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Keywords:

  • diffusion process;
  • environmental variance;
  • expected relative fitness;
  • Haldane–Jayakar model;
  • long-run growth rate;
  • stochastic environment

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), inline image, where r is the mean Malthusian fitness in the average environment and inline image is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane–Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

Wright's adaptive topography is one of the conceptual foundations of evolutionary biology (Wright, 1932, 1937, 1969; Simpson, 1953; Lande, 1976, 1979, 1982; Kauffman, 1993; Arnold et al., 2001; Gavrilets, 2004). Representing a population as a point on the surface of mean fitness as a function of allele frequencies at one or more genetic loci, assuming random mating, weak selection and loose linkage (promoting approximate Hardy–Weinberg equilibrium within loci and linkage equilibrium among loci), natural selection in a constant environment causes a population to evolve uphill on this adaptive topography, increasing the mean fitness in the population. The utility of this concept has been questioned, particularly in fluctuating environments when the population appears at best to be chasing a continually moving optimum (Fisher, 1958, pp. 44–45; Crow & Kimura, 1970, p. 236; Gillespie, 1991, p. 305).

The theory of population growth in a random environment was developed by demographers ignoring genetic variation. For a large density-independent population subject to a stationary distribution of environmental states, the expected rate of increase in  ln N, the natural logarithm of population size, is known as the long-run growth rate (Cohen, 1977, 1979). With small fluctuations in age-specific vital rates of reproduction and mortality, this can be approximated as inline image, in which r is the population growth rate in the average environment, and inline image is the environmental variance in population growth rate (Appendix 1). Under these conditions, all population trajectories eventually achieve the same slope, inline image (Cohen, 1977, 1979; Tuljapurkar, 1982; Caswell, 2001; Lande et al., 2003).

Combining the above genetic and demographic theories, I previously demonstrated for allele frequency in a single-locus haploid (asexual) population, p, and for the mean phenotype of quantitative characters in a sexual population, inline image, that the expected evolution is governed by the gradient of the long-run growth rate of the population, inline image, with respect to p or inline image (Lande, 2007). Stochastic demographic theory describes density-independent growth of an age-structured population, but the main theories of fluctuating selection (reviewed in Gillespie, 1991) do not incorporate age structure. For simplicity, the present theory describes a population without age structure undergoing density-dependent population growth, but assuming that selection does not depend on population density. Extensions to density-dependent selection and age-structure will appear elsewhere.

For diploid genetic systems in a randomly mating population, I demonstrate here that the long-run growth rate again provides an adaptive topography governing the expected evolution of allele frequencies. The evolution of such a population represented as a point on this surface is expected to produce a stochastic maximization of inline image, which therefore constitutes an adaptive topography that does not change with time despite environmental fluctuations. However, the actual dynamics and stability of the process depend on the genetic system, the form of selection, and the amount of stochasticity in particular cases (Turelli, 1981; Gillespie, 1991; Lande, 2007). This is established by the theory developed below, and illustrated by particular examples including the Haldane–Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.

Theory

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

Wright's adaptive topography assumes a large randomly mating diploid population in a constant environment, as well as weak selection, and for multiple loci approximate linkage equilibrium. In deterministic models, when the strength of epistatic selection is much less than the recombination rate, random mating quickly produces a nearly constant linkage disequilibrium (called quasi-linkage equilibrium), after which gene frequency evolution increases the mean fitness (Kimura, 1965; Nagylaki, 1993). But under nonrandom mating, or with frequency-dependent fitnesses, evolution generally does not maximize mean fitness (Wright, 1949; see also Fisher, 1941; Moran, 1964; Wright, 1967, 1988; Crow & Kimura, 1970, pp. 195–236).

