Stochastic branching models allow for inference of speciation and extinction rates. In these tutorials we focus on the different types of macroevolutionary models to study diversification processes and thus the diversification-rate parameters themselves.

Macroevolutionary diversification rate estimation focuses on different key hypothesis, which may include: adaptive radiation, diversity-dependent and character diversification, key innovations, and mass extinction. We classify these hypotheses primarily into questions whether diversification rates vary through time, and if so, whether some external, global factor has driven diversification rate changes, or if diversification rates vary among lineages, and if so, whether some species specific factor is correlated with the diversification rates.

Below, we describe each of the fundamental questions regarding diversification rates.

*What is the global rate of diversification in my phylogeny?*
The most basic models estimate parameters of the birth-death process
(*i.e.*, rates of speciation and extinction, or composite parameters
such as net-diversification and relative-extinction rates)
under the assumption that rates have remained constant across lineages and through time.
This is the most basic example and should be treated as a primer and introduction into the topic.

For more information, we recommend the Simple Diversification Rate Estimation.

*Is there diversification rate variation through time in my phylogeny?*
There are several reasons why diversification rates for the entire study group can vary through time, for example:
adaptive radiation, diversity dependence and mass-extinction events.
We can detect a signal any of these causes by detecting diversification rate variation through time.

The different tutorials references below cover different scenarios for diversification rate variation through time. The common theme of these studies is that the diversification process is tree-wide, that is, all lineages of the study group have the exact same rates at a given time.

In ‘RevBayes’ we use an episodic birth-death model to study diversification rate variation through time. That is, we assume that diversification rates are constant within an epoch but may shift between episodes (Stadler (2011), Höhna (2015)). Then, we are estimating the diversification rates for each episode, and thus diversification rate variation through time.

You can find examples and more information in the Episodic Diversification Rate Estimation.

Another question in this category asks whether our study tree was impacted by a mass-extinction event (where a large fraction of the standing species diversity is suddenly lost, e.g., Höhna (2015), May et al. (2016)).

*Are diversification rates correlated with some abiotic (e.g., environmental) variable in my phylogeny?*
If we have found evidence in the previous section that diversification rates vary through time,
then we can start asking the question whether these changes in diversification
rates are driven by some abiotic (e.g., environmental) factors.
For example, we can ask whether changes in diversification rates are correlated with
environmental factors, such as environmental CO_{2} or temperature (Condamine et al. 2013; Condamine et al. 2018).

You can find examples and more information in the Environmental-dependent Speciation & Extinction Rates.

*Is there evidence that diversification rates have varied across the branches of my phylogeny?*
Have there been significant diversification-rate shifts along branches in my phylogeny,
and if so, how many shifts, what magnitude of rate-shifts and along which branches?
Similarly, one may ask what are the branch-specific diversification rates?

You can study diversification rate variation among lineages using our birth-death-shift process (Höhna et al. 2019). Examples and more information is provided in the Branch-Specific Diversification Rate Estimation.

If we have found that there is rate variation among lineage, then we could ask if diversification rates correlated with some biotic (e.g., morphological) variable. This can be addressed by using character-dependent birth-death models (often also called state-dependent speciation and extinction models; SSE models). Character-dependent diversification-rate models aim to identify overall correlations between diversification rates and organismal features (binary and multi-state discrete morphological traits, continuous morphological traits, geographic range, etc.). For example, one can hypothesize that a binary character, say if an organism is herbivorous/carnivorous or self-compatible/self-incompatible, impact the diversification rates. Then, if the organism is in state 0 (e.g., is herbivorous) it has a lower (or higher) diversification rate than if the organism is in state 1 (e.g., carnivorous) (Maddison et al. 2007).

You can find examples and more information in the Background on state-dependent diversification rate estimation.

We begin this section with a general introduction to the stochastic birth-death branching process that underlies inference of diversification rates in RevBayes. This primer will provide some details on the relevant theory of stochastic-branching process models. We appreciate that some readers may want to skip this somewhat technical primer; however, we believe that a better understanding of the relevant theory provides a foundation for performing better inferences. We then discuss a variety of specific birth-death models, but emphasize that these examples represent only a tiny fraction of the possible diversification-rate models that can be specified in RevBayes.

Our approach is based on the *reconstructed evolutionary process*
described by (Nee et al. 1994); a birth-death process in which only sampled,
extant lineages are observed. Let $N(t)$ denote the number of species at
time $t$. Assume the process starts at time $t_1$ (the ‘crown’ age of
the most recent common ancestor of the study group, $t_\text{MRCA}$)
when there are two species. Thus, the process is initiated with two
species, $N(t_1) = 2$. We condition the process on sampling at least one
descendant from each of these initial two lineages; otherwise $t_1$
would not correspond to the $t_\text{MRCA}$ of our study group. Each
lineage evolves independently of all other lineages, giving rise to
exactly one new lineage with rate $b(t)$ and losing one existing lineage
with rate $d(t)$ ( and
). Note that although each lineage evolves
independently, all lineages share both a common (tree-wide) speciation
rate $b(t)$ and a common extinction rate $d(t)$
(Nee et al. 1994; Höhna 2015). Additionally, at certain times,
$t_{\mathbb{M}}$, a mass-extinction event occurs and each species
existing at that time has the same probability, $\rho$, of survival.
Finally, all extinct lineages are pruned and only the reconstructed tree
remains ().

