How many species have mass M?

Within large taxonomic assemblages, the number of species with adult body mass M is characterized by a broad but asymmetric distribution, with the largest mass being orders of magnitude larger than the typical mass. This canonical shape can be explained by cladogenetic diffusion that is bounded below by a hard limit on viable species mass and above by extinction risks that increase weakly with mass. Here we introduce and analytically solve a simplified cladogenetic diffusion model. When appropriately parameterized, the diffusion-reaction equation predicts mass distributions that are in good agreement with data on 4002 terrestrial mammal from the late Quaternary and 8617 extant bird species. Under this model, we show that a specific tradeoff between the strength of within-lineage drift toward larger masses (Cope’s rule) and the increased risk of extinction from increased mass is necessary to produce realistic mass distributions for both taxa. We then make several predictions about the evolution of avian species masses.

💡 Research Summary

The paper tackles a long‑standing pattern in macro‑evolution: across large taxonomic groups the distribution of species by adult body mass is highly skewed, with a long right tail extending many orders of magnitude beyond the modal mass. The authors propose that this shape emerges from two interacting processes. First, lineage splitting (cladogenesis) generates a stochastic “diffusion” of log‑mass values: each daughter species inherits its parent’s mean mass but experiences a small random shift. This is captured mathematically by a diffusion term with coefficient D and a drift term μ that represents the average tendency toward larger size (Cope’s rule). Second, extinction risk is assumed to increase with body size; the authors model this as a mass‑dependent mortality rate λ(x)=λ₀ e^{βx}, where x=ln M. A hard lower bound at the physiologically viable minimum mass M_min is imposed, preventing the diffusion from crossing into impossible size ranges.

Combining these ingredients yields a reaction‑diffusion equation for the probability density p(x,t):

∂p/∂t = D ∂²p/∂x² – μ ∂p/∂x – λ₀ e^{βx} p,

with reflecting/absorbing boundary at x_min. Solving for the stationary distribution (∂p/∂t=0) gives an analytic form:

p*(x) ∝ exp