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The mountain bumblebees of the subgenus Alpigenobombus Skorikov, 1914, are uniquely distinctive because the females have enlarged mandibles with six large, evenly spaced teeth, which they use to bite holes in long-corolla flowers for nectar robbing. Recognition of species in this subgenus has been uncertain, with names used in various combinations. To revise the species, we examined COI-like barcodes for evidence of species’ gene coalescents using MrBayes and PTP and we compare the coalescent groups with morphological variation for integrative assessment. While we seek to include only orthologous barcodes (the ‘good’) and exclude all of the more strongly divergent barcode-like numts (the ‘bad’), for some nominal taxa only low-divergence numts could be obtained (the ‘ugly’). For taxa with no orthologous sequences available, using a minimum number of the lowest divergence numts did yield coalescent candidates for species that were consistent with morphologically diagnosable groups. These results agree in recognising 11 species within this subgenus, supporting: (1) recognising the widespread European Bombus mastrucatus Gerstaecker, 1869 stat. rev. as a species separate from the west Asian B. wurflenii Radoszkowski, 1860 s. str.; (2) the recently recognised B. rainai Williams, 2022, as a species separate from B. kashmirensis Friese, 1909, within the western Himalaya; (3) the recognition once again of B. sikkimi Friese, 1918 stat. rev. and B. validus Friese, 1905 stat. rev. as species separate from B. nobilis Friese, 1905 s. str. within the eastern Himalaya and Hengduan regions; (4) confirming the recognition of B. angustus Chiu, 1948, B. breviceps Smith, 1852 s. lat., B. genalis Friese, 1918, and B. grahami (Frison, 1933) as separate species within the Himalaya, China, and Southeast Asia; (5) recognising the conspecificity of the nominal taxa (not species) channicus Gribodo, 1892 (Southeast Asia) and dentatus Handlirsch, 1888 (Himalaya) as parts of the species B. breviceps s. lat. (southern and eastern China); and (6) recognising the conspecificity of the rare taxon beresovskii (Skorikov, 1933) syn. n. as part of the species B. grahami within China. Nectar robbing by bumblebees is reviewed briefly and prospects for future research discussed.
The hypnorum-complex of bumblebees (in the genus Bombus Latreille, 1802) has been interpreted as consisting of a single widespread Old-World species, Bombus hypnorum (Linnaeus, 1758) s. lat., and its closely similar sister species in the New World, B. perplexus Cresson, 1863. We examined barcodes for evidence of species’ gene coalescents within this species complex, using the closely related vagans-group to help calibrate Poisson-tree-process models to a level of branching appropriate for discovering species. The results support seven candidate species within the hypnorum-complex (Bombus taiwanensis Williams, Sung, Lin & Lu, 2022, B. wolongensis Williams, Ren & Xie sp. nov., B. bryorum Richards, 1930, B. hypnorum, B. koropokkrus Sakagami & Ishikawa, 1972, and B. hengduanensis Williams, Ren & Xie sp. nov., plus B. perplexus), which are comparable in status to the currently accepted species of the vagans-group. Morphological corroboration of the coalescent candidate species is subtle but supports the gene coalescents if these candidates are considered near-cryptic species.
We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $\alpha$-stable branching mechanisms. The random ancestral partition is then a time-changed $\Lambda$-coalescent, where $\Lambda$ is the Beta-distribution with parameters $2-\alpha$ and $\alpha$, and the time change is given by $Z^{1-\alpha}$, where $Z$ is the total population size. For $\alpha = 2$ (Feller's branching diffusion) and $\Lambda = \delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For $\alpha =1$ and $\Lambda$ the uniform distribution on $[0,1]$, this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen-Sznitman coalescent.
We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly & Kurtz (1999); the other is based on direct calculations with generators.
In this paper we prove asymptotic normality of the total length of external branches in Kingman's coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with n black balls. Empty it in n steps according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers Uk, 0 < k < n, of red balls after k steps exhibit an unexpected property: (U0, ... ,Un) and (Un, ... ;U0) are equal in distribution.