### Equal-rates Markov Models

The simplest evolutionary model is known as an equal-rates Markov model (ERM). Here we start with one species (at time zero) that splits into two (at time one). Each of these splits into two more species at time two, making four in total. This process continues in an exponential fashion (i.e. 1, 2, 4, 8, 16, 32…) with each doubling taking place in the same amount of time (hence the term ‘equal-rates’).Of course many real trees don’t work like this. Some species do not split, others go extinct and others still split more rapidly. All these deviations from the ‘equal rates’ assumption may fall within the assumptions of a random walk. In other words, in reality one expects evolution to be quixotic and opportunistic, and a perfect ERM tree would never arise. But, how far does the tree have to deviate from the ERM perfect expectation to be deemed sufficiently different that some additional process might be involved?

A **random walk** encompasses the kind of random fluctuations that might happen in real evolution – a flurry of speciation in one lineage, a batch of extinctions in another – but which are not entirely unexpected. If the tree pattern though diverges beyond the expectations of the random walk, then there may be evidence for a **trend**, say, a directional pattern of evolution where a cause might be sought. An example might be that large species survive better than small ones, and this deviates significantly from a random walk, and so this is a real trend to larger size within the clade, and interesting biological causes might be sought.

Statistically significant deviations from the random-walk expectations of a classic ERM are called **diversification shifts**. Computer programs allow diversification shifts to be identified by processing a complete phylogenetic tree, and assessing the amount of splitting at each **node**, or branching point. So, in our dinosaur supertree, with 420 species, we had 419 branching points, but only 11 of these showed evidence of diverging sufficiently from the ERM expectation that they could be identified as diversification shifts.

### More information

An excellent introduction to Equal-rate Markov models, and diversification shifts, is Nee (2006), and the standard software then to do the calculations was SymmeTree (Chan & Moore 2005).

- Chan, K. M. A. and Moore, B. R. 2005. SymmeTree: whole-tree analysis of differential diversification rates.
*Bioinformatics*21, 1709-1710. - Nee, S. 2006. Birth-death models in macroevolution.
*Annual Reviews of Ecology and Systematics*37, 1-26.

The supertree, and our further macroevolutionary studies are developed further here.