## Abstract

Fluctuations in age structure caused by environmental stochasticity create autocorrelation and transient fluctuations in both population size and allele frequency, which complicate demographic and evolutionary analyses. Following a suggestion of Fisher, we show that weighting individuals of different age by their reproductive value serves as a filter, removing temporal autocorrelation in population demography and evolution due to stochastic age structure. Assuming weak selection, random mating, and a stationary distribution of environments with no autocorrelation, we derive a diffusion approximation for evolution of the reproductive value weighted allele frequency. The expected evolution obeys an adaptive topography defined by the long-run growth rate of the population. The expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Simulations of the age-structured model verify the accuracy of the diffusion approximation. We develop statistical methods for measuring the expected selection on the reproductive value weighted allele frequency in a fluctuating age-structured population.

THE evolutionary dynamics of age-structured populations were formalized by Charlesworth (1980, 1994) and Lande (1982) on the basis of earlier ideas of Fisher (1930, 1958), Medawar (1946, 1952), and Hamilton (1966), showing that the strength of selection on genes affecting the vital rates of survival or fecundity depends on their age of action (reviewed by de Jong 1994; Charlesworth 2000). Fisher defined the reproductive value of individuals of a given age as their expected contribution to future population growth, determined by the age-specific vital rates. This has the property that in a constant environment the total reproductive value in a population always increases at a constant rate. The total population size, however, undergoes transient fluctuations as the stable age distribution is approached, and the total population size only asymptotically approaches a constant growth rate (Caswell 2001).

Environmental stochasticity creates continual fluctuations in age structure, producing temporal autocorrelation in population size and in allele frequencies, which seriously complicate demographic and evolutionary analyses. Fisher (1930, 1958, p. 35) suggested for analysis of genetic evolution that individuals should be weighted by their reproductive value to compensate for deviations from the stable age distribution. Here we apply this suggestion to study weak fluctuating selection in an age-structured population in a stochastic environment.

One of the central conceptual paradigms of evolutionary biology was described by Wright (1932). His adaptive topography represents a population as a point on a surface of population mean fitness as a function of allele frequencies. Assuming weak selection, random mating, and loose linkage (implying approximate Hardy–Weinberg equilibrium within loci and linkage eqilibrium among loci), natural selection in a constant environment causes the population to evolve uphill of the mean fitness surface (Wright 1937, 1945, 1969; Arnold *et al.* 2001; Gavrilets 2004). Evolution by natural selection thus tends to increase the mean fitness of a population in a constant environment.

Lande (2007, 2008) generalized Wright's adaptive topography to a stochastic environment, allowing density-dependent population growth but assuming density-independent selection, showing that the expected evolution maximizes the long-run growth rate of the population at low density, . Here *r* is population growth rate at low density in the average environment and is the environmental variance in population growth rate among years, which are standard parameters in stochastic demography (Cohen 1977, 1979; Tuljapurkar 1982; Caswell 2001; Lande *et al.* 2003). In this model of stochastic evolution the adaptive topography describing the expected evolution is derived by expressing *r* and as functions of allele frequencies with parameters being the mean Malthusian fitnesses of the genotypes and their temporal variances and covariances. These results are based on diffusion approximations for the coupled stochastic processes of population size and allele frequencies in a fluctuating environment.

Diffusion approximations are remarkably accurate for many problems in evolution and ecology (Crow and Kimura 1970; Lande *et al.* 2003). Because a diffusion process is subject to white noise with no temporal autocorrelation, the approximation is most accurate if the noise in the underlying biological process is approximately uncorrelated among years. Despite temporal autocorrelation in total population size produced by age-structure fluctuations, the stochastic demography of age-structured populations over timescales of a generation or more can nevertheless be accurately approximated by a diffusion process (Tuljapurkar 1982; Lande and Orzack 1988; Engen *et al.* 2005a, 2007). The success of the diffusion approximation for total population size occurs because the noise in the total reproductive value is nearly white, with no temporal autocorrelation to first order, and the log of total population size fluctuates around the log of reproductive value with a return time to equilibrium on the order of a few generations (Engen *et al.* 2007). Hence the diffusion approximation is well suited to describe the stochastic dynamics of total reproductive value as well as total population size.

