Comparative methods used to study patterns of evolutionary change in a continuous trait on a phylogeny range from Brownian motion processes to models where the trait is assumed to evolve according to an Ornstein-Uhlenbeck (OU) process. Although these models have proved useful in a variety of contexts, they still do not cover all the scenarios biologists want to examine. For models based on the OU process, model complexity is restricted in current implementations by assuming that the rate of stochastic motion and the strength of selection do not vary among selective regimes. Here, we expand the OU model of adaptive evolution to include models that variously relax the assumption of a constant rate and strength of selection. In its most general form, the methods described here can assign each selective regime a separate trait optimum, a rate of stochastic motion parameter, and a parameter for the strength of selection. We use simulations to show that our models can detect meaningful differences in the evolutionary process, especially with larger sample sizes. We also illustrate our method using an empirical example of genome size evolution within a large flowering plant clade. K E Y W O R D S : Brownian motion, comparative method, continuous characters Hansen model, Ornstein-Uhlenbeck.Single-rate Brownian motion works reasonably well as a model for evolution of traits. It models drift, drift-mutation balance, and even stabilizing selection toward a moving optimum (Hansen and Martins 1996). However, a single parameter model can certainly not explain the evolution of traits across all life. There have been extensions to the model, such as a single Ornstein-Uhlenbeck (OU) process that has a constant pull toward an optimum value, a multiple mean OU process with different possible means for different groups (Hansen 1997;Butler and King 2004), and multiple rate Brownian motion processes allowing different rates of evolution on different branches (O'Meara et al. 2006;Thomas et al. 2006). These models, while useful, still do not cover all the scenarios biologists want to examine. For example, existing models with a value toward which species are being pulled all have a fixed strength of pull over the entire history of the group. It is possible to allow the rate of stochastic motion to vary, or the value of the attractor to vary, but not for both to vary. Such restrictions on model complexity may make sense when phylogenies are limited to a few dozen taxa. However, in an era where phylogenies can have over 55,000 taxa (Smith et al. 2011), we may be so bold as to attempt to fit models that vary both rates and means of the evolutionary process. This article develops and implements such models. Hansen (1997) described a model where quantitative characters are assumed to evolve according to an OU process. The Hansen model, as it has become known, expresses the amount of change in a quantitative trait along each branch in a phylogeny and is given by the stochastic differential equation: Generalizing the Hansen Model(1) Equation ...

Almost all species are subject to continuous attack by parasites and pathogens. Because parasites and pathogens tend to have shorter generation times and often experience stronger selection due to interaction than their victims do, it is frequently argued that they should evolve more rapidly and thus maintain an advantage in the evolutionary race between defence and counter-defence. This prediction generates an apparent paradox: how do victim species survive and even thrive in the face of a continuous onslaught of more rapidly evolving enemies? One potential explanation is that defence is physiologically, mechanically or behaviourally easier than attack, so that evolution is less constrained for victims than for parasites or pathogens. Another possible explanation is that parasites and pathogens have enemies themselves and that victim species persist because parasites and pathogens are regulated from the top down and thus generally have only modest demographic impacts on victim populations. Here we explore a third possibility: that victim species are not as evolutionarily impotent as conventional wisdom holds, but instead have unique evolutionary advantages that help to level the playing field. We use quantitative genetic analysis and individual-based simulations to show that victims can achieve such an advantage when coevolution involves multiple traits in both the host and the parasite.