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3.1: Introduction to Brownian Motion

Squamates, the group that includes snakes and lizards, is exceptionally diverse. Since sharing a common ancestor between 150 and 210 million years ago (Hedges and Kumar 2009), squamates have diversified to include species that are very large and very small; herbivores and carnivores; species with legs and species that are legless. How did that diversity of species’ traits evolve? How did these characters first come to be, and how rapidly did they change to explain the diversity that we see on earth today? In this chapter, we will begin to discuss models for the evolution of species’ traits.

Imagine that you want to use statistical approaches to understand how traits change through time. To do that, you need to have an exact mathematical specification of how evolution takes place. Obviously there are a wide variety of models of trait evolution, from simple to complex. For example, you might create a model where a trait starts with a certain value and has some constant probability of changing in any unit of time. Alternatively, you might make a model that is more detailed and explicit, and considers a large set of individuals in a population. You could assign genotypes to each individual and allow the population to evolve through reproduction and natural selection.

In this chapter – and in comparative methods as a whole – the models we will consider will be much closer to the first of these two models. However, there are still important connections between these simple models and more realistic models of trait evolution (see chapter five).

In the next six chapters, I will discuss models for two different types of characters. In this chapter and chapters four, five, and six, I will consider traits that follow continuous distributions – that is, traits that can have real-numbered values. For example, body mass in kilograms is a continuous character. I will discuss the most commonly used model for these continuous characters, Brownian motion, in this chapter and the next, while chapter five covers analyses of multivariate Brownian motion. We will go beyond Brownian motion in chapter six. In chapter seven and the chapters that immediately follow, I will cover discrete characters, characters that can occupy one of a number of distinct character states (for example, species of squamates can either be legless or have legs).