DOI: 10.1534/genetics.109.105841
Reproductive Value and Fluctuating Selection
in an Age-Structured Population
Steinar Engen,*
,1Russell Lande
†,‡and Bernt-Erik Sæther
‡*Centre for Conservation Biology, Department of Mathematical Sciences and‡Centre for Conservation Biology, Department of Biology, Norwegian University of Science and Technology, N-7491 Trondheim, Norway and†Division of Biology, Imperial College London,
Silwood Park Campus, Ascot, Berkshire SL5 7PY, United Kingdom Manuscript received June 3, 2009
Accepted for publication July 15, 2009
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.
T
HE evolutionary dynamics of age-structured pop-ulations 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 Jong1994; 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 in-creases at a constant rate. The total population size, however, undergoes transient fluctuations as the stable age distribution is approached, and the total popula-tion size only asymptotically approaches a constant growth rate (Caswell 2001).
Environmental stochasticity creates continual fluctu-ations in age structure, producing temporal autocorre-lation in popuautocorre-lation size and in allele frequencies, which seriously complicate demographic and evolution-ary 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 selec-tion in an age-structured populaselec-tion in a stochastic environment.
One of the central conceptual paradigms of evolu-tionary 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, ran-dom 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;
Arnoldet al.2001; Gavrilets2004). Evolution by
nat-ural 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 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, ˜rr s2
e=2. Here ris
pop-ulation growth rate at low density in the average environment and s2
e is the environmental variance in
population growth rate among years, which are stan-dard parameters in stochastic demography (Cohen
1977, 1979; Tuljapurkar1982; Caswell2001; Lande et al. 2003). In this model of stochastic evolution the adaptive topography describing the expected evolution is derived by expressingrands2
e as functions of allele
1Corresponding author:Department of Mathematical Sciences, Norwe-gian University of Science and Technology, N-7491 Trondheim, Norway. E-mail: [email protected]
frequencies with parameters being the mean Malthu-sian 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 (Crowand
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 approx-imately 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 tempo-ral autocorrelation to first order, and the log of total population size fluctuates around the log of reproduc-tive 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
fluctu-ating 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 withniindividuals in stageilet the dynamics of the population column vector
n be governed by the stochastic projection matrix L
giving the population vector in the next yearLn. The expected projection matrix in the average environment is denoted asl. 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 sur-vival 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 (Lefkovitch1965; Caswell2001).
The stochastic projection matrices are assumed to be independent and identically distributed through time. SubscriptslandLare used to indicate dependence on the expected matrixland the stochastic matrixL. Let the column vectoruland row vectorvlwith components
uli and vli, i¼1;2;. . .;k, be the right and left
eigen-vector oflassociated with the dominant real eigenvalue
ll, defined bylul¼llulandvll¼llvl. If the eigenvectors are scaled so thatPiuli¼1 andPiulivli¼1, thenulis the stable age distribution and vl is the vector of reproductive values for the stages.
The total reproductive value in the population is
VL¼vln¼
P
inivli. In a constant environment, the total reproductive value grows at a constant rate by an annual multiplicative factor lleven 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
DlnVL lnll1ln½11l1l vleLul; ð1Þ
whereeL¼Ll. This shows that the log of reproductive value approximates a random walk with no temporal autocorrelation in the noise. First-order Taylor expan-sion of the noise terms yields the mean and variance of annual changes in total reproductive value under small noise
E½DlnVL ¼˜rL ¼rl
s2L
2 ; ð2Þ
where rl ¼ ln ll is the growth rate (or Malthusian parameter) in the average environment,˜rL is the long-run growth rate, and
s2L¼var½DlnVL ¼l2l
X
ijab
vliuljvlaulbcov½eLij;eLab ð3Þ
is the environmental variance in total reproductive value (Tuljapurkar1982; Caswell2001; Landeet al.
2003; Engenet al.2007). HereeLijdenotes the element
in the ith row and jth column of eL. The total re-productive value equals the total population sizeNLif the population is exactly at its stable age distribution, and Engen et al.(2007) showed that lnNLundergoes
the success of the diffusion approximation for lnVL (and lnNL), identified simply as the Wiener process with infinitesimal mean and variance˜rL ands2L (Landeand Orzack1988).
