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Time in evolution is arithmetic, with units of equal measure added repeatedly. Time is directional in the sense that units are added but never subtracted. The time scale of a generating process may be different from the time scale of an observation, and the distinction is important. Natural selection and random change are generation-scale processes acting on the parade of phenotypes produced in each generation. Generation times of organisms vary in length from minutes to decades. Close correlation with body size in groups like mammals enables unknown generation times to be estimated from size. Differences in organisms over spans of ecological and evolutionary time represent an accumulation of changes happening on a time scale of generations. Microevolution and macroevolution parallel ecological and evolutionary time, but these terms are also used to distinguish evolution within species from evolution between species and higher taxa. Evolutionary change is often imagined and modeled on a ‘game board’ or simulation space of form and time that are equivalent in length and width, but these may differ greatly.
Variation is essential for natural selection in evolution. Attempts to quantify biological variation in the nineteenth century focused on its resemblance to distributions of measurement error in astronomy and physics, but in biology variable populations evolve from variable populations: the expectation is not an average with error, but a full distribution of variation in each successive generation. The normality of biological variation is geometric rather than arithmetic: biological variation is lognormal rather normal, and individual differences are differences of proportion. Logarithms employed to transform counts to proportions can be chosen to reflect halving and doubling (log2), standardized deviations (ln or loge), or orders of magnitude (log10). Comparisons of populations in standard deviation units incorporate dimension and remove its effect, making standard deviations the preferred units for expressing the similarities and differences of variable populations.
The theory of punctuated equilibria promoted by Niles Eldredge and Stephen Jay Gould in 1972 re-ignited the long-standing conflict between traditionalists who believe that species are fixed in form and Lamarck-Darwin progressivists who envision species changing through time. The history of Poecilozonites developed by Gould is slightly complicated, but there was no special ‘punctuation’ in the speciation nor any static ‘equilibrium’ in Gould’s original interpretation. The history of Phacops developed by Eldredge is similarly slightly complicated, but there was no special ‘punctuation’ in the speciation, nor any statistically-justified ‘equilibrium’ in Eldredge’s interpretation. The Metrabdotos species lineages described by Cheetham et al. in 2007 are lineages in stasis. Species pairs diverged rapidly, yielding punctuated patterns, but the divergences can all be explained by directional selection at the median step rate for field studies documented here in Chapter 8. Empirical step rates are high and change by natural selection is much faster than many paleontologists appreciate.
A random walk is a pattern of change or movement in successive steps, with the change at each step governed by chance, independent of what came before. Time t is the independent variable in random walks and in evolution. Other measures such as displacement d or cumulative displacement s are dependent variables because they change as a function of time. An evolutionary random walk is a time series with successive si drawn from repeated iteration of si + 1 = si ± d0, where d0 is a random variable drawn from a normal distribution with mean μ and standard deviation σ.
Rates are ratios. The step rate r0, moving to the left or right, is ±r0 = ±d0 / i0, which defines the relationship of d0 and i0. Individual random walks are by nature unpredictable and idiosyncratic, with a wide range of possible behaviors. Random walks aggregated as Brownian diffusion share some common characteristics: (1) terminal values are normally distributed with most random walks near their starting point and few in each tail; (2) the variance of the aggregate increases as a constant proportion of time; and (3) the standard deviation of the aggregate increases as the square root of time.
Response to artificial or natural selection can be expressed in terms of a selection differential in measurement units, or a selection intensity in standard deviation units. Heritability determines the proportional relationship of response to selection in both cases. Truncation is the most efficient form of directional selection. The response is independent of population size and the response curve increases indefinitely. Gradient selection is less efficient. The response is independent of population size but the response curve for a linear gradient is limited at R < 0.2 ∙ h2 ∙ σ. Stabilizing selection requires a balance of truncation or gradient selection (or both). Stability in an evolutionary time series can also be achieved through a succession of directional corrections, with the response in each generation determined by the difference between the mean at the time and some optimal value. Random drift is acutely sensitive to sample size and inherently self-balancing. Consequently, random drift has much less power to move a mean or to constrain it than any form of mass selection.
Analysis of 34 paleontological studies in the fossil record yields no step rates quantifying change from one generation to the next, and thus cannot inform our understanding of generation-to-generation change by natural selection. However, temporal scaling of base rates found in fossil studies yields an LRI intercept and residuals consistent with the step rates found in selection experiments and field studies. Some 84% of the variance in evolutionary rates is determined by variance in interval lengths (rate denominators), meaning rates in the fossil record cannot be compared to rates in field studies without temporal scaling. Stasis predominates in the fossil record but change is evident too. Rapid change in a fossil study is change lying above a line fit to an empirical LRI distribution of base rates and their corresponding intervals: change lying above Y = −0.895 ∙ X − 0.615, where Y is log10 of the rate r and X is log10 of the corresponding interval i. When rate numerators are constrained to a range three orders of magnitude smaller than the range of their denominators, inverse scaling is inevitable.
Transformation of biological species from one to another through many generations was a radical thesis proposed by Jean-Baptiste Lamarck in 1809 and forcefully opposed by Charles Lyell in 1832. Lyell, with Carl Linnaeus and others, believed that species, once created, have distinguishing characters in common that remain the same and never vary. In the Origin of Species in 1859 Charles Darwin supported and extended the thesis of Lamarck and contradicted the reactionary antithesis of Lyell. The debate between "transformationists" and "punctuationists" extends to the present day. The "Modern Synthesis" of evolution in the mid-19th century was at best a partial synthesis, with "quantum evolution" proposed to describe and explain rapid shifts of species from one equilibrium state to another. J. B. S. Haldane calculated very slow rates of evolutionary change from the fossil record and raised the possibility of micromutation as an alternative to natural selection to explain long-term evolution. The challenge before us is reconciliation of the Lamarck-Darwin thesis of slow gradual change in species with the Lyell-Linnaeus antithesis involving mysterious origins followed by stasis.
