12.4: Diversity-Dependent Models
Time-dependent models in the previous section are often used as a proxy to capture processes like key innovations or adaptive radiations (Rabosky 2014) . Many of these theories suggest that diversification rates should depend on the number of species alive in a certain time or place, rather than time (Phillimore and Price 2008; Etienne and Haegeman 2012; Etienne et al. 2012; Rabosky 2013; Moen and Morlon 2014) . Therefore, we might want to define speciation rate in a truly diversity dependent manner rather than using time as a proxy:
\[ \lambda(t) = \lambda_0 (1 - \frac{N_t}{K}) \label{12.8}\]
Since speciation rate now depends on number of lineages rather than time, we can't plug this expression into our general formula (Morlon et al. 2011) . Instead, we can use the approach outlined by Etienne et al. (2012) and Etienne et al. (2016) . This approach focuses on numerical solutions to differential equations moving forward through time in the tree. The overall idea of the approach is similar to Morlon, but details differ; likelihoods from Etienne et al. (2012) should be directly comparable to all the likelihoods presented in this book provided that the conditioning is the same and they are multiplied by the total number of topological arrangements, ( n + 1)!, to get a likelihood for the tree rather than for the branching times. Etienne's approach can also deal with incomplete sampling under a uniform sampling model.
As an example, we can fit a basic model of diversity-dependent speciation to our phylogenetic tree of lungless salamanders introduced above. Doing so, we find a ML estimate of λ 0 = 0.099, μ = 0, and K = 979.9, with a log-likelihood of 537.3 and an AIC of -1068.7. This is a substantial improvement over any of the time-varying models considered above, and evidence for diversity dependence among lungless salamanders.
Both density- and time-dependent approaches have become very popular, as time-dependent diversification models are consistent with many ecological models of how multi-species clades might evolve through time. For example, adaptive radiation models based on ecological opportunity predict that, as niches are filled and ecological opportunity “used up,” then we should see a declining rate of diversification through time (Etienne and Haegeman 2012; Rabosky and Hurlbert 2015) . By contrast, some models predict that species create new opportunities for other species, and thus predict accelerating diversification through time (Emerson and Kolm 2005) . These are reasonable hypotheses, but there is a statistical challenge: in each case, there is at least one conceptually different model that predicts the exact same pattern. In the case of decelerating diversification, the predicted pattern of a lineage-through-time plot that bends down towards the present day can also come from a model where lineages accumulate at a constant rate, but then are not fully sampled at the present day (Pybus and Harvey 2000) . In other words, if we are missing some living species from our phylogenetic tree and we don’t account for that, then we would mistake a constant-rates birth death model for a signal of slowing diversification through time. Of course, methods that we have discussed can account for this. Some methods can even account for the fact that the missing taxa might be non-random, as missing taxa tend to be either rare or poorly differentiated from their sister lineages (e.g. often younger than expected by chance; Cusimano and Renner 2010; Brock et al. 2011) . However, the actual number of species in a clade is always quite uncertain and, in every case, must be known for the method to work. So, an alternative explanation that is often viable is that we are missing species in our tree, and we don't know how many there are. Additionally, since much of the signal for these methods comes from the most recent branching events in the tree, some "missing" nodes may simply be too shallow for taxonomists to call these things "species." In other words, our inferences of diversity dependence from phylogenetic trees are strongly dependent on our understanding of how we have sampled the relevant taxa.
Likewise, a pattern of accelerating differentiation mimics the pattern caused by extinction. A phylogenetic tree with high but constant rates of speciation and extinction is nearly impossible to distinguish from a tree with no extinction and speciation rates that accelerate through time.
Both of the above caveats are certainly worth considering when interpreting the results of tests of diversification from phylogenetic data. In many cases, adding fossil information will allow investigators to reliably distinguish between the stated alternatives, although methods that tie fossils and trees together are still relatively poorly developed (but see Slater and Harmon 2013) . And many current methods will give ambiguous results when multiple models provide equivalent explanations for the data - as we would hope!