Three modes of evolution? Remarks on rates of evolution and time scaling

Rates of evolution get smaller when they are measured over longer time intervals. As first shown by Gingerich, rates of morphological change measured from fossil time series show a robust minus-one scaling with time span, implying that evolutionary changes are just as large when measured over a hundred years as when measured over a hundred-thousand years. On even longer time scales, however, the scaling shifts toward a minus-half exponent consistent with evolution behaving as Brownian motion, as commonly observed in phylogenetic comparative studies. Here, I discuss how such scaling patterns arise, and I derive the patterns expected from standard stochastic models of evolution. I argue that observed shifts cannot be easily explained by simple univariate models, but require shifts in mode of evolution as time scale is changing. To illustrate this idea, I present a hypothesis about three distinct, but connected, modes of evolution. I analyze the scaling patterns predicted from this, and use the results to discuss how rates of evolution should be measured and interpreted. I argue that distinct modes of evolution at different time scales act to decouple micro- and macroevolution, and criticize various attempts at extrapolating from one to the other.