Continual changes in environment cause both selection and population size to fluctuate. Theories of stochastic selection and stochastic demography can be unified by developing a genetic demography for populations in fluctuating environments. Consider a continuous-time model of a single genetic locus with two alleles, A0 and A1 at frequencies p0 = p and p1 = 1 − p, or multiple alleles, with 1 − p the total frequency of all alleles other than A0. Random mating and weak selection are assumed so that the population is approximately in Hardy–Weinberg equilibrium. The ordered genotype AiAj therefore has frequency pipj, with a Malthusian fitness at low population density that is symmetric, rij = rji. The Malthusian fitness of each genotype is assumed to be reduced by the same amount, −f(N), with increasing population density, N, so that population growth may be density dependent but selection is not.

Classical theories of deterministic population dynamics and evolution reveal that in a constant environment, under random mating and weak selection

  • image( (1a))
  • image( (1b))

where r = ∑ijpipjrij is the mean fitness in the population and ri = ∑jpjrij is the mean fitness of allele Ai (Crow & Kimura, 1970, chaps 1, 5.9). The final form of eqn (1b) is the continuous-time version of Wright's adaptive topography, demonstrating that allele frequency evolution in a constant environment always occurs in the direction of increasing mean fitness (Kimura, 1958; Crow & Kimura, 1970, chap. 5.9).

 Instead of using the abundance of the ordered diploid genotype AiAj, denoted as Nij = pipjN, it is sufficient to obtain results from the dynamics of allele abundance, denoted for allele Ai as 2Ni where Ni = piN. Employing eqns (1a) and (1b), this obeys

  • image

Environmental stochasticity is introduced to this classical deterministic model by adding white noise to the Malthusian fitness of genotype AiAj, which then becomes

  • image

Assuming a stationary distribution of environmental states with no serial correlation, let E[dBij(t)] = 0 and E[dBij(t)dBkl(t)] = cijkl dt (Karlin & Taylor, 1981, pp. 340–343, 347). The covariance is symmetric among diploid genotypes, cijkl = cklij, and among alleles within genotypes, cijkl = cjikl = cijlk.

 The stochastic differential equation for Ni is then

  • image( (2a))

Using the Ito stochastic calculus (Turelli, 1977), the corresponding diffusion process has infinitesimal mean and infinitesimal covariance

  • image( (2b))
  • image( (2c))

where N denotes the vector of allele abundances.

 The method of analysis relies on the Ito transformation formulas (Karlin & Taylor, 1981; Gillespie, 1991, pp. 157–158) which can be applied to derive a coupled pair of diffusion processes for the natural logarithm of total population size,  ln N =  ln ∑iNi, and allele frequency, pi = Ni/∑jNj. Infinitesimal moments of the transformed diffusion are calculated from those of the original diffusion for the Ni and the derivatives of  ln N and p with respect to Ni. The resulting dynamical equations can be expressed in a compact and revealing form.

First, I describe the genetic components of the long-run growth rate and the environmental variance of the population, as these appear in all of the infinitesimal moments of the transformed diffusion process. The expected rate of increase in  ln N at low population density is the long-run growth rate of the population, inline image (eqns 3a and 4a), which has two components. The mean Malthusian fitness in the average environment, r is a quadratic function of p, which can be written using the symmetric matrix of Malthusian fitnesses r with elements rij and the column vector of allele frequencies, pT = (p,1 − p), where the superscript ‘T’ denotes transposition (eqn  3b). The environmental variance in population growth rate, inline image, is a quartic function of p, which, however, can also be written as a quadratic form using the matrix C with element CIJ being the covariance between Malthusian fitnesses of unordered diploid genotypes I and J, and the column vector of Hardy–Weinberg frequencies for the three (unordered) diploid genotypes A0A0, A0A1, A1A1 in proportions PT = [p2,2p(1 − p),(1 − p)2], indexed by the number of A1 alleles, I and J  ∈  {0,1,2} (eqn 3c),

  • image( (3a))

where

  • image( (3b))
  • image( (3c))

The infinitesimal moments of the coupled diffusion processes for population size and allele frequency are

  • image( (4a))
  • image( (4b))
  • image( (4c))
  • image( (4d))
  • image( (4e))

The infinitesimal mean and variance of  ln N (eqns 4a and 4b) appear in standard demographic form (Tuljapurkar, 1982; Caswell, 2001; Lande et al., 2003), but now incorporating genetic variation explicitly in inline image and inline image (eqns 3a–3c).

