# 6.1: Introduction to Non-Brownian Motion

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Detailed studies of contemporary evolution have revealed a rich variety of processes that influence how traits evolve through time. Consider the famous studies of Darwin’s finches, Geospiza, in the Galapagos islands carried out by Peter and Rosemary Grant, among others (e.g. Grant and Grant 2011). These studies have documented the action of natural selection on traits from one generation to the next. One can see very clearly how changes in climate – especially the amount of rainfall – affect the availability of different types of seeds (Grant and Grant 2002). These changing resources in turn affect which individuals survive within the population. When natural selection acts on traits that can be inherited from parents to offspring, those traits evolve.

One can obtain a dataset of morphological traits, including measurements of body and beak size and shape, along with a phylogenetic tree for several species of Darwin’s finches. Imagine that you have the goal of analyzing the tempo and mode of morphological evolution across these species of finch. We can start by fitting a Brownian motion model to these data. However, a Brownian model (which, as we learned in Chapter 3, corresponds to a few simple scenarios of trait evolution) hardly seems realistic for a group of finches known to be under strong and predictable directional selection.

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. In this chapter I will discuss these approaches and what they can tell us about evolution. I will also describe how moving beyond Brownian motion can point the way forward for statistical comparative methods.

In this chapter, I will 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.

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