This article extends Lande's (2008) model of fluctuating selection without age structure by deriving a diffusion approximation for the evolution of an age-structured population in a stochastic environment. Assuming weak selection at all ages, random mating, and a stationary distribution of environments with no temporal autocorrelation, we show that the main results of the model remain valid, provided that the model parameters are expressed in terms of means, variances, and covariances of age-specific vital rates and that allele frequencies are defined by weighting individuals of different age by their reproductive value, as suggested by Fisher (1930, 1958). We perform simulations to verify the accuracy of the diffusion approximation and outline statistical methods for estimating the expected selection acting on the reproductive value weighted allele frequency.

## STOCHASTIC DEMOGRAPHY AND REPRODUCTIVE VALUE

In a stage-structured population with *n _{i}* individuals in stage

*i*let the dynamics of the population column vector

*n*be governed by the stochastic projection matrix

*L*giving the population vector in the next year

*Ln*. The expected projection matrix in the average environment is denoted as

*l*. Projection matrices for an age-structured population have nonzero elements in the first row representing age-specific annual fecundities and on the subdiagonal representing age-specific annual survival probabilities (Leslie 1945, 1948). For a stage-structured population nonzero elements may also occur on the diagonal representing stage-specific annual survival without transition and below the subdiagonal or above the diagonal representing nonadjacent stage transitions (Lefkovitch 1965; Caswell 2001).

The stochastic projection matrices are assumed to be independent and identically distributed through time. Subscripts *l* and *L* are used to indicate dependence on the expected matrix *l* and the stochastic matrix *L*. Let the column vector *u _{l}* and row vector

*v*with components

_{l}*u*and

_{li}*v*, , be the right and left eigenvector of

_{li}*l*associated with the dominant real eigenvalue λ

_{l}, defined by

*lu*= λ

_{l}*and*

_{l}u_{l}*v*= λ

_{l}l*. If the eigenvectors are scaled so that and , then*

_{l}v_{l}*u*is the stable age distribution and

_{l}*v*is the vector of reproductive values for the stages.

_{l}The total reproductive value in the population is . In a constant environment, the total reproductive value grows at a constant rate by an annual multiplicative factor λ_{l} even under departures from the stable age distribution, although the total population size grows at this rate only asymptotically as a stable age distribution is approached (Fisher 1930; Leslie 1948; Caswell 2001). Engen *et al.* (2007) showed that the first-order approximation to the annual change in reproductive value is(1)where ε_{L} = *L* − *l*. This shows that the log of reproductive value approximates a random walk with no temporal autocorrelation in the noise. First-order Taylor expansion of the noise terms yields the mean and variance of annual changes in total reproductive value under small noise(2)where *r _{l}* = ln λ

_{l}is the growth rate (or Malthusian parameter) in the average environment, is the long-run growth rate, and(3)is the environmental variance in total reproductive value (Tuljapurkar 1982; Caswell 2001; Lande

*et al.*2003; Engen

*et al.*2007). Here ε

_{Lij}denotes the element in the

*i*th row and

*j*th column of ε

_{L}. The total reproductive value equals the total population size

*N*if the population is exactly at its stable age distribution, and Engen

_{L}*et al.*(2007) showed that ln

*N*undergoes stationary fluctuations around ln

_{L}*V*with a return time to equilibrium of about one generation. That the process ln

_{L}*V*has approximately white noise explains the success of the diffusion approximation for ln

_{L}*V*(and ln

_{L}*N*), identified simply as the Wiener process with infinitesimal mean and variance and (Lande and Orzack 1988).

_{L}Note that the stable age distribution and reproductive value vector appearing in the approximations are those associated with the dominant eigenvalue of the expected projection matrix *l* = *EL* and not the expected value of the vectors associated with the stochastic matrix *L* (Tuljapurkar 1982; Lande and Orzack 1988). This occurs because the theory assumes small noise, exploring by first-order expansions how the stochastic dynamics deviate from the dynamics in a constant environment determined by the dominant eigenvalue of *l* and the corresponding eigenvectors *u _{l}* and

*v*.