The infinitesimal mean of p (eqn 4c) obtained directly from the Ito transformation formulas is proportional to the difference in the expected relative fitnesses of the alleles, inline image, as in the deterministic model (Crow & Kimura, 1970, chaps 1, 5.9),

  • image( (5a))

where

  • image( (5b))

The expected relative fitness of an allele, inline image, is a weighted average of the expected relative fitnesses of diploid genotypes containing it, inline image. Thus, the expected relative fitness of a diploid genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with the population growth rate.

 Differentiation of the long-run growth rate also produces the selection gradient acting on allele frequency, inline image, which with eqn (5a) yields the gradient formula (4c) for the infinitesimal mean of p. This demonstrates that the expected evolution of allele frequency is in the direction of increasing inline image as a function of p. With multiple alleles eqn (4c) applies to the frequency of each one, provided that the partial derivatives with respect to p are interpreted as holding constant the relative frequencies of all other alleles (Wright, 1969; Crow & Kimura, 1970). The long-run growth rate of the population therefore constitutes an adaptive topography for allele frequency. Remarkably, this adaptive topography, depicted as a function of allele frequency, does not change with time despite environmental fluctuations, assuming the latter have a stationary distribution.

Environmental fluctuations cause the expected relative fitness of a genotype to be frequency dependent (eqn 5b). Wright (1949, 1969) noted that with frequency-dependent fitnesses, the mean fitness in a population generally is not maximized by evolution. This accords with the observation that under fluctuating selection, the expected evolution does not maximize the mean expected relative fitness, inline image, but rather the long-run growth rate of the population (eqns 3a and 4a).

The infinitesimal variance of p (eqn 4d) involves a non-negative quadratic form based on the covariance matrix of genotypic fitnesses, which can be identified as the variance of the selection coefficient acting on allele frequency. The infinitesimal mean and variance of p constitute an autonomous system, independent of N, because selection is assumed to be density independent. The dynamics of allele frequency are nevertheless coupled to those of the population size through the impact of genetic variation on the expected population dynamics (eqns 3a–3c, 4a and 4b) and through the infinitesimal covariance between  ln N and p (eqn 4e).

With multiple loci, assuming epistatic selection much smaller than the recombination rate (Kimura, 1965), approximate linkage equilibrium will prevail and eqns (4a)–(4e) apply simultaneously for each locus, if the diploid genotypic fitnesses are averaged over the genetic background given the current allele frequencies at other loci (Wright, 1937, 1967, 1969). For example, with two diallelic loci, the genotype AiAjBkBl has Malthusian fitness in the average environment rijkl with i,j,k,l ∈ {0,1}. Letting p be the frequency of allele A0 and q be the frequency of allele B0, we can construct the vectors of allele frequency and Hardy–Weinberg genotype frequencies for the second locus as for the first (above eqns 3a–3c), qT = (q,1−q) and QT = [q2,2q(1−q), (1−q)2], elements of which appear in the following formulas

  • image( (6a))
  • image( (6b))

Averaging across genotypes at the second locus eliminates the third and fourth subscripts in eqn (6a), whereas in eqn (6b) it eliminates the second and fourth subscripts. In the expected Malthusian fitness rijkl, the subscripts ij designate alleles at the first locus and subscripts kl represent alleles at the second locus; by contrast, CIKJL is the covariance of fitness between two-locus diploid genotypes IK and JL. The indices I, J, K, L  ∈  {0,1,2} count the number of A1 and B1 alleles in the genotype, e.g. C0112 is the covariance between genotypes A0A0B0B1 and A0A1B1B1. The components of the long-run growth rate of the population, inline image, are then

  • image( (7a))
  • image( (7b))

The present theory of fluctuating selection is incomplete for multiple alleles at a locus, and for multiple loci, in that the infinitesimal covariance of allele frequencies has not been specified, only the infinitesimal mean and variance of each allele frequency (eqns 4c and 4d). Describing a selection gradient for multiple alleles at a locus requires the application of nonlinear geometry (Akin, 1979; Shahshahani, 1979), which will be carried out elsewhere. The current results are nevertheless useful for analysing fluctuating selection in systems of one or more diallelic loci, as illustrated for two classical examples below.

