DISCUSSION

In both Pacycephalosaurus and Homalocephale, maximum von Mises and compressive stress fall below the 300 MPa yield stresses of bone (Table 1), even in high-force collisions. Based on this criterion for head-butting suitability, the high safety factors corroborate our hypothesis that the domes could withstand stresses of combat in these pachycephalosaurs. Notably, the highest rate of deceleration occurs in Homalocephale at the highest tested closing speed and shortest deceleration distance (approximately 76 g). This rate is within the range at which colliding American football players experience concussions (94 +/- 28 g: Viano et al. 2007) and suggests that the animals would have preferred lower collision speeds. We conclude that cranial mechanics of these animals permitted high-energy, head-to-head combat, and that ultimate collision speeds were constrained by other factors.

As with other analyses of complex behavior in fossil animals, we can test pachycephalosaurs' capability for head-butting and even characterize their potential performance. However, whereas biomechanics might support or contradict hypotheses of capability, our ability to test hypotheses of specific behaviors will remain asymptotic without trace evidence. Pathologies are common on the dorsal surfaces of pachycephalosaur domes (University of Alberta and Royal Tyrrell Museum specimens, P. Bell and E. Snively, unpublished data) and are conceivably the result of impact trauma, but may also be age-related. Yet the domes of some pachycephalosaurs are an unusual case, in which a feature not seen in extant vertebrates is uniquely suitable for a postulated behavior (in this case cranial impacts against resistant structures). By cranial and post-cranial evidence (Galton 1971, Maryánska and Osmólska 1974, Sues 1978), our finite element results, and analogy with intraspecific combat in modern forms, we find inference of head-based combat in some pachycephalosaurs to be reasonable and compelling.

Whereas our conclusions about the possibility of pachycephalosaur head-butting differ from those of Goodwin and Horner (2004), our results parallel their caution about the compression resistance role of trabeculae. Stress would diminish greatly before it reached the cancellous zone II within some pachycephalosaur domes. However, a lower density and elastic modulus ensured that zone II would experience higher compressive strain than surrounding zones, in which compact bone predominates. (This general result is predictable from material properties, but without FEA our deductive omniscience fails for postulating finer patterns of stress and strain.) These results indicate that trabeculae angled relative to the compressive force would experience some elastic deformation and recoil, as originally proposed by Alexander (1989) but at lower strain magnitudes.

Our phylogenetic analysis suggests escalating adaptation towards head-butting in some but not all lineages of pachycephalosaurs. The domes most suitable for head-butting co-occur with tubercular ornamentation in Tylocephale, Prenocephale, and Pachycephalosaurus. This association suggests that species differentiation was not dependent solely on the presence of a dome in these taxa, and that the dome had a function in addition to its species recognition role. FEA confirms the domes' suitability for hypothesized intraspecific combat. However, crania became less adept for high-energy head-butting in Stygimoloch (Goodwin et al. 1998) and Dracorex, taxa with the most outlandish tubercular ornamentation (Bakker et al. 2006). This evident decrease in performance conforms to suggestions of Goodwin et al. (1998) and Maryánska et al. (2004) that cranial thickenings and ornamentation in these animals evolved through flank- and head-butting adaptations to apomorphic display functions, in parallel with the evolution of horn and antler morphologies postulated for artiodactyls (Geist 1966). If the phylogeny here is correct, we suggest that head-butting increased in selective importance among high-domed forms, but that this function decreased in importance in at least one lineage of derived pachycephalosaurs.

Alternatively, strong evidence that low-domed specimens were subadults of contemporaneous high-domed taxa (Horner et al. 2007, Sullivan 2007) suggests an ontogeny of agonism for pachycephalosaurines. Such an ontogenetic scenario mirrors Geist's (1966) evolutionary scenario for artiodactyl agonism, and consolidates the current biomechanical results with the histological findings of Goodwin and Horner (2004). Flat-headed subadults of Pachycephalosaurus would have engaged in shoving and flank-butting matches, and display with enlarged cranial ornamentation. During a stage of rapid growth of the dome, the resilient and easily remodeled trabecular zone would have facilitated head-butting as suggested by mechanical results in this study. Finally, large adults would rely on intimidation, display, and flank-butting, as the dome became massive and its acellular histology (Goodwin and Horner 2004) reduced its capacity for remodeling and repair.

Both evolutionary and ontogenetic scenarios are predictive and have contradictable components. For example, impact pathologies (if they can be so identified) are predicted as more commonly evident in high- than low-domed specimens, in individuals of moderate size. Further sensitivity analyses may reveal a range of soft-tissue morphologies that would enable or prohibit head-butting in large pachycephalosaurs, with keratin acting to protect the acellular osseous dome. Whereas we cannot envision reconstructing the integument with known accuracy, future workers may devise methods that surmount this obstacle and enable tests for head-butting at respective stages in each scenario.

The current study analyzes mechanical response of simplified models of pachycephalosaur domes, at opposing poles of size and morphology for the clade. Refined models of Homalocephale and Pachycephalosaurus domes collisions (especially dynamic models) would better approximate in vivo stresses and strains under putative loadings, but are unlikely to shift the relative performance of their respective thickened crania. Yet our Homalocephale model in particular is greatly simplified, and results for it should be viewed with caution. Ideally, 3D finite element models of entire skulls (Rayfield 2007) of pachycephalosaurs, based on CT scans with material properties derived from X-ray attenuation data (Wroe et al. 2007), will be constructed across ontogenetic and phylogenetic spectra of these animals. Collision velocities based on quantitatively estimated speeds for pachycephalosaurs would improve upon the arbitrary velocities chosen here.

As noted above, the head-butting hypothesis is falsifiable if the keratinous covering of the dome can be established as precluding such combat. If surface vascularization suggests differential keratinous growth atop different regions of the dome surface, the resulting shapes may have prevented dome-dome contact.

Regardless of the shape of soft tissue covering the domes of pachycephalosaurs, species recognition (Goodwin and Horner 2004) is compatible with combat behavior as a function of their thickened crania and must remain a primary hypothesis (without intraspecific recognition, such combat would be less likely). However, unless research on pachycephalosaurian integument and postcrania indicate otherwise, the present mechanical results uphold the possibility of head-butting behavior.