Inferring Bounded Evolution in Phenotypic Characters from Phylogenetic Comparative Data.

Our understanding of phenotypic evolution over macroevolutionary timescales largely relies on the use of stochastic models for the evolution of continuous traits over phylogenies. The two most widely used models, Brownian motion and the Ornstein-Uhlenbeck (OU) process, differ in that the latter includes constraints on the variance that a trait can attain in a clade. The OU model explicitly models adaptive evolution toward a trait optimum and has thus been widely used to demonstrate the existence of stabilizing selection on a trait. Here we introduce a new model for the evolution of continuous characters on phylogenies: Brownian motion between two reflective bounds, or Bounded Brownian Motion (BBM). This process also models evolutionary constraints, but of a very different kind. We provide analytical expressions for the likelihood of BBM and present a method to calculate the likelihood numerically, as well as the associated R code. Numerical simulations show that BBM achieves good performance: parameter estimation is generally accurate but more importantly BBM can be very easily discriminated from both BM and OU. We then analyze climatic niche evolution in diprotodonts and find that BBM best fits this empirical data set, suggesting that the climatic niches of diprotodonts are bounded by the climate available in Australia and the neighboring islands but probably evolved with little additional constraints. We conclude that BBM is a valuable addition to the macroevolutionary toolbox, which should enable researchers to elucidate whether the phenotypic traits they study are evolving under hard constraints between bounds.