Examples

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

The complexity of fluctuating selection on a diploid genetic system precludes a general analytic solution; so, results must be derived for particular cases (Turelli, 1981; Gillespie, 1991, p. 154). Models of fluctuating selection on a single genetic locus showed that with discrete nonoverlapping generations the geometric mean fitnesses of genotypes, or in continuous time their long-run growth rates, determine whether a rare allele will invade a population, regardless of the correlations in fitnesses among genotypes (Dempster, 1955; Haldane & Jayakar, 1963; Gillespie, 1973, 1991, p. 147; Tuljapurkar, 1982; Metz et al., 1992; Charlesworth, 1994; Caswell, 2001). At a diallelic locus, if the heterozygote has a higher long-run growth rate than either homozygote, then each allele invades when rare, and polymorphism persists. In the language of diffusion theory, if both 0 and 1 are entrance boundaries, then a stationary distribution of p exists (Karlin & Taylor, 1981, pp. 241–242).

However, the long-run growth rates of genotypes do not determine the expected change in allele frequency between successive generations, which defines natural selection and the expected relative fitnesses of genotypes in a polymorphic population. The present theory reveals that the expected relative fitness of a genotype within a population should be defined by its Malthusian fitness in the average environment minus the covariance between its fitness and the population growth rate (eqn 5b). Evolution in a fluctuating environment may therefore involve trade-offs among genotypes in their means and variances (Haldane & Jayakar, 1963; Tuljapurkar, 1982), and also their correlations (Gillespie, 1973, 1991).

The behaviour of a stochastic process near its boundaries, although important, does not entirely describe the process. Thus, an invasion analysis using the long-run growth rates of rare genotypes can give a misleading or incomplete picture of the evolution in a polymorphic population. Furthermore, sample paths of diffusion processes tend to accumulate in regions of low stochasticity. The expected selection gradient, and the stochasticity of selection, as functions of allele frequency (eqns 4c and 4d), are both necessary to fully describe evolution in a fluctuating environment. To illustrate these points, I analyse some particular one- and two-locus models.

Haldane–Jayakar model

Consider a symmetric version of the Haldane & Jayakar (1963) model of fluctuating selection at a diallelic locus. All three genotypes are assumed to have the same Malthusian fitness in the average environment, r0. The heterozygote fitness remains constant, whereas the fitnesses of the two homozygote genotypes have variance c and correlation, ρ. A correlation of ρ = −1 between homozygote fitnesses describes random directional selection, whereas ρ = +1 describes random over-dominance and under-dominance. Previous analyses of the Haldane–Jayakar model were restricted to conditions for the maintenance of a polymorphism, rather than the form of a polymorphism that does persist (Haldane & Jayakar, 1963; Gillespie, 1991, p. 229; Nagylaki, 1992, pp. 65–71). Here, I derive the form of the stationary distribution of allele frequency for the Haldane–Jayakar model.

In this model, r = r0 is constant and the environmental variance is

  • image( (8a))

The infinitesimal mean and variance of p, abbreviated as Mp and Vp, obtained from eqns (4c) and (4d), after rearrangement are

  • image( (8b))
  • image( (8c))

A stationary distribution of allele frequency exists for this model because the long-run growth rate of the heterozygote exceeds that of either homozygote, so that each allele invades when rare (see Table 1). The stationary distribution can be derived from Wright's formula,

  • image

where ‘const’ is a constant such that the distribution integrates to 1. Solution is facilitated by first transforming the diffusion process for allele frequency to the logit scale,

  • image

(see Appendix). Using the Ito transformation formulas, solving for the stationary distribution of y, and transforming this back to allele frequency gives

  • image( (8d))
Table 1.   Expected relative fitnesses and density-independent long-run growth rates of genotypes in the Haldane–Jayakar model.
GenotypeExpected relative fitnessLong-run growth rate
  1. All genotypes have the same Malthusian fitness in the average environment, r0. The two homozygote genotypes have fitness variance c and correlation ρ, whereas the heterozygote has constant fitness. The frequency of allele A0 is p and the population mates randomly.

