EARLY BURSTS OF EVOLUTION ARE RARE? LET'S GET THE RATES RIGHT!

Paleontologists have classically viewed adaptive radiations as involving an early burst of diversification from an ancestral stock, with rapid divergence in morphology followed by slower diversification and divergence. A recent study of many extant animal groups published by Harmon et al. in Evolution found that adaptive radiations are rare compared to random walk models. Here I compare an early burst model for the evolution of size to a random walk model. The Harmon et al. evidence included (1) clade age; and (2) net ‘variance’ in trait means per million years (m.y.) in units of within-population standard deviations (s.d.). Forty-two clades were found to have a mean rate of divergence in body size of 0.74 s.d./m.y.; an average time depth (time scale) of 24 m.y.; and consequently a mean divergence of 17.8 s.d. units. Average generation (gen.) time is 1.71 years, so the average rate of divergence corrected from Harmon et al. is D_7.15 = 4.33 × 10^-7 s.d./gen. on a time scale of 1.40 × 10⁷ gen.

The dispersion of a random-walk evolutionary time series increases in proportion to the square root of time. For most lineages (95%) to lie within 8.9 s.d. of the starting value (a range of 17.8 s.d.), a rate of dispersion, D₀ on a time scale of one generation, is required. Here 1.96 × D₀ × √(1.40 × 10⁷) = 17.8 s.d. Solving, D₀ can be as small as 2.42 × 10^-3 s.d./gen. (a much higher rate than the Harmon et al. 4.33 × 10^-7 s.d./gen.). If D₀ is as high as 0.10 s.d./gen., a rate commonly found in scaling studies, then the time required to fill a range of 17.8 s.d. is 8.25 × 10³ gen. (much less than 1.40 × 10⁷ gen.).

What happens when the range of possible variation is filled? When a range is filled, diversification is necessarily much slower for the remaining 1.40 × 10⁷ generations. We can quibble about details, but any reasonable rates will yield an early burst of diversification with rapid evolution, followed by a long interval of much slower diversification to yield what we see today. Harmon et al.’s average rate of divergence D_7.15 = 4.33 × 10^-7 s.d./gen. is a rate on a time scale irrelevant to the evolutionary process. D_7.15 = 4.33 × 10^-7 s.d./gen. is also a rate averaged when it should have been scaled: average rates are meaningless when their time scales are different.