Global Completeness of the Bat Fossil Record

Abstract

Bats are unique among mammals in their use of powered flight and their widespread capacity for laryngeal echolocation. Understanding how and when these and other abilities evolved could be improved by examining the bat fossil record. However, the fossil record of bats is commonly believed to be very poor. Quantitative analyses of this record have rarely been attempted, so it has been difficult to gauge just how depauperate the bat fossil record really is. A crucial step in analyzing the quality of the fossil record is to be able to accurately estimate completeness. Measures of completeness of the fossil record have important consequences for our understanding of evolutionary rates and patterns among bats. In this study, we applied previously developed statistical methods of analyzing completeness to the bat fossil record. The main utility of these methods over others used to study completeness is their independence from phylogeny. This phylogenetic-independence is desirable, given the recent state of flux in the higher-level phylogenetic relationships of bats. All known fossil bat genera were tabulated at the geologic stage or sub-epoch level. This binning strategy allowed an estimate of the extinction rate for each bat genus per bin. Extinction rate—together with per-genus estimates of preservation probability and original temporal distributions—was used to calculate completeness. At the genus level, the bat fossil record is estimated to be 12% complete. Within the order, Pteropodidae is missing most of its fossil history, while Rhinolophoidea and Vespertilionoidea are missing the least. These results suggest that 88% of bats that existed never left a fossil record, and that the fossil record of bats is indeed poor. Much of the taxonomic and evolutionary history of bats has yet to be uncovered.

Appendices

Appendix 1

Table 2 Compilation of bat fossil genera used in this study. Temporal records of bat genera were tabulated as present (1) or absent (0) in each discrete time interval. Sources, shown below the table, record the first and last occurrences of each genus (excluding Recent), plus additional intervening occurrences when available (non-exhaustive). Secondary sources were used in cases where primary sources were unavailable. Stratigraphic bin abbreviations as follows: Eo = Eocene, Oligo = Oligocene, Mio = Miocene, P-P = Pliocene through Pleistocene, Rec = Recent; Ypr = Ypresian, Lut = Lutetian, Bart = Bartonian, Pria = Priabonian, Rup = Rupelian, Chat = Chattian, Aqui = Aquitanian, Burd = Burdigalian, M Mio = Middle Miocene, L Mio = Late Miocene. Ma = Millions of Years Ago. Records for Recent bats were not used in statistical calculations

Appendix 2

Statistical calculations for Vespertilionoidea, following the method of Foote and Raup (1996).

Using the data for all vespertilionoid bats (Appendix 1), we first computed the total number of temporal bins in which each genus occurs. There are 11 stratigraphic bins (excluding the Recent), beginning with the Ypresian (early Eocene) and ending with the Plio-Pleistocene. Of the 56 vespertilionoid genera that occur as fossils, 38 occur in only one bin, five range through two bins, eight occur in three bins, one occurs in four bins, two occur in five bins, and one each occurs in six (Myotis) and eight (Tadarida) bins. No vespertilionoid bats lasted exactly seven bins, or in any more than eight bins.

To estimate preservation probability, R, we calculate the FreqRat (which estimates R) following the formula:

$$ R \approx FreqRat = {{f(2)^2 } \mathord{\left/ {\vphantom {{f(2)^2 } {\left[ {f(1)f(3)} \right]}}} \right. } {\left[ {f(1)f(3)} \right]}} $$

where f(x) is the frequency of genera with observed range-durations of x bins. For Vespertilionoidea, $R \approx {{FreqRat = 5^2 } \mathord{\left/ {\vphantom {{FreqRat = 5^2 } {\left[ {\left( {38} \right)(8)} \right] = 0.082}}} \right. } {\left[ {\left( {38} \right)(8)} \right] = 0.082}}$.

To account for differential original temporal distributions of bat genera, R must be scaled according to original durations. Estimated preservation probability is used to calculate an estimate of the probability P ₁ (T) that a taxon is preserved at least once given a true original duration T. The probability that a genus with original duration T is not at all preserved equals (1-R) ^T. Thus, the probability that it is preserved at least once equals:

$$ P_1 (T) = 1 - \left( {1 - R} \right)^T $$

For example, to calculate the probability that a genus is preserved given a true original duration of three bins, the equation became $P_1 (3) = 1 - \left( {1 - 0.082} \right)^3 = 0.0227$. This same procedure was used for all possible values of T (i.e., T = 1-∞, though in this and subsequent calculations it is usually sufficient to take out to just several hundred).

We next estimated extinction rate, q, which was to be used to compute original genus durations. We used linear regression on the ln-transformed frequencies of genera with a range of two or more bins. The slope of the linear regression line equals q (the extinction rate) with the sign reversed. Next, original durations were calculated, assuming a constant extinction rate, q, and an exponential distribution of original durations. q was calculated from the ln-slope of range-frequencies (omitting taxa that occur in only one bin) as the slope with the sign reversed. For Vespertilionoidea, the ln-slope of range-frequency distributions is −0.3226, so the extinction rate q equals 0.3226.

Using our estimate for q, we calculated the probability, h(T), that the original distribution equaled T as:

$$ h(T) = e^{{ - q\left( {T - 1} \right)}} - e^{- qT} $$

Using T of 3 again, we see that $h(3) = e^{{ - \left( {0.3226} \right)\left( {3 - 1} \right)}} - e^{{ - \left( {.3226} \right)(3)}} = 0.145$. Again, this procedure was calculated for all values of T.

Finally, to calculate completeness, we computed the sum, for all original durations, T, of the product between h(T) (probability of having original duration T) and P ₁ (T) (probability of being preserved at least once during an interval given an original duration T) as:

$$ P_p = \sum\limits_{T = 1}^{\infty } {h(T)P_1 (T)} $$

As T increased, the completeness for that number of bins decreases rapidly, so this summation was to only be taken to several hundred rows in a spreadsheet. Doing this for Vespertilionoidea to 900 rows gave a completeness of 24.5% (rounded and included in Table 1). Summing the completeness calculations up to only 50 rows produced the same overall completeness value (to eight digits), indicating that it is sufficient to compute the summation using a finite set of values for T that is substantially less than 1000.