Testing for Depéret's Rule (Body Size Increase) in Mammals using Combined Extinct and Extant Data

Abstract

Whether or not evolutionary lineages in general show a tendency to increase in body size has often been discussed. This tendency has been dubbed "Cope's rule" but because Cope never hypothesized it, we suggest renaming it after Depéret, who formulated it clearly in 1907. Depéret's rule has traditionally been studied using fossil data, but more recently a number of studies have used present-day species. While several paleontological studies of Cenozoic placental mammals have found support for increasing body size, most studies of extant placentals have failed to detect such a trend. Here, we present a method to combine information from present-day species with fossil data in a Bayesian phylogenetic framework. We apply the method to body mass estimates of a large number of extant and extinct mammal species, and find strong support for Depéret's rule. The tendency for size increase appears to be driven not by evolution toward larger size in established species, but by processes related to the emergence of new species. Our analysis shows that complementary data from extant and extinct species can greatly improve inference of macroevolutionary processes.

Keywords: Body size; Cope's rule; macroevolution; paleontology; phylogenetics.

Figure 1.

Updating of polytomies and branch lengths. Capital A–D are tip species, E, F, and G are internal nodes. Horizontal arrows indicate a node's range of possible branching times. Black arrows indicate the branch around which topology is altered as explained in the main text.

Figure 2.

Illustration of evolution of body size by directional Brownian motion. Natural logarithmic body size of a species, which is $m_{\text{0}}$ at time $t_{\text{0}}$ follows at time $t_{\text{1}}$ a normal distribution with mean $d(t_{\text{1}}-t_{\text{0}})$ and variance $s^{\text{2}}(t_{\text{1}}-t_{\text{0}})$. $s^{\text{2}}$ is the intrinsic rate of Brownian motion, and $d$ is the strength of Depéret's rule. One trajectory of body size change is shown in bold, and 100 equally probable alternative trajectories in gray.

Figure 3.

Prior and posterior distributions of the tendency for body size increase $d$. The horizontal axis at the top shows proportional change (% body size increase per myr) on an ordinary scale. Posterior distributions of MCMC samples are expressed as probability density, calculated using kernel density estimation. (Histograms serve as illustration only.) Note that because the prior distribution has much greater variance than the posterior distributions, its probability density is much lower than the peaks of the posterior distributions. Left panel: $d$ estimated without fossil information (light gray bars), with fossil information (dark gray bars), and the gradual component of $d$ (no bars), estimated with fossil information alongside with the cladogenetic component $d_{\text{c}}$ (shown only on the right panel). Right panel: prior and posterior distribution of $d_{\text{c}}$. Note that Bayes factor is calculated as the ratio of prior to posterior at $d=0$ (vertical dashed line), where the posterior obtained without fossil information (light gray bars) has far higher density than the prior, while the posterior obtained with fossil evidence (dark gray bars) has much lower density than the prior.

Figure 4.

Estimates of the strength of Déperet's rule, $d$, as a function of the number of fossil body mass estimates. We randomly sampled 10, 25, 50, 100, 200, 300, 400, and 500 of the fossil body mass estimates, and estimated $d$. We did this twice, so for every number of species two estimates of $d$ are shown. The estimate using all 553 fossil estimates is also shown. (Standard deviations around these estimates are approximately 0.0011, independent of the number of fossil species sampled, and therefore not shown.) Trendline by eyeballing.