The whale and the shark

Updated the Physeter-based version of Livyatan based on a larger dataset. For details see here: www.deviantart.com/theropod1/a…

Note: 22/03/23: No changes to the comparison, but a quick note on whether or not it is fair to compare them this way:
   regarding Livyatan:
For now, the only individual of Livatan for which a reasonably scientific body size estimate can realistically be made is the holotype. However, by now a handful of isolated teeth have surfaced, which can at least give us a tentative idea as to whether we are dealing with a particularly large or small individual. Compiling isolated teeth with known diameters, this is how they compare to the holotype:

As you can see, all the referred teeth are within the size range of the holotype, with none clearly larger or smaller. Note that it is important to consider that less than half of the holotype’s teeth are actually preserved, with the smallest (as based on the size of the tooth sockets) not being present, so only looking at the preserved teeth will give a wrong representation of their size. Hence, missing teeth were estimated based on the alveolar diameter or (if the former was unavailable) tooth position.
The mean diameter of the holotype teeth does end up somewhat larger than the isolated teeth, about 99 mm vs 91 mm, so there’s a tendency to say that the holotype is a bit larger than the average of all isolated teeth. However all of the isolated specimens are well within the size range of those in the holotype, and the difference in mean sizes is not statistically significant (wilcoxon test p=0.11), meaning the holotype may very well be an approximately average-sized individual based on current data.

refs: 

Govender, R. 2019. Early Pliocene fossil cetaceans from Hondeklip Bay, Namaqualand, South Africa. Historical Biology: 1–20.

Lambert, O., Bianucci, G. and De Muizon, C. 2016. Macroraptorial sperm whales (Cetacea, Odontoceti, Physeteroidea) from the Miocene of Peru. Zoological Journal of the Linnean Society 179 (2): 404–474.

Loch, C.S., Gutstein, C.S., Pyenson, N.D. and Clementz, M.T. 2019. But did it eat other whales? New enamel microstructure and isotopic data on Livyatan, a large physeteroid from the Atacama region, northern Chile. SVP 79th Annual Meeting: Program and Abstracts: 159–160.

Piazza, D.S., Agnolín, F.L. and Lucero, S. 2019. First record of a macroraptorial Sperm Whale (Cetacea, Physeteroidea) from the Miocene of Argentina. Revista Brasileira de Paleontologia 21 (3): 276–280.

as well as:

Watmore, K.I. and Prothero, D.R. 2023. Gigantic Macroraptorial Sperm Whale Tooth (cf. Livyatan) from the Miocene of Orange County, California. : 2023.01.25.525567.

regarding O. megalodon:
Another caveat is that sharks do not replace their teeth at an equal rate throughout their life. While there are no studies on Lamnids, let alone C. carcharias (or Otodontids, for obvious reasons), it is known from other galeomorph sharks such as Carcharhinus milberti (Wass 1973) that tooth replacement is faster in younger sharks. In the case of C. milberti, functional tooth life varied by approximately a factor of 2 between juveniles (18 days) and adults (36 days), i.e. the smallest sharks lost twice as many teeth in the same time as the largest.

Would this have a major impact on our estimates of average sizes if we accounted for it?
To test this I introduced a weighting factor varying linearly from 1 (for the smallest individuals) to 2 (for the largest), which gives a higher (twice higher) weight to the largest individuals than to the smallest in calculating the mean, in order to account for the different probability of losing teeth, roughly based on Wass’ data for C. milberti (where there appears to be such a linear relationship).
The results might be disappointing to some, because this only increases the mean total length in the entire sample by about half a metre (from 1055 to 1107 cm TL). It may be even more disappointing when we only look at the adults, because here it results in a difference in means of just 39 cm (from 13.78 to 14.17 m) if we add the variation in the weighting factors for adults on top of the "adulthood weighting factor" 1 for all certain adults and ranging from 0 to one for individuals in the size range within which the sharks attain adulthood) already used to estimate the adult average.

That being said, we are comparing fossil samples here, and fossil samples are always subject to certain filters compared to living organisms. For instance, C. carcharias has a type III survivorship curve (Burgess et al. 2014), which can likely by extension also be assumed for O. megalodon. That means that while it’s true that young sharks at a given time have a higher probability of shedding a tooth, they also have a higher probability of dying and entering the fossil record that way, so that the tooth record may well represent a good proxy for what a fossil record based on exclusively dead individuals would look like.
Which, funnily enough, is exactly what we are comparing between all other fossil organisms, with nobody griping about it. So maybe we should disregard this whole issue altogether as a bit of an unnecessary complexity at this point. Either way, I think I have demonstrated that it doesn’t have a very large effect, and we still don’t get average megalodons outsizing the Livyatan holotype.

Note: 20/03/23: Following a poster I did on the aforementioned side project analyzing O. megalodon body size data (see: DOI 10.13140/RG.2.2.17983.79525), here are some small updates to the size estimates for the shark.
They did not change much since the last revision; what change there is is down to refined estimates of the proportion of the toothrow occupied by inter-crown spacing (now ~22%, based on the published 3D model by the NHM).

The estimate for the mean length of adult sharks is now 13.8 m, based on a weighted mean of sharks above the maximum length at maturity given a weight of 1, sharks below the minimum length at maturity given a weight of 0, and linear interpolation between the two.

The dentition of GHC 1 represents the largest relatively complete tooth set for megalodon, significant as this is a specimen we can be more . Still, the 90% CI ranges from 14.65-16.63 m. If we want the largest individual that we can be relatively confident in, it is this, because here we have an almost complete tooth set giving us the size of the jaw.

