For AARs, we relied on the dispersal–extinction–cladogenesis (DEC) model, implemented in Lagrange (Ree et al., 2005; Ree & Smith, 2008). It uses the information contained in genetic branch lengths and allows the incorporation of changing dispersal probabilities across areas and time. We devised two time-slice models, one with bins of 0–30, 30–90 and 90–150 Ma, and the other with bins of 0–30, 30–60, 60–90 and 90–150 Ma (Table S3). The oldest bin captures the plate tectonic situation before the breakup of West Gondwana, and the youngest the period during which the Central American land bridge and South-East Asia formed. The middle bins tried to capture connectivity via the North Atlantic land bridge and Antarctica, and differed in the way in which India connects to Eurasia (Table S3). Our nine operational geographical units were Eurasia (A), Africa (B), Madagascar (C), South-East Asia and India (D), Australia (E), North and Central America (F), South America (G) and Antarctica (H). To accommodate the mostly globally distributed water-associated and free-floating taxa, we created a ninth category, ‘water-associated’ (I), assigned to the marine Alismatales families Cymodoceaceae, Posidoniaceae, Ruppiaceae and Zosteraceae, the freshwater aquatics Alismataceae, Aponogetonaceae and Potamogetonaceae, plants of marshy coastal habitats (Juncaginaceae), and free-floating Araceae (Lemnoideae, Pistia). Remusatia was coded as present only in area D because most of its species occur in the Himalayan foothills and the Western Ghats of India; the widespread R. vivipara, occurring in Africa, Asia and Australia, is especially adapted to bird dispersal. Similarly, Pothos, occurring on Madagascar with one widespread species (P. scandens), was only coded for areas A, D and E, where it has several species, and Sauromatum was only coded for areas A and D, although it also has one species (S. venosum) ranging from Africa to tropical China. Spathiphyllum was coded for South America and Central North America, where 44 species occur, but not for South-East Asia, where S. commutatum and S. solomonense occur on the Philippines and New Guinea; its monospecific sister group, Holochlamys, is endemic on New Guinea. In the absence of molecular phylogenetic evidence on the relationships of the two Asian Spathiphyllum species, we felt it unwise to code this genus as present in Asia because its two Asian species may turn out instead to belong to Holochlamys.
To create a tree that included all families of Alismatales (the order to which the Araceae family belongs), we manually added one representative per family to a newick tree file obtained from BEAST. The topology and divergence times were constrained to match the results of the large monocot chronogram of Janssen & Bremer (2004). The enlarged tree had 145 tips (132 Araceae and 13 outgroups) and became the input tree for Lagrange. To create Python script input files (with the tree of choice and the area connectivity probability matrices; Table S3), we used the Lagrange online configurator tool. Ancestral areas were limited to maximally two, and a relative probability of > 66.6% was considered to be strong support for an ancestral range scenario.
To integrate fossil ranges into the reconstructions, we added them manually in the newick chronogram (as performed for the Alismatales outgroups, above). Each fossil was inserted along the stem lineage of the group to which it had been assigned, with its age determining where it was placed. In addition, each fossil was given either a short branch length (1 Ma), simulating an extinct range, or a long branch length (the fossil’s age), simulating a range occupied for a long time (until today). When fossils are given short branch lengths, the DEC model, which considers branch lengths as proportional to time, will treat any range shifts indicated by their geographical occurrences as evidence for rapid geographical change (and, conversely, for long branch lengths). One of the seven fossils used for clock calibration, Limnobiophyllum scutatum, was not used in the AAR because it was too young to be assigned to the relevant stem lineage, whereas three others that had not been used as constraints were added because they contributed geographical information (Lysichiton austriacus, Orontium mackeii and Keratosperma allenbyense; Table S2).