Note that the stable age distribution and reproductive value vector appearing in the approximations are those associated with the dominant eigenvalue of the ex-pected projection matrixl¼ELand not the expected value of the vectors associated with the stochastic matrix
L(Tuljapurkar1982; Landeand Orzack1988). This
occurs because the theory assumes small noise, explor-ing by first-order expansions how the stochastic dynam-ics deviate from the dynamdynam-ics in a constant environment determined by the dominant eigenvalue of l and the corresponding eigenvectorsulandvl.
Analyzing allele frequency evolution requires consid-eration of two or more correlated age-structured pro-cesses. Along with the stochastic projection matrix L
described above, let M be the stochastic projection matrix for a different genotype with expectationmin the average environment. Generally, the environments may have distinct effects on different genotypes so that the environmental correlations between corresponding elements inLandMare 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
cov½DlnVL;DlnVM ¼ ðlllmÞ1 X
ijab
vliuljvmaumbcov½eLij;eMab:
ð4Þ
We employ the simplified notation C(L, M) for this covariance so that the environmental variances associ-ated with the two stochastic projection matrices ares2
L¼
C(L, L) and s2
M ¼ C(M, M). The joint process for lnVL and lnVM can then be approximated by a two-dimensional Wiener process with infinitesimal covari-anceC(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 approxima-tions 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 viuj (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. Engenet al.(2007) pointed out that these
approxima-tions 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 (Engenet al.2005a), occurs because
the process lnVLapproximates a random walk, which is known to be accurately described as a Brownian motion (or Wiener) diffusion process (Karlinand Taylor1981).
This argument also justifies using a two-dimensional diffusion for (lnVL, lnVM) 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.
STOCHASTIC EVOLUTION WITHOUT AGE STRUCTURE
For a population without age structure in a constant environment, with no density regulation, weak selec-tion, and random mating, classical theory for continu-ous-time models reveals that the population sizeNgrows approximately as
dN
dt ¼rN; ð5Þ
wherer ¼PiPjpipjrijis the mean Malthusian fitness in the population,piis the frequency of alleleAi, andrijis the Malthusian fitness of genotypeAiAj. The correspond-ing rate of allele frequency evolution is approximately
dpi
dt ¼piðrirÞ ¼
pið1piÞ 2
@r
@pi
; ð6Þ
where ri ¼
P
jpjrij is the mean fitness of allele Ai (Fisher1930; Crowand Kimura1970). The final form
of Equation 6 is Wright’s (1937, 1969) adaptive
topog-raphy in continuous time.
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,
˜r¼r s2
e=2, wheres 2
e is the environmental variance in
population growth rate. This result is derived from the stochastic fitness rij 1 dBij(t)/dt, where the Bij(t) are Brownian motions with E[dBij(t)] ¼ 0 and E[dBij(t)
dBab(t)] ¼ cijabdt (Karlin and Taylor 1981). The sto-chastic differential equation for Ni is then dNi¼
riNidt1Ni
P
jpjdBijðtÞ. Using the Ito stochastic calculus (Turelli1977), the infinitesimal covariance between
lnNi and lnNj is Pabpapbciajb and the environmental variance in population growth rate takes the form
s2e¼ 1 dt
X
ij
pipjcov½dlnNi;dlnNj ¼
X
ijab
pipjpapbcijab:
ð7Þ
Finally, for simplicity, consider only two alleles A0
andA1, writing 2Ni¼2Npifor the abundance ofAiand
change in allele frequency can be expressed using an expected selection coefficient, w˜0w˜1, in a form
analogous to the classical deterministic model (Crow
and Kimura1970),
Edp
dt ¼pð1pÞðw˜0w˜1Þ; ð8Þ
where the expected fitness of alleleAi,w˜i ¼
P
jpjw˜ij, is the weighted average expected fitness of genotypes containing it, w˜ij¼rij
P
abpapbcijab. 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 (Lande2008).