To condition the probability of observing the branching times on the survival of both lineages that descend from the root, we divide by $P(N(T) > 0 | N(0) = 1)^2$. Then, the probability density of the branching times, $\mathbb{T}$, becomes

\[\begin{aligned} P(\mathbb{T}) = \frac{\overbrace{P(N(T) = 1 \mid N(0) = 1)^2}^{\text{both initial lineages have one descendant}}}{ \underbrace{P(N(T) > 0 \mid N(0) = 1)^2}_{\text{both initial lineages survive}} } \times \prod_{i=2}^{n-1} \overbrace{i \times b(t_i)}^{\text{speciation rate}} \times \overbrace{P(N(T) = 1 \mid N(t_i) = 1)}^\text{lineage has one descendant}, \end{aligned}\]and the probability density of the reconstructed tree (topology and branching times) is then

\[\begin{aligned} P(\Psi) = \; & \frac{2^{n-1}}{n!(n-1)!} \times \left( \frac{P(N(T) = 1 \mid N(0) = 1)}{P(N(T) > 0 \mid N(0) = 1)} \right)^2 \nonumber\\ \; & \times \prod_{i=2}^{n-1} i \times b(t_i) \times P(N(T) = 1 \mid N(t_i) = 1) \label{eq:tree_probability} \end{aligned}\]We can expand Equation ([eq:tree_probability]) by substituting $P(N(T) > 0 \mid N(t) =1)^2 \exp(r(t,T))$ for $P(N(T) = 1 \mid N(t) = 1)$, where $r(u,v) = \int^v_u d(t)-b(t)dt$; the above equation becomes

\[\begin{aligned} P(\Psi) = \; & \frac{2^{n-1}}{n!(n-1)!} \times \left( \frac{P(N(T) > 0 \mid N(0) =1 )^2 \exp(r(0,T))}{P(N(T) > 0 \mid N(0) = 1)} \right)^2 \nonumber\\ \; & \times \prod_{i=2}^{n-1} i \times b(t_i) \times P(N(T) > 0 \mid N(t_i) = 1)^2 \exp(r(t_i,T)) \nonumber\\ = \; & \frac{2^{n-1}}{n!} \times \Big(P(N(T) > 0 \mid N(0) =1 ) \exp(r(0,T))\Big)^2 \nonumber\\ \; & \times \prod_{i=2}^{n-1} b(t_i) \times P(N(T) > 0 \mid N(t_i) = 1)^2 \exp(r(t_i,T)). \label{eq:tree_probability_substitution} \end{aligned}\]For a detailed description of this substitution, see Höhna (2015). Additional information regarding the underlying birth-death process can be found in Thompson (1975) [Equation 3.4.6] and Nee et al. (1994) for constant rates and Höhna (2013), Höhna (2014), Höhna (2015) for arbitrary rate functions.

To compute the equation above we need to know the rate function, $r(t,s) = \int_t^s d(x)-b(x) dx$, and the probability of survival, $P(N(T)!>!0|N(t)!=!1)$. Yule (1925) and later Kendall (1948) derived the probability that a process survives ($N(T) > 0$) and the probability of obtaining exactly $n$ species at time $T$ ($N(T) = n$) when the process started at time $t$ with one species. Kendall’s results were summarized in Equation (3) and Equation (24) in Nee et al. (1994)

\[\begin{aligned} P(N(T)\!>\!0|N(t)\!=\!1) & = & \left(1+\int\limits_t^{T} \bigg(\mu(s) \exp(r(t,s))\bigg) ds\right)^{-1} \label{eq:survival} \\ \nonumber \\ P(N(T)\!=\!n|N(t)\!=\!1) & = & (1-P(N(T)\!>\!0|N(t)\!=\!1)\exp(r(t,T)))^{n-1} \nonumber\\ & & \times P(N(T)\!>\!0|N(t)\!=\!1)^2 \exp(r(t,T)) \label{eq:N} %\\ %P(N(T)\!=\!1|N(t)\!=\!1) & = & P(N(T)\!>\!0|N(t)\!=\!1)^2 \exp(r(t,T)) \label{eq:1} \end{aligned}\]An overview for different diversification models is given in Höhna (2015).

## Phylogenetic trees as observations

The branching processes used here describe probability distributions on phylogenetic trees. This probability distribution can be used to infer diversification rates given an “observed” phylogenetic tree. In reality we never observe a phylogenetic tree itself. Instead, phylogenetic trees themselves are estimated from actual observations, such as DNA sequences. These phylogenetic tree estimates, especially the divergence times, can have considerable uncertainty associated with them. Thus, the correct approach for estimating diversification rates is to include the uncertainty in the phylogeny by, for example, jointly estimating the phylogeny and diversification rates. For the simplicity of the following tutorials, we take a shortcut and assume that we know the phylogeny without error. For publication quality analysis you should always estimate the diversification rates jointly with the phylogeny and divergence times.

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