia/SA <0.001 Africa/SE As 0.150 Asia/NA <0.001 Africa/SA 0.003 SE As/SA 0.006 Africa/NA 0.098 SE As/NA <0.001 Asia/SE As <0.001 SA/NA <0.001 Significance assessed using a Bonferroni-corrected a level of 0.005, indicated in bold. See Fig. 6 for abbreviations. considerable overlap is apparent, but some interesting trends are also visible. First, all geographic areas incorporate species from three quadrants of morphospace. Second, Asia has a completely different trend than the other geographical areas, primarily because of the presence of Gavialis. Third,South America and Southeast Asia are mirror images of each other along the trend of the primary cluster, with South America trending toward the extreme short/broad morphotype and Southeast Asia trending toward the extreme long/narrow morphotype. Finally, Africa stretches along the entire length of the primary cluster and North America sits centrally. With respect to disparity, we see that Asia is responsible for the majority of morphological variation with 40% of the total disparity, followed by South America at 23%, and Southeast Asia, Africa, and North America at successively lesser amounts (Table 4). In addition, the partial disparity of Asia is significantly different from all other geographic regions and that of South America is significantly different from all the remaining geographic regions except Southeast Asia (Table 5). Finally, the partial disparities of Africa/Southeast Asia, Africa/North
CROCODILE SKULL SHAPE AND BIOMECHANICS 851 America, and Southeast Asia/South America are not significantly different from each other (Table 5). There may be some relationship between the number of species in a geographical area and partial disparity in these results (i.e., areas with more species may be more likely to have a greater amount of morphological variation), but this relationship does not completely account for the results. For example, Asia has one less species than North America, but possesses much higher partial disparity because one of its species (G. gangeticus) has a highly divergent morphology. Finite Element Analysis Peak stress PC endpoints. Color-coded stress distribution plots for the PC end points (PC1 5 20.15 and 0.22; PC2 5 20.10 and 0.06) illustrate the pattern of peak stress in the skull during bilateral biting (Fig. 7A), unilateral biting (Fig. 7B), and lateral loading of the snout (Fig. 7C). There is not a noticeable difference between the bilateral and unilateral models. As such, the unilateral bite does not induce a marked torsional response in our 2D models although there is slight asymmetry in the stress patterning. Furthermore, the pattern of stress distribution is very similar between the four PC endpoint models (within the confines of each loading condition), although the magnitude of stress changes. Stress patterns during a bilateral and unilateral bite (Fig. 7A,B) suggest that stress increases through the skull in a posterior direction during biting, peaking around the orbits and temporal fenestrae (if we ignore the erroneous localized peak at the quadrates that is caused by their proximity to the anchoring constraints). Examination of the stress magnitudes (along the midline) along the snout during a bilateral bite (Fig. 8A; see Appendix B) shows that PC1neg is the strongest skull morphotype with a maximum snout stress of 30,900 Pa; this is followed by PC2neg (48,300 Pa), PC2pos (55,400 Pa), and finally PC1pos (178,000 Pa). The difference in maximal peak stress is not substantial between PC1neg, PC2pos, and PC2neg, but it is 3 6 times greater in the PC1pos morphotype. Conversely, during lateral loading of the snout (Fig. 7C), the pattern of stress distribution increases posteriorly along the lateral margin of the skull, peaking lateral to the obits and infratemporal fenestrae (approximately at the level of the jugal), and then moving medially and anteriorly to surround the orbits and infratemporal fenestrae. Examination of the stress magnitudes along the lateral margin of the snout during lateral loading (Fig. 8B; see Appendix B) shows that stress gradually increases through the snout in all morphotypes except PC1pos, which shows a marked decrease in stress values. PC1neg is the strongest skull morphotype with a maximum stress value through the snout being 432 Pa; this is followed by PC2neg (1,090 Pa), PC2pos (1,760 Pa), and PC1pos (13,500 Pa). Again, the difference in maximal peak stress is not substantial between PC1neg, PC2pos, and PC2neg, but it is 8 31 times greater in the PC1pos morphotype. Peak stress species. Calculating stress through the snouts of all 23 species during a bilateral bite and the application of a lateral force (Fig. 9; see Appendix C) shows that C. latirostris has the strongest skull shape and T. schlegelii has the weakest. All species increase in stress posteriorly along the snout, except for G. gangeticus, which decreases in stress during lateral loading. Although the majority of species maintain their ranking on the stress spectrum for all the different load types, a few species increase or decrease in rank under differing load conditions. For example, during a bilateral bite A. mississippiensis, A. sinensis, and C. siamensis experience lower stress values posteriorly through the snout, whereas C. crocodylus, C. yacare, and P. trigonatus experience greater stresses. During lateral loading, A. mississippiensis and O. tetraspis experience lower stress values posteriorly through the snout, whereas C. crocodylus, C. moreletii, C. yacare, and P. trigonatus experience greater stresses. Taking the average stress value along the snout and ranking each species (Table 6) demonstrates a close correspondence between skull strength and the distribution of species within morphospace. The short/broad morphotype is the strongest with an average rank of 6.5 6.7 out of 23; this is followed by the short/narrow morphotype (avg. rank 5 11.5 11.83/23), the long/narrow morphotype (avg. rank 5 19.67 19.83/23), and the long/broad morphotype (avg. rank 5 21 22/23). Although we might predict that the long/broad morphotype is stronger than the long/narrow morphotype (based on the analysis of the PC end points), it is actually the weakest quadrant of morphospace. This, however, is the result of being occupied by only one species, G. gangeticus, which sits at the extreme limits of this quadrant. In addition, A. mississippiensis and A. sinensis rank much higher than other species occupying the same quadrant of morphospace and, as a result, separate the short/broad morphotype into two clusters. Shape and Causal Variables Regressions of the raw data PC scores against stress and drag (Table 7) show a strong and significant linear relationship between snout length (PC1) and bilateral stress, lateral stress, and drag (r 2 5 0.781; P < 0.0001). With respect to snout width (PC2), bilateral stress and drag show a nonsignificant relationship, but lateral stress shows a significant relationship (P 5 0.043) (Table 7). However, if PC1 is removed from the regression, PC2 no longer has a significant effect on lateral stress.
852 S.E. PIERCE ET AL. Fig. 7. Von Mises stress magnitude and distribution in crocodile skulls represented by the extreme shapes along PC1 (20.15 and 0.22) and PC2 (20.10 and 0.06). A: LC 1, bilateral bite. B: LC 2, unilateral bite. C: LC 3, lateral load to the snout. Units are Pa. See text for details. A similar result is found when the raw data are corrected for phylogeny using independent contrasts (Table 8): snout length (PC1) shows a strong and significant linear relationship with function, whereas snout width (PC2) shows no relationship. Re-running these analyses using Pearson productmoment correlation tests (parametric) or Kendall s s (nonparametric rank-order correlation test)