Speciation and Extinction Rates

The birth-death process

Speciation and extinction are modeled using a birth-death process $N(t)$ which encodes the number of species (or "lineages") at alive at time $t$. This is a continuous time Markov process with two parameters:

• $\lambda:$ birth rate (per lineage per unit of time)
• $\mu:$ death rate (per lineage per unit of time)

Under this model, if one begins with a single ancestral lineage (i.e., $N(0)=1$), there are two possible events that can occur: either the lineage goes extinct, or it speciates (i.e., splits into two daughter lineages). The time until the next event is an exponential random variable with rate λ+μ. When the event occurs,

• the probability that it is a speciation is $\frac{\lambda}{\lambda+\mu}$;
• the probability that it is an extinction is $\frac{\mu}{\lambda+\mu}$.

Supposing that the lineage speciates, the process begins anew, and is run simultaneously for each of the two daughter lineages. This procedure is continued until a stop criteria is reached: either (1) when a certain time has elapsed, (2) when a certain number of species is reached, or (3) when all species have gone extinct.

For more details, see 1. Here is a visual example of a tree being built using such a process:

(In this figure, the simulation starts at the first speciation event, which is why the tree begins with two lineages rather than one.)

In this tutorial, we focus on the simple birth-death process, which assumes that the parameters $\mu$ and $\lambda$ are (1) constant throughout time, and (2) are the same for all lineages in the tree. The first assumption is not appropriate for persisting over long geologic time scales (e.g., due to mass extinction events), and the second assumption is unrealistic for large, highly diverse clades. Both of these assumptions can be relaxed.

Connection to exponential growth models

We may reparameterize the birth-death process using the following two parameters:

• $r = \lambda - \mu$ (the "diversification rate")
• $\epsilon = \mu/\lambda$ (the "relative extinction rate" - this tells us how quickly new lineages turnover)

By standard results in probability theory (see 2 and 3), the expected number of lineages at time $t$ is

$$ E(t) = N(0)e^{rt}. $$

Thus, on average, we expect the number of lineages to grow exponentially if $r>0$. Conversely, if $r<0$, the number of species will, on average, decline exponentially; with enough time, if $r\leq 0$, all lineages will go extinct with probability one.

Note that even if $r>0$, there is still a chance that all lineages will go extinct.

Lineages Through Time (LTT) Plots

A commonly-used way to summarize the history of diversification is through a lineages through time (LTT) plot, which plots the number of lineages as a function of time.

When the number of lineages is large and birth/death rates are constant over time, the LTT plot is expected to be linear on a log scale (as shown below).

In practice, this "straight line" relationship is often not observed. Interpreting deviations from this pattern, however, is fraught with difficulties and caution is warranted when attempting to make inferences from birth-death trees. There are two reasons for this. First, when only extant taxa are observed, birth-death trees do not contain enough information to uniquely identify both $\mu$ and $\lambda$, if these rate parameters are allowed to vary over time (see 4; but also 5). Hence, an observed pattern may be consistent with multiple diversification histories.

Second, birth-death trees are affected by nontrivial sources of bias. One example of this is the "push of the past" 6. This is a form of sampling bias which arises because the clades we study are atypical in that they survived to the present. Clades that survived for a long time are more likely to have experienced, by chance, an unusually large number of speciations early on in their history (since the ones that didn't were far more likely to have completely died off). The same is true of clades which are highly diverse in the present day (i.e., which have many tips), and this source of bias is known as the "large clade effect" (see 7). The effect of these biases is shown in the following plot:

Note how the diversification rate appears to decrease over time, despite constant rate parameters.

References

[1] Luke J. Harmon, Phylogenetic Comparative Methods (2019). Chapter 10, https://lukejharmon.github.io/pcm/chapter10_birthdeath/#section-10.2-the-birth-death-model

[2] William Feller, An Introduction to Probability Theory and its Applications, vol. 1, 7th ed., John Wiley & Sons, (1957). Pages 407-411.

[3] David G. Kendall, On the Generalized 'Birth-and-Death' Process, Ann. Math. Statist. 19(1): 1-15 (March, 1948). DOI: 10.1214/aoms/1177730285

[4] Stilianos Louca and Matthew W. Pennell, Extant timetrees are consistent with a myriad of diversification histories, Nature (2020). DOI: 10.1038/s41586-020-2176-1

[5] Brandon Legried and Jonathan Terhorst, A class of identifiable phylogenetic birth–death models, Proc. Natl. Acad. Sci. U.S.A. (2022) DOI: 10.1073/pnas.2119513119

[6] Sean Nee, Robert M. May, and Paul H. Harvey, The reconstructed evolutionary process, Phil. Trans. R. Soc. London. B (1994). DOI: 10.1098/rstb.1994.0068

[7] G. E. Budd and R. P. Mann, History is written by the victors: The effect of the push of the push of the past on the fossil record, Evolution (2018) DOI: 10.1111/evo.13593