  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.