A0A0r0 − [p2 + ρ(1 − p)2]cr0 − c/2
A0A1r0r0
A1A1r0 − [ρp2 + (1 − p)2]cr0 − c/2

For ρ = −1, the stationary distribution of allele frequency is simply the uniform distribution, φ(p) = 1. This is a special case of Gillespie's stochastic additive scale–concave fitness function (SAS-CFF) model for additive effects between two alleles with equal variance and perfect negative correlation (Gillespie, 1991, p. 190). For higher correlations, the stationary distribution becomes more concentrated around p = 1/2, as depicted in Fig. 1. With a perfect positive correlation between homozygote fitnesses, ρ = +1, the stationary distribution is the Dirac delta function, δ(p − 1/2), an infinite spike with unit area at p = 1/2, which acts as an absorbing barrier eventually reached by all trajectories. The correlation between homozygotes thus has a profound influence on the dynamics of polymorphism in the Haldane–Jayakar model. This occurs mainly because a larger value of ρ decreases the infinitesimal variance at intermediate frequencies, which for ρ = +1 vanishes at p = 1/2 (see Fig. 1).

image

Figure 1.  Long-run growth rate of the population, inline image, infinitesimal mean and variance, Mp and Vp, and stationary distribution, φ(p), of allele frequency, p, in the Haldane–Jayakar model for different values of ρ, the fitness correlation between the two homozygote genotypes (eqns 8a–8d).

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Because of its utility in determining the outcome of selection, such as whether a polymorphism can be maintained, many authors have interpreted the geometric mean fitness or long-run growth rate as a surrogate for genotypic fitness, or as fitness itself. In the Haldane–Jayakar model, where the long-run growth rate of the heterozygote exceeds that of either homozygote, this interpretation suggests that a net heterozygote advantage maintains the polymorphism. However, this is not necessarily correct, depending on the value of ρ. To understand how selection in a fluctuating environment operates one should instead examine the expected relative fitnesses of the genotypes obtained from the expected change in allele frequency (eqn 5b), as in Table 1.

Figure 2 illustrates that when ρ = −1, the expected relative fitnesses do not display a heterozygote advantage at any allele frequency, and that of the heterozygote lies exactly between the two homozygotes at every frequency, with the rarest homozygote always having the highest expected relative fitness. In this case, the polymorphism is maintained solely by frequency-dependent selection favouring rare genotypes and the rare allele. A higher correlation between homozygotes produces a stronger tendency to maintain the polymorphism, due to a combination of heterozygote advantage and low stochasticity at some allele frequencies. When homozygote fitnesses vary independently, ρ = 0, the expected relative fitnesses display heterozygote advantage at every allele frequency except p = 0 or 1. When the homozygotes are positively correlated, for example ρ = +1, a heterozygote advantage is expected at every frequency.

image

Figure 2.  Expected relative fitnesses, inline image, of genotypes AiAj as functions of the frequency, p, of allele A0 in the Haldane–Jayakar model for different values of ρ, the fitness correlation between the two homozygote genotypes (Table 1).

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Two-locus quantitative character model

An adaptive topography of fluctuating selection on quantitative characters, describing the expected evolution of the mean phenotype, was previously derived by Lande (2007), assuming normal distributions of phenotypes and breeding values with genetic variance maintained by mutation and recombination. Here, I apply a similar method at a more detailed level to investigate whether fluctuating directional selection, or fluctuating stabilizing and disruptive selection, can maintain genetic variance at multiple loci in a quantitative character.

