In brief, the physical and biotic conditions of growth vary on all time-scales, with rarer events of greater magnitude making a progressively smaller contribution to the overall variance. Because the environment changes on all time-scales, allele frequencies will also show trends on all scales of calendar time. At the shortest scales, there is strong fluctuating selection that can cause appreciable shifts in allele frequency within a generation, or a few generations. Longer term changes in conditions will create fluctuating selection over longer periods, which if the period is long enough will be perceived as a directional trend. Long-term evolution is usually depicted in two alternative ways: either as a gradual change driven by chronic weak selection or as an abrupt change following a long interval of stasis. I am presenting here a third interpretation that seems to me more consistent with observations of natural selection in open populations: selection is generally rather strong and fluctuates on all time-scales such that abrupt changes can occur over short periods of time and gradual directional change occurs over long periods of time. This process is brought to a halt when the species becomes extinct, at which point it expresses directional change corresponding to the long-term environmental change on the time-scale of its longevity. This is not a new theory: for example, Stanley & Yang (1987)
found that morphological characters in fossil series of bivalves fluctuated strongly over time with little directional trend, which they attributed to ‘zigzag selection’. However, it has not yet become established as the usual mode of evolution at all time-scales, although Leroi (2000)
takes a very similar view in arguing that the scale independence of adaptive change constitutes a basic evolutionary principle.
This process can be modelled by coupling a multi-scale model of environmental change with an evolutionary model. In the environmental model, any particular category of event occurs at intervals of some given number of years, and during this interval the event continues to have the same effect on conditions of growth. More extreme events occur at longer intervals. An event in the category of events that occur every year will be an exponential random variable with some small average value; an event in the category that occur every 2 years will likewise be exponentially distributed but with somewhat greater average, and so forth. The current state of the environment is then the sum of the effects of events in all categories. Its absolute value is largely determined by rare events of a large magnitude, whereas its variation from year to year is usually attributable to smaller and more frequent kinds of events. Conditions will thereby change on all time-scales, and will fluctuate around an overall trend, no matter what period of time is chosen for a survey. This generates an increase in environmental variance over time that continues indefinitely.
The evolutionary model simulates a population of N
diploid individuals bearing a given number of loci affecting a quantitative character, to which the environmental variation σ2E
also contributes. New genetic variation is contributed by mutation to the average effect at each locus at rate u
per locus per replication. The current state of the environment defines the optimal character state, with the fitness of an individual determined by the distance of its phenotype from the optimum, using a Gaussian function of width ω2
. Individuals produce offspring in proportion to their fitness. The evolving population will then track the changing environment, always lagging somewhat behind. Its dynamics will be governed mainly by the mutation supply rate, the heritability and the strength of selection, which depend on N
. Any result may be obtained, including extinction, but biologically reasonable parameter values will satisfy the following criteria:
- the mutation supply rate Nu < 1 per generation (if we are interested primarily in large multicellular organisms),
- the average distance between the population mean and the optimum should be about 1 phenotypic standard deviation (Estes & Arnold 2007, fig. 8; Bell 2008, p. 164),
- the coefficient of variation should be between 0.05 and 0.2 (for morphological characters) (Simpson et al. 1960; Mouse Phenome Project 2008),
- the width of the fitness function ω2 10–40 (Estes & Arnold 2007, p. 236),
- the average heritability should not be far from 0.5 (Weigensberg & Roff 1996),
- the exponent z of the power law relating environmental variance to elapsed time should not be far from 0.5 (Vasseur & Yodzis 2004), and
- and mean fitness should be adequate to perpetuate the population at all times.
These together ensure that the simulation is biologically realistic.
An example of a simulation that meets these criteria is shown in , which shows the pattern of environmental change at different time-scales and the response of the mean character state. I have not yet attempted an extensive exploration of how parameter values affect outcomes, but simulations following the restrictions listed above have always produced qualitatively similar results consistent with the conclusions drawn here. We can now ask how the amount of evolutionary change corresponds with observed values by computing the exponent zP of the power law for the increase in phenotypic variance through time. The corresponding power law for the absolute difference in character values has zD = zP/2. The environment itself, and hence the optimal character state, follows a power law with exponent zE, which will vary by chance among simulations between about 0.2 and 0.7. If the population were able to track the changing environment precisely, through intense selection and high mutation supply rate, the mean phenotype would always correspond with the optimum, so that zP = zE. In practice, the population always lags behind the fluctuating optimum because selection can act only by modifying the genotypic distribution of the previous generation. The consequence of this lag is that genetic change is always more coarse-grained than environmental change, so that zP > zE. As zE is substantial, this implies that zD will also be substantial, and in practice usually exceeds 0.2. The multi-scale model thus predicts that the amount of evolutionary change will steadily increase with elapsed time, which seems inconsistent with the very extensive data that support zD ≈ 0.