_{l}Analyzing allele frequency evolution requires consideration of two or more correlated age-structured processes. Along with the stochastic projection matrix *L* described above, let *M* be the stochastic projection matrix for a different genotype with expectation *m* in the average environment. Generally, the environments may have distinct effects on different genotypes so that the environmental correlations between corresponding elements in *L* and *M* are less than one. In addition to the long-run growth rates and environmental variances for the two processes, we then also need to incorporate in the model an environmental covariance between the two processes that can be expressed using the above first-order approximation to the noise term in Equation 1 as(4)We employ the simplified notation *C*(*L*, *M*) for this covariance so that the environmental variances associated with the two stochastic projection matrices are and . The joint process for ln *V _{L}* and ln

*V*can then be approximated by a two-dimensional Wiener process with infinitesimal covariance

_{M}*C*(

*L*,

*M*).

The approximations in Equations 2 and 3, first derived by Tuljapurkar (1982) for population size and here derived more simply using reproductive value, are asymptotically exact as the temporal variances in vital rates approach zero. However, these approximations have good accuracy for coefficients of variation in vital rates up to 30% and for larger coefficients of variation in vital rates that have small sensitivities *v _{i}u_{j}* (Lande

*et al.*2003). Lande and Orzack (1988) first employed these results in a diffusion approximation for the total population size in a stochastic environment. Engen

*et al.*(2007) pointed out that these approximations were more accurate for the total reproductive value that contains all information on future population size as originally shown by Fisher (1930) for a constant environment. This result was based on showing that the log of total population size fluctuates around the log of total reproductive value with a characteristic return time to equilibrium of about one generation. The accuracy of these diffusion models, which has been confirmed by stochastic simulations (Engen

*et al.*2005a), occurs because the process ln

*V*approximates a random walk, which is known to be accurately described as a Brownian motion (or Wiener) diffusion process (Karlin and Taylor 1981). This argument also justifies using a two-dimensional diffusion for (ln

_{L}*V*, ln

_{L}*V*) with the covariance given by Equation 4 derived from a first-order approximation to the noise term in Equation 1 as done for the variances.

_{M}## STOCHASTIC EVOLUTION WITHOUT AGE STRUCTURE

For a population without age structure in a constant environment, with no density regulation, weak selection, and random mating, classical theory for continuous-time models reveals that the population size *N* grows approximately as(5)where is the mean Malthusian fitness in the population, *p _{i}* is the frequency of allele

*A*, and

_{i}*r*is the Malthusian fitness of genotype

_{ij}*A*. The corresponding rate of allele frequency evolution is approximately(6)where is the mean fitness of allele

_{i}A_{j}*A*(Fisher 1930; Crow and Kimura 1970). The final form of Equation 6 is Wright's (1937, 1969) adaptive topography in continuous time.

_{i}Lande (2008) showed that the above results hold also for fluctuating selection with environmental variances and covariances in genotypic fitnesses, provided that in Equation 5 and in the final form of Equation 6 *r* is replaced by the long-run growth rate of the population, , where is the environmental variance in population growth rate. This result is derived from the stochastic fitness *r _{ij}* +

*dB*(

_{ij}*t*)/

*dt*, where the

*B*(

_{ij}*t*) are Brownian motions with

*E*[

*dB*(

_{ij}*t*)] = 0 and

*E*[

*dB*(

_{ij}*t*)

*dB*(

_{ab}*t*)] =

*c*(Karlin and Taylor 1981). The stochastic differential equation for

_{ijab}dt*N*is then . Using the Ito stochastic calculus (Turelli 1977), the infinitesimal covariance between ln

_{i}*N*and ln

_{i}*N*is and the environmental variance in population growth rate takes the form(7)

_{j}Finally, for simplicity, consider only two alleles *A*_{0} and *A*_{1}, writing 2*N _{i}* = 2

*Np*for the abundance of

_{i}*A*and

_{i}*p*

_{0}=

*p*= 1 −

*q*for the frequency of

*A*

_{0}. The expected change in allele frequency can be expressed using an expected selection coefficient, , in a form analogous to the classical deterministic model (Crow and Kimura 1970),(8)where the expected fitness of allele

*A*, , is the weighted average expected fitness of genotypes containing it, . The expected fitness of a genotype can therefore be defined as its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population (Lande 2008).