Consider two diallelic loci with equal additive effects on a quantitative trait, assuming random mating and approximate Hardy–Weinberg equilibrium and linkage equilibrium. This model was often used by Wright (1935a,b, 1956, 1967, 1969) to analyse stabilizing selection in a constant environment, and is employed here to analyse fluctuating selection as shown in Table 2. All genotypes are assumed to have the same Malthusian fitness in the average environment; so, r = r0 is constant. At any time, the fitness function in Table 2 is a straight line with a random slope. This corresponds to the Haldane–Jayakar model of random directional selection (ρ = −1). The covariance between the fitnesses of any pair of genotypes is

  • image( (9a))
Table 2.   Two-locus quantitative character model of fluctuating directional selection.
GenotypePhenotypeFitness
  1. Each A1 and B1 allele add one unit to the background phenotype z0. The density-independent Malthusian fitness in the average environment is the same for all genotypes, r0. The fitness function is a straight line with a random slope (1/dt)dB(t) representing white noise with mean (1/dt)E[dB(t)] = 0 and variance (1/dt)E[dB(t)2]=c. The four underlined double homozygotes correspond to pure-breeding populations of a single genotype at the corners of the allele frequency space in Fig. 3.

A0A0B0B0z0inline image
A0A1B0B0, A0A0B0B1z0 + 1inline image
A1A1B0B0, A0A1B0B1,A0A0B1B1z0 + 2r0
A1A1B0B1, A0A1B1B1z0 + 3inline image
A1A1B1B1z0 + 4inline image

Substituting this in eqns (7a) and (7b) gives the long-run growth rate of the population as a function of the frequencies of the A0 and B0 alleles at the two loci, denoted as p and q respectively (Fig. 3),

  • image( (9b))
image

Figure 3.  Topographic (above) and three-dimensional (below) views of the long-run growth rate of the population, inline image, as a function of frequencies of alleles A0 and B0, respectively, p and q, at two diallelic loci affecting a quantitative trait under random directional selection (Table 2, eqns 9a and 9b) or random stabilizing and disruptive selection (eqns 10a and 10b).

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If, instead, the two halves of the fitness function, on either side of the intermediate phenotype, have opposite slopes, then in Table 2 the minus signs on the white noise terms for fitnesses of the two smallest phenotypes become plus signs. This describes random stabilizing and disruptive selection, analogous to the Haldane–Jayakar model of random over-dominance and under-dominance (ρ = +1). The covariance between the fitnesses of any pair of genotypes is then

  • image( (10a))

which with eqns (7a) and (7b) gives the long-run growth rate of the population,

  • image( (10b))

Inferences from the long-run growth rates of the genotypes can be contrasted with those gained from the adaptive topography, the long-run growth rate of the population as a function of allele frequencies.

Long-run growth rates of the genotypes show a pattern of stabilizing selection, being r0 for the intermediate phenotype including the double heterozygote and the two mixed double homozygotes (underlined in Table 2), r0 − c/2 for singly heterozygous phenotypes, and r0 − 2c for the extreme homozygotes (underlined in Table 2). Because the two mixed double homozygotes with intermediate phenotype have a long-run growth rate higher than that of any single heterozygote, they cannot be invaded by alternate alleles at either loci; they therefore constitute alternative stable equilibria with no polymorphism. Likewise, the two extreme double homozygotes can be invaded by alternative alleles at either locus, and so are unstable equilibria. This still allows the possibility of invasion from an edge equilibrium, where a rare allele at one locus invades, while the second locus is polymorphic. Even if one could determine the stability of all edge equilibria, which could be a daunting task in two or more dimensions, its relationship to the dynamics in the interior of the allele frequency space may be unclear. Long-run growth rates of the genotypes therefore have limited practical utility for analysis of multiple-locus genetic systems.

The adaptive topography for random directional selection (eqn 9a, Fig. 3) features a ridge connecting the two mixed homozygotes which can therefore be seen to be neutrally stable in two dimensions, not stable as indicated by invasion analysis for rare alleles at one locus at a time. Simultaneous changes in allele frequency at both loci along the ridge (p + q = 1) maintain the mean phenotype at the intermediate value and the long-run growth rate of the population also remains constant along the ridge. Stochasticity in selection will eventually drive both loci toward complementary quasi-fixations at one end of the ridge or the other.

