A second model considered by Hansen and Martins (1996) describes the circumstance where traits change in a punctuated manner. 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 (for example, major geographic dispersal or vicariance events, or the evolution of certain traits).
We have developed peak shift models by integrating OU models and reversible-jump MCMC (Uyeda and Harmon 2014). The mathematics of this model are beyond the scope of this book, but follow closely from the description of the multi-rate Brownian motion model described in the section “variation in rates of trait evolution across clades,” above. In this case, when we change model parameters, we move among OU regimes, and can alter the OU model parameters σ2 or α. The approach can be used to either identify parts of the tree that are evolving in separate regimes or to test particular hypotheses about the drivers of evolution.