_{i}On the logit scale, *y* = ln[*p*/(1 − *p*)], the infinitesimal mean defining the diffusion approximation for *y* takes its simplest form,(9)with indexes *a* and *b* summed over 0 and 1 so that *p _{a}* =

*p*

^{1−a}

*q*and similarly for the infinitesimal variance(10)Equations 9 and 10 define the diffusion approximation for

^{a}*y*by substituting

*p*=

*e*/(1 +

^{y}*e*). The diffusion for

^{y}*p*is found by the reverse transformation (Karlin and Taylor 1981), giving the infinitesimal mean and variance(11)(12)

In stochastic models of genetic drift in a finite population, when using the Ito calculus to compute the infinitesimal mean and variance of a diffusion approximation, it is often assumed that the environment influences each genotype identically (Engen *et al.* 2005a; Shpak 2007). However, in models of fluctuating selection (Lande 2007, 2008) stochastic environments exert distinct influences on the demography of different genotypes, producing a positive environmental variance contributing to the infinitesimal variance β(*p*) driving stochastic changes in allele frequencies even in populations sufficiently large to neglect genetic drift. The next section demonstrates that this model can be applied to an age-structured population by interpreting model parameters in terms of age-specific vital rates and calculating allele frequency by weighting individuals of different age by their reproductive value. The final section outlines statistical methods for estimating the expected selection coefficient acting on the reproductive value weighted allele frequency.

## STOCHASTIC EVOLUTION WITH AGE STRUCTURE

We proceed to analyze the stochastic model with multiple alleles *A _{i}* when the population has stages 1, 2, …,

*k*. Individuals of the ordered genotype

*A*define a projection matrix

_{i}A_{j}*L*=

_{ij}*L*that fluctuates through time with mean

_{ji}*l*and no temporal autocorrelation. For a Leslie matrix model (Caswell 2001), the elements in the first row of the matrices expressing fecundity are half the mean number of offspring produced by the genotype at each age in the given year, that is, the mean number of copies of each allele transmitted to offspring. The subdiagonal elements are their survival probabilities. The long-run growth rate associated with the stochastic matrix for each genotype

_{ij}*L*, denoted as , is that of a hypothetical pure population of genotype

_{ij}*A*. Each genotype

_{i}A_{j}*A*also has an environmental variance

_{i}A_{j}*C*(

*L*,

_{ij}*L*), and every pair of genotypes

_{ij}*A*and

_{i}A_{j}*A*has an environmental covariance

_{a}A_{b}*C*(

*L*,

_{ij}*L*). Fluctuating selection is generated by the environmental fluctuations in the matrices

_{ab}*L*provided that these are not all perfectly correlated, while differences between the expected matrices

_{ij}*l*cause selection in the average environment.

_{ij}We assume random mating among reproductive individuals of all stages and weak selection at all stages such that differences among corresponding elements of the matrices *L _{ij}* among genotypes are small. All stages therefore remain close to Hardy–Weinberg equilibrium. The population also is assumed to undergo density-independent growth and to be sufficiently large to ignore random genetic drift, so that genotypic projection matrices are independent of population density. Let

*X*denote the (column) vector describing the number of

_{i}*A*alleles in individuals at each stage. The stochastic projection matrices for allelic numbers

_{i}*X*are then approximately with mean . The (row) vector of reproductive values for allele

_{i}*A*is the left eigenvector,

_{i}*v*, associated with the dominant eigenvalue of

_{i}*l*.

_{i}The total reproductive value of alleles *A _{i}* in the age-structured population is , where the summation over

*a*covers all component stages of the vectors. When the projection matrix

*X*fluctuates in time, the dynamics of the log of total reproductive value have no temporal autocorrelation to the first order, as in the case of no age structure. Over a period short enough for

_{i}*p*to remain nearly constant, the joint process ln

_{i}*V*for all the alleles can accordingly be described as a multivariate Wiener process with infinitesimal means that are the long-run growth rates associated with the matrices and infinitesimal covariances

_{i}*C*(

*L*,

_{i}*L*) as above.

_{j}Under weak selection at all stages, the dominant eigenvalues and eigenvectors are similar for all matrices *l _{ij}*. The Malthusian fitness of allele

*A*in the average environment, given by the dominant eigenvalue of the matrix , can then be approximated by a linear function in the

_{i}*p*, that is, . This linear approximation is very accurate under weak selection, justifying the application of Fisher's formula (the first form of Equation 6) to an age-structured population in a constant environment, as proven rigorously by Charlesworth (1980, 1994). From Equation 4 it also follows that , where

_{j}*c*=

_{iajb}*C*(

*L*,

_{ia}*L*).

