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

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, Brownian motion with reflective limits, that they anticipated might be of use in testing hypotheses about a particular sort of constraint on phenotypic character evolution. Since their analytic solution for the probability function under this bounded evolutionary scenario was not practical to evaluate for reasonably-sized trees, Boucher and Demery (2016) also identified a creative technique for computing the likelihood of their model. The basis of this methodology derives from the convergence of an equal-rates, symmetric, ordered Markov chain and continuous stochastic diffusion in the limit as the number of steps in our chain goes to {infty} (or, alternatively, as their widths decrease towards zero). We refer to this convergence in the limit as the discretized diffusion approximation or (more compactly) the discrete approximation. We realized that this discrete approximation of Boucher and Demery (2016) unlocked a 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 of discrete and continuous traits in which the discrete trait influences the rate of evolution of our continuous character, as well as a 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.