6.5: Early Burst Models

Last updated
Save as PDF

Page ID: 21611

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Adaptive radiations are a slippery idea. Many definitions have been proposed, some of which contradict one another (reviewed in Yoder et al. 2010). Despite some core disagreement about the concept of adaptive radiations, many discussions of the phenomenon center around the idea of “ecological opportunity.” Perhaps adaptive radiations begin when lineages gain access to some previously unexploited area of niche space. These lineages begin diversifying rapidly, forming many and varied new species. At some point, though, one would expect that the ecological opportunity would be “used up,” so that species would go back to diversifying at their normal, background rates (Yoder et al. 2010). These ideas connect to Simpson’s description of evolution in adaptive zones. According to Simpson (1945), species enter new adaptive zones in one of three ways: dispersal to a new area, extinction of competitors, or the evolution of a new trait or set of traits that allow them to interact with the environment in a new way.

One idea, then, is that we could detect the presence of adaptive radiations by looking for bursts of trait evolution deep in the tree. If we can identify clades, like Darwin’s finches, for example, that might be adaptive radiations, we should be able to uncover this “early burst” pattern of trait evolution.

The simplest way to model an early burst of evolution in a continuous trait is to use a time-varying Brownian motion model. Imagine that species in a clade evolved under a Brownian motion model, but one where the Brownian rate parameter (σ²) slowed through time. In particular, we can follow Harmon et al. (2010) and define the rate parameter as a function of time, as:

\[σ^2(t)=σ_0^2e^{bt} \label{6.7}\]

We describe the rate of decay of the rate using the parameter b, which must be negative to fit our idea of adaptive radiations. The rate of evolution will slow through time, and will decay more quickly if the absolute value of b is large.

This model also generates a multivariate normal distribution of tip values. Harmon et al. (2010) followed Blomberg's "ACDC" model (2003) to write equations for the means and variances of tips on a tree under this model, which are:

\[ \begin{array}{l} \mu_i(t) = \bar{z}_0 \\ V_i(t) = \sigma_0^2 \frac{e^{b T_i}-1}{b} V_{ij}(t) = \sigma_0^2 \frac{e^{b s_{ij}}-1}{b} \end{array} \label{6.8}\]

Again, we can generate a vector of means and a variance-covariance matrix for this model given parameter values (\(\bar{z}_0\), σ², and b) and a phylogenetic tree. We can then use the multivariate normal probability distribution function to calculate a likelihood, which we can then use in a ML or Bayesian statistical framework.

For mammal body size, the early burst model does not explain patterns of body size evolution, at least for the data considered here (\(\hat{\bar{z}}_0 = 4.64\), \(\hat{\sigma}^2 = 0.088\), \(\hat{b} = -0.000001\), lnL = −78.0, AIC_c = 162.6).