_{jb}The assumption of weak selection is required to approximate the growth rate *r _{i}* of the total reproductive value of

*A*alleles as a linear function of allele frequencies

_{i}*p*, as in deterministic and stochastic models with no age structure (Equations 6 and 8). Hence, we require that the matrices

_{i}*l*for the different genotypes are sufficiently close for this approximation to be valid. In practice, when estimates of the matrices are available and these indicate that selection may be too strong for the theory to be applicable, it is recommended to check numerically the linearity of the

_{ij}*r*as functions of the

_{i}*p*.

_{j}Adopting the above approximations, the stochastic process for the reproductive values , conditioned on the allele frequencies, is the same as the model for without age structure outlined in a previous section. The total reproductive value in the population, , thus grows at the rate . Furthermore, for given allele frequencies *p _{i}*, the reproductive value weighted allele frequencies(13)have the same dynamics as in Lande (2008). Complete formulas for the joint dynamics of population size and allele frequencies in terms of infinitesimal means, variances, and covariances are given by Lande (2008, Equations 3a, 3b, and 4a–4e).

For an age-structured population, the expected evolution is(14)On the left side the expected evolution of *P _{i}* refers to reproductive value weighted frequencies defined by Equation 13, while on the right side the

*p*are unweighted allele frequencies from individual counts. Transient fluctuations due to temporal autocorrelations in the log of allelic numbers, ln

_{i}*N*, caused by stochastic age structure are to first order absent from the dynamics of the log of total reproductive value of the allele, ln

_{i}*V*. The allele frequency,

_{i}*p*, therefore fluctuates around the reproductive value weighted allele frequency

_{i}*P*. On average, unconditionally with respect to age structure,

_{i}*EdP*=

_{i}*Edp*, and the above model therefore also defines, in the unconditional sense, the adaptive topography for the

_{i}*p*. Conditioned on the age structure, the expected evolution of

_{i}*P*obeys Wright's adaptive topography, whereas the expected evolution of

_{i}*p*differs somewhat. If the population is exactly at the stable age distribution, then

_{i}*P*=

_{i}*p*and

_{i}*EdP*=

_{i}*Edp*. Given an age structure that deviates from the stable age distribution, only

_{i}*P*is expected to follow Wright's adaptive topography, while

_{i}*p*instead evolves to track the path of

_{i}*P*.

_{i}A sample path of a simulated age-structured model with two alleles compares the stochastic processes for *P* and *p* in Figure 1, showing that the actual allele frequency *p* displays transient fluctuations around the path of the reproductive value weighted allele frequency *P*. A large difference between *P* and *p* indicates that the age structure deviates much from the stable age distribution. Given an unstable age distribution, the expected change in *p* may deviate substantially from that of *P*, which follows Wright's adaptive topography. Note, however, that even *P* has stochastic fluctuations governed by the environmental variances and covariances defined by the matrices *L _{ij}*. For

*p*<

*P*there is a positive selection component for

*p*in addition to that given by Wright's adaptive topography, whereas for

*p*>

*P*this component is negative. The strength of this component depends on the timescale for return to equality of log reproductive value and log population size. This timescale depends on the ratio between the norms of the subdominant and dominant eigenvalues of the expected projection matrices, typically being on the order of one or a few generations (Caswell 2001; Lande

*et al.*2003). Under weak selection all projection matrices have nearly the same eigenvalues, implying that the above ratio also determines the timescale for return to equality of

*p*and

*P*.

These results indicate that an accurate diffusion approximation for the reproductive value weighted allele frequency *P* can be obtained by using the infinitesimal mean and variance from the model without age structure (Equations 11 and 12) by substituting *P* for *p* throughout. Parameters of the model must also be expressed using statistics of the age-specific vital rates, identifying the Malthusian fitness of a genotype in the average environment, *r _{ij}*, as log of the leading eigenvalue of its mean projection matrix,

*l*, and identifying the environmental covariance between genotypes

_{ij}*c*as

_{ijab}*C*(

*L*,

_{ij}*L*). The adaptive topography (Equation 6 with in place of

_{ab}*r*) and the expected fitness of an allele or a genotype (Equation 8), derived by Lande (2008) for a diploid population without age structure, then remain valid for the age-structured model applied to the reproductive value weighted allele frequency.