On the logit scale,y¼ln[p/(1p)], the infinitesimal mean defining the diffusion approximation forytakes its simplest form,
mðyÞ ¼X
a
ðr0ar1aÞpa 1 2
X
ab
ðc0a0bc1a1bÞpapb ð9Þ
with indexesaandbsummed over 0 and 1 so thatpa¼
p1aqaand similarly for the infinitesimal variance
nðyÞ ¼X
ab
ðc0a0b1c1a1b2c0a1bÞpapb: ð10Þ
Equations 9 and 10 define the diffusion approximation foryby substitutingp¼ey/(11ey). The diffusion forpis
found by the reverse transformation (Karlin and
Taylor 1981), giving the infinitesimal mean and
variance
aðpÞ ¼pð1pÞmðyÞ11
2pð1pÞð12pÞnðyÞ ð11Þ
bðpÞ ¼p2ð1pÞ2nðyÞ: ð12Þ
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 environ-ment influences each genotype identically (Engenet al.
2005a; Shpak2007). However, in models of fluctuating
selection (Lande2007, 2008) stochastic environments
exert distinct influences on the demography of differ-ent genotypes, producing a positive environmdiffer-ental variance contributing to the infinitesimal variance
b(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 interpret-ing model parameters in terms of age-specific vital rates and calculating allele frequency by weighting individu-als of different age by their reproductive value. The final section outlines statistical methods for estimating the expected selection coefficient acting on the reproduc-tive value weighted allele frequency.
STOCHASTIC EVOLUTION WITH AGE STRUCTURE
We proceed to analyze the stochastic model with multiple alleles Ai when the population has stages 1,
2,. . .,k. Individuals of the ordered genotypeAiAjdefine
a projection matrix Lij ¼ Lji that fluctuates through time with meanlijand no temporal autocorrelation. For a Leslie matrix model (Caswell2001), 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 probabili-ties. The long-run growth rate associated with the stochastic matrix for each genotypeLij, denoted as ˜rij, is that of a hypothetical pure population of genotype
AiAj. Each genotype AiAj also has an environmental varianceC(Lij,Lij), and every pair of genotypesAiAjand
AaAb has an environmental covariance C(Lij, Lab). Fluctuating selection is generated by the environmental fluctuations in the matricesLijprovided that these are not all perfectly correlated, while differences between the expected matriceslijcause selection in the average environment.
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 Lijamong genotypes are small. All stages therefore remain close to Hardy–Weinberg equilib-rium. The population also is assumed to undergo density-independent growth and to be sufficiently large to ignore random genetic drift, so that genotypic pro-jection matrices are independent of population density. Let Xi denote the (column) vector describing the number of Aialleles in individuals at each stage. The stochastic projection matrices for allelic numbersXiare then approximately Li ¼
P
jpjLij with mean li¼
P
jpjlij. The (row) vector of reproductive values for alleleAi is the left eigenvector, vi, associated with the dominant eigenvalue ofli.
The total reproductive value of allelesAiin the age-structured population is Vi¼viXi¼
P
aviaXia, where the summation overacovers all component stages of the vectors. When the projection matrix Xi 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 forpito remain nearly constant, the joint process lnVi for all the alleles can accordingly be described as a multivariate Wiener process with infinitesimal means that are the long-run growth rates˜riassociated with the matrices Li¼PjpjLij and infinitesimal covariances
C(Li,Lj) as above.
