13.2: A State-Dependent Model of Diversification
The models that we will consider in this chapter include trait evolution and associated lineage diversification. In the simplest case, we can consider a model where the character has two states, 0 and 1, and diversification rates depend on those states. We need to model the transitions among these states, which we can do in an identical way to what we did in Chapter 7 using a continuous-time Markov model. We express this model using two rate parameters, a forward rate q 01 and a backwards rate q 10 .
We now consider the idea that diversification rates might depend on the character state. We assume that species with character state 0 have a certain speciation rate ( λ 0 ) and extinction rate ( μ 0 ), and that species in 1 have potentially different rates of both speciation ( λ 1 ) and extinction ( μ 1 ). That is, when the character evolves, it affects the rate of speciation and/or extinction of the lineages. Thus, we have a six-parameter model (Maddison et al. 2007) . We assume that parent lineages give birth to daughters with the same character state, that is that character states do not change at speciation.
It is straightforward to simulate evolution under our state-dependent model of diversification . We proceed in the same way as we did for birth-death models, by drawing waiting times, but these waiting times can be waiting times to the next character state change, speciation, or extinction event. In particular, imagine that there are n lineages present at time t , and that k of these lineages are in state 0 (and n − k are in state 1). The waiting time to the next event will follow an exponential distribution with a rate parameter of:
\[ρ = k(q_{01} + λ_0 + μ_0)+(n − k)(q_{10} + λ_1 + μ_1) \label{13.1}\]
This equation says that the total rate of events is the sum of the events that can happen to lineages with state 0 (state change to 1, speciation, or extinction) and the analogous events that can happen to lineages with state 1. Once we have a waiting time, we can assign an event type depending on probabilities. For example, the probability that the event is a character state change from 0 to 1 is:
\[p_{q_{01}} = (n ⋅ q_{01})/ρ\label{13.2}\]
And the probability that the event is the extinction of a lineage with character state 1 is:
\[p_{μ_1} = \dfrac{(n − k)⋅μ_1}{ρ} \label{13.3}\]
And so on for the other four possible events.
Once we have picked an event in this way, we can randomly assign it to one of the lineages in the appropriate state, with each lineage equally likely to be chosen. We then proceed forwards in time until we have a dataset with the desired size or total time depth.
An example simulation is shown in Figure 13.1. As you can see, under these model parameters the impact of character states on diversification is readily apparent. In the next section we will figure out how to extract that information from our data.