To test the accuracy of the diffusion approximation we compared it to simulations of age-structured populations for two distinct cases: the transitional probability distribution for allele frequency under fluctuating selection of a consistent direction leading to quasi-fixation and selection of fluctuating direction producing a stationary distribution of allele frequency (Haldane and Jayakar 1963; Lande 2008). Simulated quantiles of *p* for an advantageous allele on the way to fixation closely match those from the diffusion approximation for reproductive weighted allele frequency *P*, as shown in Figure 2. Histograms of simulated stationary distributions of allele frequency *p* also agree well with stationary distributions of the reproductive value weighted frequency *P* derived from the diffusion approximation, as illustrated in Figures 3 and 4 for a model in which all three genotypes have the same expected life history, but the heterozygote has the advantage of a smaller environmental variance in fecundity than the homozygotes.

## MEASUREMENT OF SELECTION

Measuring the expected selection coefficient acting on the reproductive value weighted allele frequency (Equation 8) requires estimation of parameters in the infinitesimal mean and variance of the diffusion approximation (Equations 9–12). The first step is to estimate the mean projection matrix for each genotype in the average environment by recording mean values of vital rates through time. From each mean projection matrix *l _{ij}* an estimate of the log of the dominant eigenvalue

*r*and the corresponding eigenvectors can be computed. With known allele frequencies this gives the first term of Equation 9 and it remains to estimate the environmental variances and covariances

_{ij}*c*(Equation 4). Engen

_{ijab}*et al.*(2009) developed statistical methods to estimate all age-specific components of environmental variance and demographic variance of an age-structured population, using the concept of individual reproductive value, assuming no temporal autocorrelations in the projection matrices. However, here we require only environmental components so that the estimation can be performed in a simpler way. Writing

*G*= and , Equation 4 takes the form .

_{ij}Considering one particular pair of elements (*ij*) and (*ab*) in two genotypic projection matrices and omitting subscripts, it is sufficient to show how to estimate cov(*G*, *H*). These environmental variance and covariance components can be estimated by recording vital rates and from samples of individuals through time. If the element is a survival rate, we replace ε in the definition of *G* and *H* by the indicator of survival being 1 for individual survival and 0 for death, and if the element is a fecundity, we replace ε by half the number of offspring the individual produces in the given year. Using subscripts *t* and τ to denote two different times, we have(15)Consequently, each term of the form cov(*G*, *H*) can be estimated by computing the mean over all possible combinations of individual records for (*ij*) and (*ab*) at times *t* and τ for all combinations of *t* ≠ τ. This leads to estimates of *c _{ijab}* and the infinitesimal mean and variance of the diffusion. Uncertainties can be found by bootstrapping, resampling observed individuals with replacement. Since we are here dealing with environmental variances and covariances generated by temporal environmental fluctuations, the bootstrap sampling should be preformed in a way that reflects the temporal structure of the data. For details, see Engen

*et al.*(2009).

## DISCUSSION

Fisher (1930, 1958) developed the concept of reproductive value to describe the expected contribution by individuals of a given age to future growth of a population in a constant environment and suggested that allele frequencies should be calculated by weighting individuals of different age by their reproductive value. This was done in the context of his fundamental theorem of natural selection, which has often been considered obscure (Price and Smith 1972; Crow 2002), perhaps explaining why his suggestion has been largely neglected in population genetics, with few exceptions. The reproductive value weighted allele frequency has been used to derive the effective size of an age-structured population (Felsenstein 1971), and to model selection in a structured population (Taylor 1990, 2009; Grafen 2006), in a constant environment. The reproductive value of the age class in which a new allele is introduced into a population, *e.g.*, by immigration, affects its probability of fixation by random genetic drift (Emigh and Pollak 1979), or by drift and selection (Athreya 1993), in a constant environment.

Fisher's reproductive value is a deterministic concept, constructed to compensate for transient deviations from the stable age distribution, although he must have realized that such deviations are continually generated by changing environments. We previously extended the application of reproductive value to an important demographic property of a population in a stochastic environment, showing that the log of total reproductive value in a large density-independent population approximates a random walk with no temporal autocorrelation in the noise, provided that the population projection matrices are temporally uncorrelated (Engen *et al.* 2007). The diffusion approximation for the log of total population size is quite accurate for a density-independent age-structured population in a random environment (Lande and Orzack 1988; Engen *et al.* 2005a). This occurs because this process actually approximates a diffusion for the log of total reproductive value (Engen *et al.* 2007). For a given total reproductive value, the log of total population size has no additional predictive power, fluctuating around the log of total reproductive value with a timescale of about one generation for return to equality of these two variables.