Under weak selection at all stages, the dominant eigenvalues and eigenvectors are similar for all matrices
matrixli ¼Ppjlij, can then be approximated by a linear function in the pj, that is, ri Pjpjrij. 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
fol-lows thatCðLi;LjÞ
P
abpapbciajb, whereciajb¼C(Lia,Ljb). The assumption of weak selection is required to approximate the growth rateriof the total reproductive value ofAialleles as a linear function of allele frequen-ciespi, as in deterministic and stochastic models with no age structure (Equations 6 and 8). Hence, we require that the matrices lij 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 therias functions of thepj. Adopting the above approximations, the stochastic process for the reproductive values ðV1;V2;. . .;VkÞ, conditioned on the allele frequencies, is the same as the model for ðN1;N2;. . .;NkÞ without age structure out-lined in a previous section. The total reproductive value in the population, V ¼PiVi, thus grows at the rate
r ¼Pijpipjrij. Furthermore, for given allele frequencies
pi, the reproductive value weighted allele frequencies
Pi¼Vi
. X
j
Vj ð13Þ
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
EdPi
dt ¼
pið1piÞ 2
@˜r
@pi
: ð14Þ
On the left side the expected evolution ofPirefers to reproductive value weighted frequencies defined by Equation 13, while on the right side the pi are un-weighted allele frequencies from individual counts. Transient fluctuations due to temporal autocorrelations in the log of allelic numbers, lnNi, caused by stochastic age structure are to first order absent from the dynamics of the log of total reproductive value of the allele, lnVi. The allele frequency,pi, therefore fluctuates around the reproductive value weighted allele frequency Pi. On average, unconditionally with respect to age structure,
EdPi¼Edpi, and the above model therefore also defines, in the unconditional sense, the adaptive topography for thepi. Conditioned on the age structure, the expected evolution of Pi obeys Wright’s adaptive topography, whereas the expected evolution ofpidiffers somewhat. If the population is exactly at the stable age distribution,
thenPi¼piandEdPi¼Edpi. Given an age structure that deviates from the stable age distribution, only Pi is expected to follow Wright’s adaptive topography, while
piinstead evolves to track the path ofPi.
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 pdisplays transient fluctuations around the path of the reproductive value weighted allele frequency
P. A large difference betweenPandpindicates that the age structure deviates much from the stable age distri-bution. Given an unstable age distribution, the ex-pected change inpmay deviate substantially from that ofP, which follows Wright’s adaptive topography. Note, however, that even P has stochastic fluctuations gov-erned by the environmental variances and covariances defined by the matricesLij. Forp,Pthere is a positive selection component forpin 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 time-scale depends on the ratio between the norms of the subdominant and dominant eigenvalues of the ex-pected projection matrices, typically being on the order of one or a few generations (Caswell2001; Landeet 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 ofpandP.
Figure1.—A single sample path from a simulated model
These results indicate that an accurate diffusion approximation for the reproductive value weighted allele frequency P can be obtained by using the in-finitesimal mean and variance from the model without age structure (Equations 11 and 12) by substitutingP
forpthroughout. 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, rij, as log of the leading eigen-value of its mean projection matrix,lij, and identifying the environmental covariance between genotypescijabas
C(Lij,Lab). The adaptive topography (Equation 6 with˜r in place ofr) 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 pop-ulations for two distinct cases: the transitional probabil-ity distribution for allele frequency under fluctuating selection of a consistent direction leading to quasi-fixation and selection of fluctuating direction pro-ducing a stationary distribution of allele frequency (Haldaneand Jayakar1963; Lande2008). Simulated
quantiles ofpfor an advantageous allele on the way to fixation closely match those from the diffusion approx-imation for reproductive weighted allele frequencyP, as
shown in Figure 2. Histograms of simulated stationary distributions of allele frequencyp also agree well with stationary distributions of the reproductive value weighted frequency P derived from the diffusion ap-proximation, 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 ap-proximation (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 lij an estimate of the log of the dominant ei-genvaluerijand 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 covariancescijab(Equation Figure2.—Simulations from a model with five stages, with
reproduction only in the terminal stage and mean annual fe-cundities of the genotypesf00¼1.00,f01¼0.9, andf11¼0.8. Environmental variances in annual fecundities are all 0.2 and there is a common correlation 0.5 between them. All fecund-ities are lognormally distributed. The annual survival proba-bilities are constant with subdiagonal elements 0.7, 0.9, 0.9, 0.7 and 0.6 for the last diagonal element. The solid line shows quantiles for the actual allele frequencypfrom 10,000 simu-lations of the full age-structured model, while the dotted lines are quantiles for the reproductive value weighted allele fre-quencyPfrom 10,000 simulations of the diffusion approxima-tion.
Figure3.—Infinitesimal meana(P) and varianceb(P) in
4). Engenet 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 reproduc-tive value, assuming no temporal autocorrelations in the projection matrices. However, here we require only en-vironmental components so that the estimation can be performed in a simpler way. WritingGij ¼vliuljll1eLij and Hab ¼vmaumblm1eMab, Equation 4 takes the form
CðL;MÞ ¼PijabcovðGij;HabÞ.
