A direct approach to the analysis of systems with single, or commensurate delays is presented, and compared with other recently proposed methods. The finite polynomials arising in this direct method are shown to have useful sensitivity properties. Remarks are given concerning systems stable independent of delay.

The end-Devonian to mid-Mississippian time interval has long been known for its depauperate palaeontological record, especially for tetrapods. This interval encapsulates the time of increasing terrestriality among tetrapods, but only two Tournaisian localities previously produced tetrapod fossils. Here we describe five new Tournaisian tetrapods (Perittodus apsconditus, Koilops herma, Ossirarus kierani, Diploradus austiumensis and Aytonerpeton microps) from two localities in their environmental context. A phylogenetic analysis retrieved three taxa as stem tetrapods, interspersed among Devonian and Carboniferous forms, and two as stem amphibians, suggesting a deep split among crown tetrapods. We also illustrate new tetrapod specimens from these and additional localities in the Scottish Borders region. The new taxa and specimens suggest that tetrapod diversification was well established by the Tournaisian. Sedimentary evidence indicates that the tetrapod fossils are usually associated with sandy siltstones overlying wetland palaeosols. Tetrapods were probably living on vegetated surfaces that were subsequently flooded. We show that atmospheric oxygen levels were stable across the Devonian/Carboniferous boundary, and did not inhibit the evolution of terrestriality. This wealth of tetrapods from Tournaisian localities highlights the potential for discoveries elsewhere.

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