Supplementary methods and figures
Density-dependent diversification in North American wood-warblers
Daniel L. Rabosky and Irby J. Lovette
Expected lineage-through-time curves: Because our results support density-dependent declines in diversification over an alternative model where rates vary continuously through time, we attempted to identify features of the expected lineage-through-time (LTT) curves that differed among models. There are two ways in which an expected LTT curve can differ from the observed curve: (1) the curves may differ qualitatively in shape, and (2) the curves may differ in the expected number of lineages produced after a fixed amount of time. We found the expected LTT curves under the maximum likelihood parameterizations of the model as
which gives the expected number of descendent lineages at any point in time t, starting with n = 2 initial lineages at time t = 0. It must be noted that the time-varying diversification models we employed assume fixed clade age and the models do not necessarily make sense if age is not limited to the value used to derive parameter estimates. For example, speciation rates for the continuous-decline model can become negative if t > K (eqn 2.3). There is thus no requirement that the maximum likelihood parameter estimates for a given diversification model will yield the observed number of lineages in the wood-warbler phylogeny. Expected LTT curves are shown in figure S2. In the case of the continuous-decline model, the maximum likelihood parameterization does not yield the correct number of lineages (n = 25) after 1.0 time units.
Simulations: To further explore the differences in fit between density-dependent and continuous decline models, we simulated LTT curves under the maximum likelihood parameter estimates for each model. As discussed above, the time-varying diversification models we employed require simulations of fixed clade age, as the parameters of the model specify a particular time course of change in the speciation rate and are conditional on the existence of an evolutionary process of the same age as the wood-warbler tree (1.0 time units). This implies that the number of lineages in existence at the end of each simulation is itself a random variable.
Most previous studies that have simulated time-varying diversification processes have used discrete-time phylogenetic simulation algorithms (e.g., Rabosky 2006b), in which phylogenetic trees are generated by iterating over a series of time steps such that each lineage has a probability of giving birth or going extinct each time step. However, this discrete-time approach merely approximates the continuous-time diversification process, and we used a simulation procedure that permits phylogenies to be simulated in continuous time (Rabosky & Lovette 2008).
We first divided the total simulation time (1.0 time units) into 100 intervals of t = 0.01 time units. We then calculated the value of the speciation rate () for the midpoint of each interval (e.g., at t = 0.005, 0.015, 0.025…0.985, 0.995) using the maximum likelihood parameter estimates for each model and equations 2.3. Each simulation was initiated with two lineages, which had parameter1on the first time interval; after t = 0.01 time units, parameters were updated to 2 and the simulation was continued to the end of the second time interval (overall elapsed time of 0.02 time units). These sequential parameter updates were continued until the end of the simulation. Thus, while we used a discrete approximation to model variation in , the underlying simulation occurred in continuous time.
In this approach, the number of lineages in the tree at the end of each simulation is itself a random variable. We wanted to compare simulated LTT plots for the set of phylogenetic trees containing exactly 25 descendents at the end of the simulation (e.g., the same number of taxa as the wood-warbler tree). Under each simulation model, we simulated 500 trees of n = 25 taxa by retaining only those trees with exactly n = 25 taxa at the end of the simulation. Average LTT curves for each model are presented in figure S3 (a, b).
Simulated LTT curves resulting in n = 25 taxa at the end of the simulation look remarkably similar for density-dependent exponential and continuous-decline curves. However, this result must be considered in light of the fact that obtaining n = 25 taxa is much less likely under the maximum-likelihood parameter estimates for the continuous-decline model than the density-dependent model (e.g., figure S2). To demonstrate this directly, we simulated 10,000 phylogenies under maximum likelihood parameter estimates for each diversification model to look at the frequency distribution of clade size after 1.0 time units (figure S3, c, d).
Although the subset of LTT curves with n = 25 taxa appear similar among diversification models, the fitted models predict dramatically different species diversity (figure S3, c, d). Only the density-dependent model results in a pattern of expected species richness consistent with that observed in wood-warblers.
Figure S1
Figure S1. Tests for density-dependent diversification cannot simply contrast the likelihood of phylogenetic data under a constant-rate diversification model to a density dependent diversification model. To demonstrate this, we simulated sets of 5000 phylogenies under a constant speciation model with zero extinction, where phylogenies were 25%, 50%, 75%, or 100% complete at the species level [see Methods]. Each phylogeny was fitted with constant rate and density-dependent diversification models, and the difference in AIC scores tabulated as AICPB - AICDD , where AICPB and AICDD
are AIC scores under pure-birth and best-fit density dependent diversification models. Shown are means and 95% confidence intervals around the distribution of this statistic. As the level of incomplete sampling increases, density dependent models provide a better fit to the pattern of lineage accumulation. This phenomenon occurs even though phylogenies were simulated under a constant speciation model and is driven solely by the fact that incomplete taxon sampling generates spurious temporal declines in diversification rates.
Figure S2
Figure S2. Expected lineage-through-time curves (black) under maximum likelihood parameterizations of the three rate-variable diversification models considered in this study: (a) density-dependent, exponential, (b) density-dependent, linear, and (c) continuous-decline (linear). Curves represent analytical expectations assuming maximum likelihood parameter estimates for each model. Red curve is observed lineage accumulation curve for wood-warblers (e.g., figure 2a). The maximum likelihood parameterization of the continuous-decline model fails to reach the observed species diversity for wood-warblers (n = 25).
Figure S3
Figure S3. Expected lineage-through-time curves (black) for the subset of simulated phylogenies with exactly n = 25 lineages after 1.0 time units for density-dependent exponential (a) and continuous-decline (b) models. Phylogenies were simulated under maximum likelihood parameter estimates for each model. Although both of these curves appear to provide reasonable approximations of the observed lineage accumulation curve, this is in part illusory, as the maximum likelihood parameterization of the continuous-decline model is much less likely to result in the observed total number of lineages (n = 25) at the end of the simulation period. Frequency distributions of progeny lineages for phylogenetic trees simulated for 1.0 time units under these maximum likelihood parameters are shown for density-dependent exponential (c) and continuous-decline (d) models. Results are not shown for the density-dependent linear model, because the speciation rate becomes negative if the number of lineages in the simulation exceeds 25. Results for a and b based on 500 simulated phylogenies; results for c and d based on 10,000 simulated phylogenies.