Under random stabilizing and disruptive selection (eqn 9b, Fig. 3), the adaptive topography resembles that for stabilizing selection in a constant environment (Wright, 1935b, 1956, 1969, pp. 105–110). Two peaks representing alternative doubly homozygous genotypes, both corresponding to the optimum phenotype, are separated by a shallow saddle point. As with stabilizing selection in a constant environment, random stabilizing and disruptive selection thus acts to deplete additive genetic variance in a quantitative character.

The adaptive topographies in these symmetric two-locus models lack a peak in the interior of the allele frequency space, demonstrating in both cases that fluctuating selection on a quantitative character cannot maintain polygenic variation. Of course, if the expected selection were asymmetric, such that the genotype with the highest long-run growth rate was heterozygous at a single locus, then fluctuating selection could maintain polymorphism at one locus as in the Haldane–Jayakar model.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

The present theory reveals a general principle for the evolution of Mendelian populations in a fluctuating environment, extending Wright's adaptive topography for gene frequency evolution in a constant environment. Like Wright's theory, it assumes random mating, weak selection and loose linkage (to produce approximate Hardy–Weinberg and linkage equilibrium). Further, assuming a stationary distribution of environmental states, I have shown that the expected evolution of allele frequencies maximizes the long-run growth rate of a population, inline image. Under these conditions, the surface inline image as a function of allele frequencies remains constant in time, although selection continually fluctuates. Representing the population as a point on this adaptive topography, the expected rate and direction of allele frequency evolution is governed by the gradient of the long-run growth rate of the population with respect to allele frequency, inline image (eqn 4c), and fluctuating selection causes stochastic perturbations around the expected evolutionary trajectory.

Long-run growth rates of the genotypes determine the invasion of rare alleles and the existence of a stationary distribution of allele frequency, that is, whether fluctuating selection can maintain a genetic polymorphism. However, the long-run growth rates of genotypes do not predict the expected change in allele frequency between generations in a polymorphic population, which is described by the expected relative fitnesses of the genotypes. The general diploid model agrees with previous haploid and continuous quantitative character models (Lande, 2007) in revealing that the expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance between its fitness and the population growth rate (see eqns 5a and 5b). Expected relative fitnesses of genotypes generally are frequency dependent, that is, functions of allele frequency. In the Haldane–Jayakar model of fluctuating directional selection (with ρ = −1), the expected selection favours rare genotypes and hence also the rare allele (Figs 1 and 2); this occurs because the distribution of selection coefficients is symmetrical with a mean of zero, and a rare allele will increase more in frequency under positive selection than it decreases under negative selection of the same magnitude. A similar stabilizing effect is expected from fluctuating directional selection on any genetic system, but this may not be sufficient to maintain a genetic polymorphism, depending on the mode of inheritance.

For a single-locus haploid (or asexual) population, the genotype with the highest long-run growth rate will eventually prevail; so, fluctuating selection cannot maintain a polymorphism (Gillespie, 1991, chap. 4.2; Lande, 2007). For the Haldane–Jayakar model of a single diploid locus in a randomly mating population, fluctuating selection can maintain a polymorphism when the long-run growth rate of the heterozygote exceeds that of the homozygotes, allowing each allele to invade when rare. However, under fluctuating directional selection (with ρ = −1), the polymorphism is maintained purely by frequency-dependent selection, as shown by the expected relative fitnesses of the genotypes, which do not display a heterozygote advantage at any allele frequency (Table 1, Fig. 2 top panel). The infinitesimal mean and variance of allele frequency, influenced by the fitness correlation between genotypes, together determine the form of the stationary distribution of allele frequency (eqns 8a–8d, Fig. 1).

