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Expected Timefor Random Genetic Drift of a Population Between Stable Proc. Nati. Acad. Sci. USA Vol. 82, pp. 7641-7645, November 1985 Evolution Expected time for random genetic drift of a population between stable phenotypic states (paleontology/punctuated equilibria/morphological stasis/quantitative characters/adaptive topography) RUSSELL LANDE Department of Biology, The University of Chicago, Chicago, IL 60637 Communicated by David M. Raup, July 11, 1985 ABSTRACT Natural selection and random genetic drift mulated with an awareness of Wright's shifting-balance the- are modeled by using diffusion equations for the mean pheno- ory (1, 6). type of a quantitative (polygenic) character in a finite popula- By analogy with Wright's adaptive topography for gene tion with two available adaptive zones or ecological niches. frequencies, Simpson (7) discussed extensively the concept When there is appreciable selection, the population is likely to of phenotypic adaptive zones for quantitative characters. spend a very long time drifting around the peak in its original This was later formulated by Lande (8, 9), who showed that adaptive zone. With the mean phenotype initially anywhere an adaptive topography for the mean phenotype in a popula- near the local optimum, the expected time until a shift between tion can be represented by a surface of mean fitness, with phenotypic adaptive peaks increases approximately exponen- height W, in a space where the other dimensions are the tially with the effective population size. In comparison, the ex- mean values of quantitative characters in the population. Us- pected duration of intermediate forms in the actual transition ing scales of measurement (most often logarithmic) such that between adaptive peaks is extremely short, generally below the the phenotypic and genetic variation of the characters re- level of resolution in the fossil record, and increases approxi- mains nearly constant during evolution, and if the relative mately logarithmically with the effective population size. The fitnesses of the phenotypes do not change with time, weak evolutionary dynamics of this model conform to the pattern of natural selection always causes the mean phenotype to current paleontological concepts of morphological "stasis" and evolve uphill on the adaptive topography, increasing the "punctuated equilibria." mean fitness in the population until the local optimal pheno- type is attained. The widespread occurrence of convergent The shifting-balance theory of evolution developed by evolution in distantly related taxa (ref. 7, pp. 160-198; ref. Wright (reviewed in ref. 1) postulates a large population sub- 10)-for example, among marsupial and placental mam- divided into many small local populations, or demes, con- mals-indicates that for many species the adaptive topogra- nected by a low rate of migration. Stabilizing selection to- phy for morphological characters maintains a consistent pat- ward an optimal phenotype for quantitative (polygenic) char- tern of selection over long periods of time. acters, as well as the availability of more than one optimal The requirement, in Wright's theory, of a low level of mi- phenotype, creates a large number of genetic adaptive peaks gration between demes to maintain genetic variation within for the population. Wright showed that when the fitnesses of demes, for natural selection and random genetic drift, has genotypes are constant in time and alleles at different loci been criticized by Simpson (ref. 7, p. 123) and Mayr (ref. 5, are combined approximately independently, deterministic pp. 520-521) who place more emphasis on rapid evolution in changes in gene frequencies caused by natural selection al- completely isolated small populations. Lande (6) argued that ways increase the mean fitness in a population. Therefore a the high rates of spontaneous mutation observed in polygen- population can be represented as a point on a surface of ic, quantitative characters can maintain appreciable genetic mean fitness, with height W, in a space where the other di- variation even in completely isolated small populations, with mensions are gene frequencies and where natural selection effective sizes on the order of a few hundred individuals. causes the population to evolve uphill toward a local adap- The expected (or average) time for the mean phenotype in tive peak. The finite size of each local population produces a population to evolve by random genetic drift a certain dis- random genetic drift that may occasionally cause a local pop- tance away from a single adaptive peak was analyzed previ- ulation to undergo a "peak shift" (1, 2) by evolving tempo- ously (8). Barton and Charlesworth (11) gave an approximate rarily downhill on the adaptive topography, across an ina- formula for the steady rate of transitions between two phe- daptive zone, and then toward a different adaptive peak. If notypic adaptive zones, based on a formula derived from the the new adaptive peak