Unlocking a flexible set of phylogenetic models for discrete and continuous trait evolution using discretized stochastic diffusion

Abstract

The practical utility of many modern phylogenetic comparative methods can depend on how accurately mathematical models capture the evolutionary process of traits. Boucher and Demery (2016) described a new quantitative trait model for testing hypotheses about constraint on phenotypic character evolution: Brownian motion with reflective limits. Since their analytic solution for the probability function under this bounded scenario was intractable for reasonably-sized trees, Boucher and Demery (2016) also identified a creative technique for computing the likelihood of their model. The basis of this method derives from the convergence of an equal-rates, symmetric, ordered Markov chain and continuous stochastic diffusion in the limit as the number of lattice states in the chain goes to{infty} (or, alternatively, as their widths decrease towards zero). We realized that this general approach had the potential to unlock a surprisingly large number of additional models for the phylogenetic comparative analysis of discrete and continuous trait data, and we explore several of these in the present article. Specifically, we examine application of this discretized diffusion approximation to the threshold model from evolutionary quantitative genetics, to a new "semi-threshold" trait evolution model, to a joint model where the rate of evolution for a continuous trait depends on the value of a co-evolving discrete character, along with a separate model where precisely the converse is true, and to a discrete-character-dependent multi-trend trended continuous trait evolution model. We conclude with some context for the origins of our article and discussion of other possible applications of this powerful approach.