Similarly, for evolution in an age-structured population, this article demonstrates that the actual allele frequency *p* fluctuates around the reproductive value weighted allele frequency *P* on a timescale of about one generation, due to stochastic fluctuations in the age structure and temporal autocorrelation in *p* that is absent to first order from the fluctuations in *P* (Figure 1). These results together with Fisher's suggestion naturally lead to the derivation of a diffusion approximation for the reproductive value weighted allele frequency, *P*, in an age-structured population subject to density-independent fluctuating selection (Equations 11 and 12). The accuracy of this method is illustrated by comparisons of the diffusion approximation with simulations of age-structured models (Figures 2 and 4).

Distinguishing between total population size *N* and total reproductive value *V*, and between the unweighted and reproductive value weighted allele frequencies, *p* and *P*, is important for analyzing selection and evolution in a fluctuating environment, despite the timescale for autocorrelation in *N* and *p* being only about one generation. Many organisms of interest, such as large vertebrates or perennial plants, have generation times of several years or more, and for such species investigators typically measure age-specific components of fitness on an annual basis. Age-specific vital rates have been combined into total lifetime fitness on the basis of simplifying assumptions of constant population size and/or constant age structure (Fisher 1930; Charlesworth 1980; Lande 1982; Clutton-Brock 1988). Alternatively, analysis has focused on time series of age-averaged selection (Grant and Grant 2002; Sheldon *et al.* 2003; Garant *et al.* 2004) or time-averaged age-specific selection (Pelletier *et al.* 2007). Recent extension of the Price equation to include fluctuating age structure remains genetically ambiguous (Coulson and Tuljapurkar 2008). In contrast, using reproductive value weighting, and measuring the fitness of a genotype by the demographic growth rate of its total reproductive value, overcomes the problems caused by autocorrelation and correctly combines all components of fitness in a fluctuating age-structured population.

Assuming random mating and weak selection at all ages, the evolution of reproductive value weighted allele frequency, *P*, in an age-structured population follows the diffusion approximation derived by Lande (2008) for evolution of allele frequencies in a population without age structure in a fluctuating environment. The expected evolution of *P* obeys a generalization of Wright's adaptive topography (Equation 6), maximizing the long-run growth rate of the population, , as a function of allelic frequencies, where *r* is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. Contrary to common belief, the expected fitness of a genotype within a population is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population (Equation 8). For an age-structured population the model parameters must be expressed using basic statistics of age-specific vital rates of the genotypes. The Malthusian fitness of a genotype in the average environment is log of the leading eigenvalue of its mean projection matrix, and the variances and covariances of genotypic growth rates depend on patterns of variability in their vital rates (Equation 4).

The present theory assumes large population size and density-independent vital rates, which together ensure that the stochastic projection matrices are not influenced by the population vector. Genetic drift in a small age-structured population can be represented by an additional term in the infinitesimal variance of the diffusion approximation (Engen *et al.* 2005b; Shpak 2007). The model can then be used to analyze the probability of fixation and the time to fixation as the boundaries now become accessible to the diffusion process. Density-dependent selection in a fluctuating environment has been analyzed to study evolution subject to life-history trade-offs in simple models with no age structure (Lande *et al.* 2009). Extending models of density-dependent selection to age-structured populations may prove difficult because density regulation of population size generally produces complex interactions among the age classes (Lande *et al.* 2006). Only one special form of density regulation, where the population vector exerts an identical multiplicative effect on all elements of the projection matrices in a given year (Desharnais and Cohen 1986), would preserve the dynamics of the age distribution and allele frequencies, leaving our results unchanged. The present results nevertheless provide a necessary step toward a more general understanding of life-history evolution in fluctuating environments.

## Acknowledgments

This work was supported by the Norwegian University of Science and Technology through a grant to the Centre for Conservation Biology, The Research Council of Norway (Storforsk: Population genetics in an ecological perspective), and the Royal Society of London.

## Footnotes

Communicating editor: J. Wakeley

- Received June 3, 2009.
- Accepted July 15, 2009.

- Copyright © 2009 by the Genetics Society of America