is higher than the original one, this Kolmogorov forward equation for diffusion processes. Here local population may disperse more migrants and colonists I outline a method of obtaining the expected time for the than other local populations, thus spreading the new adapta- mean phenotype of a quantitative character in a finite popu- tion. lation to evolve from any initial value to any final value, Wright (2) pointed out the relevance of his theory to recent based on a formula derived from the Kolmogorov backward ideas on "punctuated equilibria" (3, 4), which holds that spe- equation. This general approach is used to investigate the cies maintain a constant phenotype during most oftheir exis- expected time until a shift of the mean phenotype between tence and that new species originate suddenly in small local- two adaptive peaks or ecological niches, as well as the ex- ized populations. In addition to qualitative observations of pected duration of intermediate stages in the transition be- morphological stasis in the fossil record of many species and tween alternative stable states. the usual lack of transitional forms between fossil species, notions of punctuated equilibria are founded in part on Mayr's (5) theory of speciation, which was itself initially for- THE MODEL The publication costs of this article were defrayed in part by page charge Assumptions. A quantitative character, z, is assumed to be payment. This article must therefore be hereby marked "advertisement" influenced by a large number of genetic loci, with roughly in accordance with 18 U.S.C. §1734 solely to indicate this fact. additive effects, and an independent environmental effect, 7641 Downloaded by guest on September 25, 2021 7642 Evolution: Lande Proc. NatL Acad Sc. USA 82 (1985) which in sum produce a phenotypic value that is normally genetic drift is distributed in the population. Genetic variance is assumed to be maintained by mutation and recombination (12, 13). By = W2N,/C W2Ned-. appropriate choice of scale of measurement (14), the pheno- +(Z) [2] typic variance, oa 2, and the heritability of the character, h2 (the fraction ofthe total phenotypic variance due to the addi- From the stationary distribution it is clear that, if there is tive portion of the genetic variance), are assumed to be near- appreciable selection, the mean phenotype of the population ly constant during the evolution of the mean phenotype, z. will spend the vast majority of the time near the highest To ensure this it is also necessary to postulate that the effec- point(s) on the adaptive topography, particularly when the tive size of the population is not very small (Ne >> 10) and effective population size is not very small. Different peaks in that selection on the additive genetic variance is weak at all the adaptive topography for the mean phenotype therefore points on the adaptive topography (15). Defining the curva- represent alternative stable states for a population, and we ture of the adaptive topography at the mean phenotype x as can investigate the expected time required for a shift be- Cx = (a2 In W/a72)X. the latter condition is tween them by random genetic drift. Expected Time Until a Shift Between Adaptive Peaks. The h22 << 11. [la] average time in generations for the mean phenotype to Ic evolve from an initial value zo to either of two arbitrary points a and /3, with a - zo c /3, is given by the following An adaptive topography for the mean phenotype, with two formula (8, 16), derived from the Kolmogorov backward adaptive zones or ecological niches, is depicted in Fig. 1. It equation of diffusion theory: is assumed that the adaptive landscape is smooth (with con- tinuous first and second derivatives), having no other adap- h2No, d2T 2 W(In dT tive peaks, with W approaching zero as z approaches ±00 so 2Ne dzi I(~a a-z =1= -1 [3a] that zodzo x0 k where T T(a,p3Izo) satisfies the boundary conditions f W2NAdZ < oo and W-2Nedz = 0 [lb] T(a,/3Ia) = T(a4,Op) = 0. [3b] for an arbitrary point k. The general solution is conveniently written in terms of the In deriving approximate expressions for the general for- stationary distribution, Eq. 2, mulae, it is further assumed that stabilizing selection at the adaptive peaks and disruptive selection in the adaptive val- T = (2Ne/h20.2) U(ZO) f +-l(y) f 4(x)dxdy ley are not negligible in comparison to random genetic drift. This condition can be stated by the requirements that v is roughly intermediate between a and b and that at these I +(y) (x)dxdy], [4] points in Fig. 1, where 11 << (b - a)2Ne, [Ic] which is consistent with the previous assumptions. u(z) = f -'(y)dy I -(y)dy. Stationary Probability Distribution of the Mean Phenotype. With these assumptions, evolution in a population with dis- / crete, nonoverlapping generations can be accurately mod- u(zo) is the probability that the mean phenotype evolves to eled by using diffusion equations to describe changes in the before reaching a, starting from zo (16, 17).
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