# 6.5: Early Burst Models

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, AICc = 162.6).