Alpha and sigma taxonomy of Lystrosaurus murrayi and L . declivis , Triassic dicynodonts ( Therapsida ) from the Karoo Basin , South Africa

PUBLISHED: 27 March 2019 Permo-Triassic dicynodont (Therapsid) fossils attributed to the genus Lystrosaurus1 are known from the South African Karoo Basin, including two Triassic taxa referred to as L. (Dicynodon) murrayi2 and L. (Ptychognathus) declivis3. Distinction of the two taxa has been recognised inter alia by Brink4, Cluver and King5, Botha and Smith6, Grine et al.7 and Botha-Brink et al.8 However, in this instance, the appropriateness of alpha taxonomy may be questioned, recognising that boundaries between these species of Lystrosaurus are not necessarily clear, especially for specimens close to the time of divergence postdating the Permo-Triassic boundary, 252 million years ago. The question has arisen as to whether only one species occurred.9 Thackeray10 proposed the following hypotheses:


•
H T -that one species existed and • H BB -that there were two taxa, clearly distinct at the species level.
Both hypotheses need to be considered in the light of a statement by Camp 11 : 'traditional data were unable to distinguish between L. murrayi and L. declivis without overlapping measurements in regions where the geometric morphometric analyses indicated they differed most'.Irrespective of which statistical methods are used for taxonomic purposes, it is essential to have some kind of definition of a species that can be applied in palaeontological contexts, to address opposing views held by 'alpha taxonomists' who assume clear boundaries between species 12 , and 'sigma taxonomists' [13][14][15] who do not.
5][16] The standard error of the m coefficient (sem) in equations of the form y=mx+c is calculated to reflect the degree of morphological similarity or dissimilarity when pairs of crania are compared.Notably, for large samples of conspecific modern taxa (used for reference purposes), there is a normal distribution of log-transformed sem values. 17 the case of conspecifics, a mean log sem value of -1.61±0.1 has been calculated from anatomical measurements obtained from more than 70 modern taxa. 14,17As it has been replicated from independent sets of data, for extinct and extant taxa 14,18 , it can be considered to reflect the typical degree of morphological variability in vertebrate taxa across evolutionary time and geographical space.As such, a mean log sem value (T) of -1.61±0.1 has been proposed as a probabilistic definition of a species applicable in palaeontological contexts. 14This morphometric approach has the potential to address taxonomic issues regarding Triassic specimens of Lystrosaurus.Examples are given here for selected specimens.
Ideally, complete and well-preserved holotypes for L. murrayi and L. declivis should be used for taxonomic purposes.Unfortunately, the holotypes for these two species described respectively by Huxley 2 and Owen 3 , and curated at the Natural History Museum in London, are not sufficiently complete to allow detailed morphometric analyses based on many cranial dimensions.On the basis of morphology alone, Broom 19 questioned whether the (incomplete) holotype of L. declivis 3 was the same species represented by the (incomplete) holotype of L. murrayi 2 .In this study, cranial measurements of other (well-preserved, more complete) specimens attributed to either L. murrayi or L. declivis are analysed with a view to identifying potential neotypes on the basis of log sem statistics.

Materials
The specimens in this study are well-preserved crania of Lystrosaurus, selected on the basis of a relatively high number of measurable dimensions among the specimens classified by Camp 11 who indicates whether there is consensus between her identifications and those of others.These specimens are: • NMNH 23349 attributed to L. murrayi (consensus identification), curated at the National Museum of Natural History, Smithsonian Institution, Washington.
• BMNH 36221 attributed to L. declivis (consensus identification), curated at the Natural History Museum in London.
• NMNH 12690, the identification of which is uncertain (L.murrayi/declivis), curated at the National Museum of Natural History, Smithsonian Institution, Washington.
These specimens are sufficiently well preserved to measure about 30 dimensions based on anatomical landmarks. 11he 20 variables common to all three specimens are listed in Table 1.

Results
Log sem values for pairwise comparisons are as follows, in increasing order:

Discussion and conclusions
Relatively low log sem values (ranging between -1.474 and -1.419) are obtained when NMNH 23349 (attributed to L. murrayi, 'consensus identification') is compared against NMNH 12690 (with uncertain status).
In relation to the probabilistic definition of a species (T=-1.61±0.1), the two specimens may be conspecific because the mean log sem value of -1.446 falls within the 95% confidence limits for a species.However, this log sem value is near the upper limit, such that in terms of degrees of variation, it is relatively close to being different at a species level.This proximity is important in the context of a view that there may not be a distinct boundary between L. murrayi and L. declivis.

On the basis of an impressive sample of almost 200 crania of Triassic
Lystrosaurus studied by Botha-Brink et al. 8 , it has been demonstrated that the mean basal skull length of specimens attributed to L. murrayi is not significantly different from the mean basal skull length dimension of specimens attributed to L. declivis 10 .The mean values (and the associated standard deviations) are almost identical, as if one species was being sampled from the same kind of populations.In addition to the morphometric data presented here, based on as many as 20 variables per specimen, this result is relevant to the possibility that no distinct boundary exists between the two taxa at a species level, in the context of both temporal and spatial variability.
Do at least some crania represent different taxa at the level of a species?High log sem values (ranging between -1.020 and -1.014) are in fact obtained when BMNH 36221 (attributed to L. declivis, 'consensus' identification) is compared against NMNH 23349 (L.murrayi, 'consensus' identification).In relation to the probabilistic definition of a species (T = -1.61±0.1),these particular well-preserved specimens have a high probability of being different at a species level as the mean log sem value of -1.017 falls undoubtedly well outside the upper 95% confidence limit for a species.As such, these results for at least two particular specimens serve to refute the hypothesis (H T ) that all specimens attributed to either L. declivis or to L. murrayi represent only one species. 10It could be concluded that L. murrayi is indeed represented by NMNH 23349, and that L. declivis is represented by BMNH 36221.
However, not all comparisons may necessarily reflect differences at a species level.In the case of closely related taxa, log sem values can serve to quantify degrees of similarity if not distinguish differences in kind.For instance, the relatively low mean log sem value of - Specimens such as NMNH 12690, whose identification has been uncertain (L.murray/declivis), may be those which fall within a spectrum of variability associated with two species which diverged close to the time of the Permo-Triassic boundary, 252 million years ago, but which may have hybridised.Hybridisation is recognised as a phenomenon that is much more common than previously thought. 15cause NMNH 23349 and BMNH 36221 are recognised here as being distinct at a species level for Triassic specimens of Lystrosaurus, on the basis of a probabilistic definition of a species (T=-1.61±0.1),and because they are sufficiently complete to allow many measurements per specimen, they are recommended as potential neotypes for L. murrayi and L. declivis, respectively.
When and if neotypes are formally described, they can be used as reference specimens to assess the probability that other Triassic fossils of Lystrosaurus do or do not belong to L. murrayi or L. declivis.The examples given above relate to sigma taxonomy [13][14][15] , where sigma is the Greek letter Σ (S for spectrum), as opposed to alpha taxonomy 12 which assumes clear boundaries between species.

Table 1 :
Cranial dimensions used in this study.Reference numbers (with definitions) are those given byCamp 11 115| Number 3/4 March/April 2019 Scientific Correspondence https://doi.org/10.17159/sajs.2019/a0296MethodsPairwisecomparisons of cranial measurements are made using least squares linear regression analysis to calculate log sem values associated with specimens attributed to L. murrayi (a 'consensus' identification for NMNH 23349) and to L. declivis (a 'consensus' identification for BMNH 36221).Log sem values are also calculated for comparisons with NMNH 12690, the identification of which is not certain.