Description: Cases of convergent evolution, particularly within ecomorphological contexts, are instructive in identifying universally adaptive morphological features across clades. Tracing of evolutionary pathways by which ecomorphological convergence takes place can further reveal mechanisms of adaptation, which may be strongly influenced by phylogeny. Ecomorphologies of carnivorous mammals represent some of the most outstanding cases of convergent evolution in the Cenozoic radiation of mammals. This study examined patterns of cranial shape change in the dog (Canidae) and hyena (Hyaenidae) families, in order to compare the evolutionary pathways that led to the independent specialization of bone-cracking hypercarnivores within each clade. Geometric morphometrics analyses of cranial shape in fossil hyaenids and borophagine canids provided evidence for deep-time convergence in morphological pathways toward the independent evolution of derived bone-crackers. Both clades contained stem members with plesiomorphic generalist/omnivore cranial shapes, which evolved into doglike species along parallel pathways of shape change. The evolution of specialized bone-crackers from these doglike forms, however, continued under the constraint of a full cheek dentition and restriction on rostrum length reduction in canids, but not hyaenids. Functionally, phylogenetic constraint may have limited borophagine canids to crack bones principally with their carnassial instead of the third premolar as in hyaenids, but other cranial shape changes associated with durophagy nevertheless evolved in parallel in the two lineages. Size allometry was not a major factor in cranial shape evolution in either lineage, supporting the interpretation of functional demands as drivers for the observed convergence. The comparison between borophagines and hyaenids showed that differential effects of alternative functional "solutions" that arise during morphological evolution may be multiplied with processes of the "macroevolutionary ratchet" already in place to further limit the evolutionary pathways available to specialized lineages.

Description: In this paper, we extend the deterministic single-group MSIRS epidemic model to a multi-group model, and we also extend the deterministic multi-group framework to a stochastic one and formulate it as a stochastic differential equation. In the deterministic multi-group model, the basic reproduction number R0 is a threshold that completely determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show that if R0〉1, then the disease will prevail, the infective condition persists and the endemic state is asymptotically stable in a feasible region. If R0⩽1, then the infective condition disappears and the disease dies out. For the stochastic version, we perform a detailed analysis on the asymptotic behavior of the stochastic model, which also depends on the value of R0, when R0〉1, we determine the asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time-averaged data. Numerical methods are used to illustrate the dynamic behavior of the model and to solve the systems.