Analysis of 14 selection experiments in 8 published studies yields 672 step rates quantifying high- and low-line selection from one generation to the next. The median step rate for this sample, in haldanes, is h0 = 0.33 standard deviations per generation on a time scale of one generation. High-line experiments selecting for increases in trait value yield virtually the same rates as low-line experiments selecting for decreases in trait value, with median h0 values of 0.32 and 0.35, respectively. Both are near the selection median of h0 = 0.33. Control lineages not subject to experimental selection have a median h0 = 0.23. Control lineages with a substantial number of generations have temporal scaling slopes showing that they are stationary rather than random, which is attributable to stabilizing natural selection. What separates stationary lineages responding to stabilizing selection from directional lineages responding to directional selection is the pattern of change in their signs. Step rates observed in selection experiments (high lines, low lines, and control lines) exceed ‘null’ rates expected for purely random change by factors ranging from 2 to 100.
A realistic evolutionary model must match the generation-to-generation time scale being modeled and employ rates that are representative on this time scale. Time in a Brownian diffusion evolutionary model is time in generations, a requirement often overlooked or ignored. Forward-modeling of Brownian diffusion shows that the time span of phylogeny’s influence on body weight in mammalian carnivores and ungulates lasts no more than about 10,000 generations, or some 57,000 years, less than one tenth of one percent of the combined group’s evolutionary history. Statistical dependence on phylogeny does not extend automatically to any or all characteristics of interest, and dependence on phylogeny may never last long for the morphological and life history traits we most often study in comparative biology.
Analysis of 57 field studies yields a total of 814 independent step rates quantifying change from one generation to the next. The median step rate for field studies is h0 = 0.15 standard deviations per generation on a time scale of one generation. Step rates following directional selection in a laboratory setting have the same range of values as step rates observed in field studies. This consistency shows: (a) experimental selection is not artificial, but represents what we see in field studies; and (b) the change we see in field studies is what we expect for experimental and hence natural selection. Rapid change in a field study is change lying above a line fit to an empirical LRI distribution of step rates and base rates: change lying above Y = −0.784 ∙ X − 0.787, where Y is log10 of the rate r and X is log10 of the corresponding interval i. Empirically, the proportion of zero rates in field studies is 1.34%, mostly due to rounding, and zero rates have negligible effect on the rate statistics presented here. The anthropologist’s ‘secular trend’ of increasing human stature has rates in the range found here for evolution by natural selection.
The Lamarck-Darwin thesis of slow and gradual, step-by-step evolutionary change and the Lyell-Linnaeus antithesis of more rapid change and no-change both share a dominant central stasis mode representing negligible change through time. Thesis and antithesis differ in the distribution of rates about the central stasis mode. Slow step-by-step evolutionary change, positive and negative, leads to a single central distribution of rates. More rapid changes imperceptible on timescales generally studied by paleontologists create what appear as distinct secondary modes, positive and negative, separated from the central stasis mode. The Lamarck-Darwin thesis of step-by-step evolutionary change, and the Lyell-Linnaeus antithesis of ‘change’ and ‘no change’ (or simply ‘no change’) are unified when rates are studied on all scales of time. Lamarck and Darwin were right about the gradual nature of evolutionary change. Darwin was wrong in claiming evolution to be slow. Rates are slow when averaged down over thousands and millions of generations, but step rates averaging 0.15 standard deviations per generation that animate evolutionary by natural selection are fast by any measure.
The long-standing Lamarck-Darwin thesis of slow and gradual evolutionary change can be reconciled with the Lyell-Linnaeus antithesis of no-change ‘equilibrium’ supported by paleontologists for more than a century. Both theses are deficient to some degree: first, evolution on the generational time scale of natural selection is gradual but not slow, and second, paleontologists working on time scales of thousands of generations cannot see change that is fast. Rates from field studies and fossil studies, combined, form a continuous distribution. Differences leading to competing Lamarck-Darwin and Lyell-Linnaeus perceptions of change are not related to the evolutionary process itself but to what we see and do not see in field studies compared to fossil studies. High step rates are the key because they facilitate rapid change in individual lineages on short time scales, with rapid stabilization of biotas on longer scales of time. Darwin was conservative and wrong to believe that evolution by natural selection is slow, intermittent, and weak. Finding evolution to be fast means the natural selection Darwin proposed is persistent, ubiquitous, and more important than we knew.
Two models of diversification are commonly recognized in evolutionary radiations: Brownian diffusion (BD) and Ornstein-Uhlenbeck diffusion (OU). A third narrowly-defined ‘early burst’ model (ER*) is a form of Brownian diffusion with a step rate decreasing through time, leading to exponential slowing of net rates of change. Darwin’s finches, Geospizini, are analyzed as a case study for comparison of BD, OU, and ER*. The step rate of evolutionary change in tarsus length for Geospizini can be estimated from temporal scaling of node differences analyzed in the context of a phylogeny. This yields an estimated step rate of h0 = 10−0.643 = 0.228 standard deviations per generation, which is close to the median step rate of h0 = 0.153 for field studies found here in Chapter 8. Forward modeling of Brownian diffusion, Ornstein-Uhlenbeck diffusion, and the narrow early burst model through 2.5 million generations of geospizine diversification indicates that BD, OU, and EB* are all are early burst models, with net disparity increasing most rapidly at the beginning of each radiation. Early bursts of evolution are common, as paleontologists and ecologists have recognized for many years.