# 6: Beyond Brownian Motion

This chapter consider four ways that comparative methods can move beyond simple Brownian motion models: by transforming the variance-covariance matrix describing trait covariation among species, by incorporating variation in rates of evolution, by accounting for evolutionary constraints, and by modeling adaptive radiation and ecological opportunity. It should be apparent that the models listed here do not span the complete range of possibilities, and so my list is not meant to be comprehensive. Instead, I hope that readers will view these as examples, and that future researchers will add to this list and enrich the set of models that we can fit to comparative data.

• 6.1: Introduction to Non-Brownian Motion
Brownian motion is very commonly used in comparative biology: in fact, a large number of comparative methods that researchers use for continuous traits assumes that traits evolve under a Brownian motion model. The scope of other models beyond Brownian motion that we can use to model continuous trait data on trees is somewhat limited. However, more and more methods are being developed that break free of this limitation, moving the field beyond Brownian motion.
• 6.2: Transforming the evolutionary variance-covariance matrix
There are three Pagel tree transformations (lambda: λ, delta: δ, and kappa: κ). I will describe each of them along with common methods for fitting Pagel models under ML, AIC, and Bayesian frameworks. Pagel’s three transformations can also be related to evolutionary processes, although those relationships are sometimes vague compared to approaches based on explicit evolutionary models rather than tree transformations (see below for more comments on this distinction).
• 6.3: Variation in rates of trait evolution across clades
There are several methods that one can use to test for differences in the rate of evolution across clades. First, one can compare the magnitude of independent contrasts across clades; second, one can use model comparison approaches to compare the fit of single- and multiple-rate models to data on trees; and third, one can use a Bayesian approach combined with reversible-jump machinery to try to find the places on the tree where rate shifts have occurred. I will explain each of these methods in t
• 6.4: Non-Brownian evolution under stabilizing selection
We can also consider the case where a trait evolves under the influence of stabilizing selection. Assume that a trait has some optimal value, and that when the population mean differs from the optimum the population will experience selection towards the optimum. As I will show below, when traits evolve under stabilizing selection with a constant optimum, the pattern of traits through time can be described using an Ornstein-Uhlenbeck (OU) model.
• 6.5: Early Burst Models
According to Simpson, species enter new adaptive zones in one of three ways: dispersal to a new area, extinction of competitors, or the evolution of a new trait or set of traits that allow them to interact with the environment in a new way. One idea is that we could detect the presence of adaptive radiations by looking for bursts of trait evolution deep in the tree. If we can identify clades that might be adaptive radiations, we can uncover this “early burst” pattern of trait evolution.
• 6.6: Peak Shift Models
One can imagine a scenario where species evolve on an adaptive landscape with many peaks; usually, populations stay on a single peak and phenotypes do not change, but occasionally a population will transition from one peak to another. We can either assume that these changes occur at random times, defining an average interval between peak shifts, or we can associate shifts with other traits that we map on the phylogenetic tree.
• 6.7: Appendix - Deriving an OU model under stabilizing selection
• 6.S: Beyond Brownian Motion (Summary)