Figure 1. Simulated evolution in a multi-scale environment. Character state is governed by alleles with additive effects at 10 loci. Mutation creates a new allele with an exponentially distributed random effect with mean 1: the mutation rate is 0.001 per locus (more ...)
There are two ways, however, in which zD
might be estimated. The first is to use the initial and final values from a large number of independent studies involving different organisms and characters. This is the method used by Gingerich (1983)
and by Kinnison & Hendry (2001)
. The second is to sample a single time-series at random intervals of time. This corresponds to the pattern actually shown by an evolving population, and is the method I have used for analysing the multi-scale model. It would be interesting to estimate zD
in this way for real time-series. Unfortunately, very few evolutionary time-series have been reported in sufficient detail. The most extensive I have found describes morphological change in a foraminiforan (Afrobolivina
) from 92 horizons in Late Cretaceous sediments (Reyment 1982
). Some less-extensive datasets involve a primate (Clyde & Gingerich 1994
), a fish (Bell et al. 1985
), ammonites (Raup & Crick 1981
) and radiolarians (Kellogg & Hays 1975
). If the rate of evolution in these studies, calculated from the initial and final character states, is plotted on log–log axes, the result is a well-marked negative regression with a slope of −0.8, which is consistent with the Gingerich rule, considering the meagre amount of data (n
= 13). If we sample the same time-series at various intervals of time, however, we obtain a quite different result: the phenotypic variance (and hence the absolute difference in character state) tends to increase over time in all cases. The Afrobolivina
series yields zP
= 0.51 and hence zD
= 0.25. The other studies yield generally larger values for zP
(). The data are too scanty to rely on for precise estimates of the pattern of evolutionary change in morphological characters, but they certainly do not support the hypothesis that zP
≈ 0. Kinnison & Hendry (2001
, table 2) also found that difference almost always increases over time in much shorter contemporary time-series of evolving populations. It appears that the Gingerich result is a consequence of combining estimates of change from many independent studies, and is not a correct description of the pattern of change within a single evolving population.
Table 1. Pattern of long-term morphological change. The variance plot is the regression of log phenotypic variance on log time interval, with slope zD and correlation coefficient r. Spatial scale of each survey is given as extent (total time interval, My) and (more ...)
Morphological change in fossil time-series has often been interpreted as a random walk. For example, Hunt (2006)
interprets the Cantius data () in terms of a random walk. Estes & Arnold (2007)
found that a random walk of the optimum did not fit the Gingerich data, primarily because it predicted a substantial increase in absolute difference over time. The random walk is a special case of the multi-scale model with zE
= 1, which is indeed inconsistent with single estimates of divergence from many independent surveys. However, more realistic models with 0 < zE
< 1 do successfully fit values sampled from within a single time-series. The displaced-optimum model of Estes & Arnold (2007)
is also a special case of the multi-scale model. As time increases, the probability that the interval includes both environmental states increases; hence, the environmental variance increases steeply (zE
> 1). For values greater than the displacement interval, however, all comparisons include both environmental states, and the variance is zero. This keeps the long-term divergence, estimated from independent surveys, close to zero. Clegg et al. (2008)
interpreted morphological change over 30 years in an island population of the passerine bird Zosterops
in terms of fluctuating selection governed by a displaced-optimum model. A multi-scale model shows only a slight increase in variance over time: for adult wing length, for example, zP
= 0.16. Hunt et al. (2008)
describe a convincing example of evolution towards a displaced optimum for a freshwater population of sticklebacks (Gasterosteus
), in which armour and pelvic spines were consistently reduced over a period of several thousand years. Indeed, the displaced-optimum model is likely to be successful in segments of a multi-scale model following some exceptionally wide environmental excursion. Over longer periods of time, however, it is likely to break down, as in the longer term Gasterosteus
series (Bell et al. 1985
; ), and indeed, in this case, the initially heavily armoured type itself replaced a more lightly armoured type.
The multi-scale model of fluctuating selection is more general than the random-walk or displaced-optimum models, generates a correct description of environmental change and is consistent with the little we know quantitatively about the pattern of change in evolving populations. This provides only weak support for the model, however, as many other theories might fit the data; for example, that morphology is directly influenced by the state of the environment. What direct evidence is there of the strong fluctuations in selection that the model requires?