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
ˆ
GandHˆ from samples of individuals through time. If the element is a survival rate, we replace e in the definition ofGandHby the indicator of survival being 1 for individual survival and 0 for death, and if the element is a fecundity, we replaceeby half the number of offspring the individual produces in the given year.
Using subscriptstandtto denote two different times, we have
covðG;HÞ ¼E1
2ðGˆtGˆtÞðHˆtHˆtÞ: ð15Þ
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 timestandtfor all combinations oft6¼t. This leads to estimates of cijab and the infinitesimal mean and var-iance of the diffusion. Uncertainties can be found by bootstrapping, resampling observed individuals with replacement. Since we are here dealing with environ-mental variances and covariances generated by tempo-ral environmental fluctuations, the bootstrap sampling should be preformed in a way that reflects the temporal structure of the data. For details, see Engenet al.(2009).
DISCUSSION
Fisher (1930, 1958) developed the concept of
re-productive 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 weight-ing 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 (Felsenstein1971), and
to model selection in a structured population (Taylor
1990, 2009; Grafen2006), in a constant environment.
The reproductive value of the age class in which a new allele is introduced into a population,e.g., by immigra-tion, affects its probability of fixation by random ge-netic drift (Emighand Pollak1979), or by drift and
selection (Athreya1993), 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 ap-proximates a random walk with no temporal autocorre-lation in the noise, provided that the popuautocorre-lation 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
Figure 4.—Stationary distributions of actual allele
fre-quency,p, and reproductive value weighted allele frequency, P. Diffusion approximations for distributions of P plotted as smooth curves were evaluated numerically using the infinitesimal mean and variance functions for the age-structured model illustrated in Figure 3, employing Wright’s
(1945, 1969) formula for the stationary distribution, ðc=bðPÞÞexp 2 ÐðaðPÞ=bðPÞÞdP, wherecis a normalization constant such that the distribution integrates to 1. Simula-tions of actual allele frequencypin the age-structured model over 106years were recorded every 100 years and plotted as
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 (Engenet 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 popula-tion, this article demonstrates that the actual allele frequency p fluctuates around the reproductive value weighted allele frequencyPon 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 inP(Figure 1). These results together with Fisher’s suggestion naturally lead to the derivation of a diffusion approxi-mation for the reproductive value weighted allele frequency, P, in an age-structured population subject to density-independent fluctuating selection (Equa-tions 11 and 12). The accuracy of this method is illustrated by comparisons of the diffusion approxima-tion with simulaapproxima-tions of age-structured models (Figures 2 and 4).
Distinguishing between total population size N and total reproductive valueV, and between the unweighted and reproductive value weighted allele frequencies,p
andP, is important for analyzing selection and evolution in a fluctuating environment, despite the timescale for autocorrelation inNandpbeing only about one gen-eration. 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 com-bined into total lifetime fitness on the basis of simpli-fying assumptions of constant population size and/or constant age structure (Fisher 1930; Charlesworth
1980; Lande 1982; Clutton-Brock 1988).
Alterna-tively, analysis has focused on time series of age-averaged selection (Grantand Grant2002; Sheldonet al.2003;
Garant et al. 2004) or time-averaged age-specific
se-lection (Pelletieret al.2007). Recent extension of the
Price equation to include fluctuating age structure re-mains genetically ambiguous (Coulsonand Tuljapurkar
2008). In contrast, using reproductive value weighting, and measuring the fitness of a genotype by the de-mographic growth rate of its total reproductive value, overcomes the problems caused by autocorrelation and correctly combines all components of fitness in a fluc-tuating 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 ex-pected evolution ofPobeys a generalization of Wright’s adaptive topography (Equation 6), maximizing the long-run growth rate of the population, ˜r¼rs2
e=2,
as a function of allelic frequencies, whereris the mean Malthusian fitness in the average environment ands2
eis
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 influ-enced 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 (Landeet al.2009). Extending models
of density-dependent selection to age-structured pop-ulations may prove difficult because density regulation of population size generally produces complex inter-actions among the age classes (Landeet al.2006). Only
one special form of density regulation, where the pop-ulation 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 fluc-tuating environments.
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.
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