This is an uncertainty also affecting each and every individual size estimate for an isolated tooth, but one that is not taken into account in the CI listed above, which is just for finding the population parameter using the sample. However, much of this uncertainty is alleviated as long as we look at a large sample, where individual variations will tend to cancel each other out.

I also added a larger background silhouette (in green) representing a very large megalodon specimen based on the size distribution of the tooth sample, at almost 19 m. This size would correspond to the 99th percentile of body sizes according to my modified methodology, or approximately to the 96th percentile based on direct sizing from the white shark data in Perez et al., so it is undoubtedly a very large specimen (the 90% CI for that quantile is 17.67-19.79 m, so in other words, here’s a propability of only 5% that the largest 1% were not at least 17.67 m long, and a probability of 5% that the smallest 99% were not at most 19.79 m long).

Mean TL for the entire sample now stands at 10.55 m (90% CI 10.29-10.81 m), still remarkably close to Pimiento & Balk’s estimate of 10.02 m. It appears that a shift in methodology from tooth crown height to dentition width-based estimates has massive implications for estimates of individual teeth (see e.g. Fig.2C on the poster, some specimens end up 5 m with one and 15 with the other method), but the cancelling-out of respective biases over such a large sample representing teeth from different positions is such that the average is only affected to a minor degree. This of course also explains why the extremes of the size distribution differ much more than the measures of central tendency, e.g. the 99th percentile of the original estimates from Pimiento & Balk was 17.55 m (90% CI 17.18-17.82), so this got 1.25 m larger.

Note: 23/01/23: Following a suggestion by user Ovleg in the comment thread below, here is an alternate version with the C. megalodon reconstruction based on Cretalamna from Greenfield (2022): www.deviantart.com/theropod1/a…

ref.:

I think there are certain open questions (e.g. How would body and fin shape scale with body size up from this very small shark to the size of O. megalodon, or even the size of a large lamnid like the great white shark) that don’t warrant replacing the other reconstruction with it (yet), but it is a valuable comparison and provides interesting inputs for considering how otodontids may have differed from lamnids morphologically. The mass figures there should be taken as vaguely indicative, seeing how they remain the same ones based on Carcharodon that are shown above at this time. A further (phylogenetically corrected) analysis of body mass and shape scaling with size in lamnids, as well as other lamniforms, would certainly be warranted though, if someone has the necessary data…

EDIT: 30/05/22:
Another rescaling of the sharks, taking into account the new published data from the last few years.
Estimate for adult average size follows methodology outlined in the last update (estimating cutoff point for average size at maturity based on ratio between average size at maturity from Caillet et al. 1985 and average size of sample from McClain et al. 2015), but on the updated megalodon dataset. The resulting estimated mean size at maturity is 10.6m, i.e. the population average likely represents an immature shark. The mean size of the 221 individuals classified as adults is 13.5 m.

Largest shark rescaled to match size of largest known dentition, GHC 1, from the Bahia Inglesa Formation (very slightly bigger than the Yorktown dentition, according to Perez et al. 2021, also coincidentally close to the 15.3m average adult size based on the Perez et al. method).
Also added the average size for the whole sample. For detailed methodology and references behind the size estimate, see: www.deviantart.com/theropod1/j…

EDIT: 30/10/19: Rescaled the sharks based on current data. I will write an in-depth post soon, but here is a teaser:

Histogram of meg tooth sizes based on dataset in Pimiento & Balk 2015

The basis is using the associated Yorktown dentition (Pimiento et al. 2010, depicted in the form of the larger shark here) as a basis for comparison and calculating the width of each tooth in Pimiento & Balk (2015) according to its assigned position. Then, using the ratio between mean size of the entire sample (McClain et al. 2015) and mean size at maturity (Caillet et al. 1985) in white sharks, we can exclude immature individuals based on their size.
Then all it takes is to know the size of the dentition (2246 mm summed crown width, I added 5% for the greater width of the roots and 15% for the interdental spacing to get 2712 mm) to estimate the corresponding bite circumferences and get a proxy that we can use to somewhat reliably estimate total length by plugging that into a regression for estimating total length of white sharks based on their bite circumference (Lowry et al. 2009, data published in supplement to Ferrón et al. 2019) to find our corresponding sizes:
Approximately 13.6 m for the average adult C. megalodon, approximately 15.5 m for the owner of the Yorktown dentition.

Expect further updates on this soon.

Refs:   

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Size comparison between Livyatan melvillei and Carcharocles megalodon, probably the largest macrophagous predators in earth history.

I will add a sperm-whale based reconstruction of Livyatan (~14.2m and ~35t) when I can get the time to do one, and possibly one based on Zygophyseter some time in the future.

EDIT: 12/05/19: Added a new megalodon reconstruction based on the body shape of the "maximum" model from here

EDIT: 24/05/16: Finally scanned the alternate reconstruction and added it.

As you see, it is somewhat shorter, but not by a huge amount (not the massive discrepancy between the upper and lower lenght estimates from the description paper), the most prominent difference between the two being chest depth and tail length, which explain the more Kogiid- or orcinine-like bulk when basing the reconstruction on a stem-physeteroid.
     ref.: Lockyer, Christina. 1991. Body composition of the sperm whale, Physeter catodon, with special reference to the possible functions of fat depots. Rit Fiskideilda 12 (2) pp. 1–24.