Gillespie (1991) explains the fundamental difference between single-locus haploid and diploid models in more technical terms. He developed the concept of c-haploid models by which the dynamical properties of simple diploid systems can be approximated by multiplying the infinitesimal mean and variance of the classical haploid model, respectively, by c and c2. For diploid models with additive fitnesses, where heterozygote fitness equals the mean fitness of the two corresponding homozygotes, this mapping is exact with c = 1/2 (Gillespie, 1991, p. 155). The ratio of the stochastic component to the deterministic component of allele frequency change is then smaller by a factor of 1/2 in the diploid model compared with the haploid model, allowing a stationary distribution of allele frequency to exist in the additive diploid model when it cannot in the haploid model.

Gillespie's SAS-CFF diploid and c-haploid models constitute a special and somewhat peculiar mechanism of fluctuating selection. Motivated in part by classical quantitative genetics and partly by consideration of enzyme function, Gillespie's models postulate a constant functional relationship between phenotype and fitness, but a changing relationship between phenotype and genotype. In otherwords, SAS-CFF models produce fluctuating selection not by changing fitnesses directly, but through genotype–environment interaction (Gillespie, 1991, pp. 144–145, 155–156, 189–190). By contrast, the present theory and that of Lande (2007) generate fluctuating fitnesses directly by changing the fitness function acting on phenotypes, and for normally distributed polygenic characters a direct effect of environment on phenotype is incorporated as phenotypic plasticity without genotype–environment interaction. Thus, except for the Haldane–Jayakar model with ρ = −1, none of the present diploid models are equivalent to those of Gillespie.

For a quantitative character influenced by two loci with equal additive phenotypic effects subject to fluctuating directional or stabilizing and disruptive selection, polygenic variation cannot be maintained (Table 2, eqns 9a, 9b, 10a and 10b). This last example resembles deterministic stabilizing selection on a quantitative character in the classical models of Wright (1935b, 1956, 1969, pp. 105–110) and Fisher (1958, pp. 118–121), which does not maintain genetic polymorphism except perhaps at a single locus. Based on analogy with deterministic models of stabilizing selection, asymmetry of the genetic system (markedly unequal phenotypic effects among loci or displacement of the mid-range phenotype from the expected optimum), or tight linkage between loci, may promote multilocus polymorphism but with limited additive genetic variance in the character (Gale & Kearsey, 1968; Barton, 1986; Bürger & Gimelfarb, 1999; Bürger, 2000, chap. VI; Willensdorfer & Bürger, 2003). In an age-structured population, strongly fluctuating selection can maintain polygenic variation but with a discontinuous phenotypic distribution (Sasaki & Ellner, 1997).

The theory and specific examples developed here for Mendelian genetic systems demonstrate the utility of the long-run growth rate of the population as an adaptive topography for the expected evolution of allele frequencies in a stochastic environment. The gradient of the long-run growth rate of the population with respect to allele frequencies indicates the expected direction of selection, and the expected relative fitnesses of the genotypes reveal the mechanism of frequency-dependent selection acting on genetic polymorphism. Understanding the evolutionary dynamics of alleles at intermediate frequency requires simultaneous analysis of the expected selection gradient and the stochasticity in selection, incorporated, respectively, in the infinitesimal mean and variance of the diffusion process for p. The present succinct description of the process, and the methods provided for deriving the infinitesimal mean and variance of allele frequency change, facilitate this analysis.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

I thank T. Coulson, B.-E. Sæther, S. Otto and anonymous reviewers for comments on the manuscript. This work was supported by grants from the US National Science Foundation and the Royal Society of London.

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  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
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Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theory
  5. Examples
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

Using the Ito transformation formulas (Karlin & Taylor, 1981, p. 173) to express the Haldane–Jayakar model (eqns 8b and 8c) on the logit scale, y =  ln [p/(1−p)], the infinitesimal mean and variance of y are

  • image
  • image

The stationary distribution of y is obtained from the formula of Wright (1931)

  • image

by substituting the infinitesimal moments above. Evaluating the exponential of the integral gives

  • image

Transforming φy(y) back to the stationary distribution of allele frequency, using

  • image

produces

  • image

where

  • image