11 Stochastic age-structured modelling: dynamics, genetics and estimation Steinar Engen, Norwegian University of Science and Technology Abstract In his book published in 1930 R.A.Fisher introduced the concept of reproductive value in relation to his fundamental theorem of natural selection, claiming that the theorem is valid also for age-structured populations provided that individuals are weighted by their reproductive value rather than using the individual numbers to define gene frequencies. This was based on the fact that the total reproductive value always grows exactly exponential in the absence of density regulation. Although Fisher defined this concept for deterministic models only, it has proved very useful also in stochastic modeling. Reproductive values can be used to define stochastic dynamics of age-structured populations and point toward simple definitions of a few main parameters, the deterministic growth rate and the demographic and environmental variances, sufficient for accurate description of the dynamics. These parameters can be estimated from individual recordings of vital rates and applied in diffusion approximations for the population size. We show that there is a simple relation between the demographic variance and genetic drift in age-structured populations and how to use this to estimate effective population size and its age- specific components from individual data of age, survival and fecundity. Reproductive values are also useful in describing stabilizing fluctuating selection on quantitative characters leading to a rather simple time series model for the stochastic evolution of mean phenotype. We also demonstrate how Fisher's concept can be used to estimate age-specific components of fluctuating selection by rather simple regression models.

22 Fisher’s reproductive value. Demographic and environmental stochasticity. Reproductive value dynamics. Individual reproductive value. Estimation. Genetic drift. Effective population size. Relations to reproductive value demographic variance. Estimation of genetic drift. Fixation of mutations in age-structured populations. Fluctuating stabilizing selection in age-structured populations Estimation of directional selection in age-structured populations.

33 Between years Among individuals, within years

44 Distribution of individual fitness W for two bird species

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66 The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time

77 The rigour of its demonstration requires that the terms employed should be used stricktly as defined; the ease of its interpretation may be increased by appropriate conventions of measurement. For example, the ratio p:q should stricktly be evaluated at any instant by enumeration, not necessarily of the census population, but of all individuals having reproductive value, weighted according to the reproductive value of each.

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11 The total reproductive value V of the population growth exactly exponential, and lnV has exactly linear growth. The age distribution approaches the stable age distribution R. A. Fisher

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13 Log of reproductive value have white noise if the matrices are temporally uncorrelated Generalization of result of Tuljapurkar 1982 Engen et al. 2005, 2007

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15 Population size Total reproductive value Age-structured population, no density regulation. (The total reproductive value serves as a filter)

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18 The distribution of the bootstrap replicates (n=1000) of the population growth rate, the demographic variance, and the environmental variance, for the population of Columbian ground squirrel. The black lines denote the estimated values and the 95 % confidence intervals are , and

19 Vindenes, Y., Engen, S. and Sæther, B.-E Individual heterogeneity in vital parameters and demographic stochasticity. American naturalist 171: Vindenes, Y., Engen, S. and Sæther, B.E Integral projection models for finite populations in a stochastic environment. Ecology 92:

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21 Effective population size determines Expected rate of random genetic drift Increase in inbreeding Loss of selectively neutral heterozygosity Loss of genes (loss og genetic diversity) Rate of fixation of beneficial mutations

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26 Allele frequeny weighted by reproductive value as proposed by Fisher in 1930.

27 According to Wright’s definition of variance effective population size

28 For details see Engen et al. 2005, Genetics

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31 Bootstrap replicates (Ratio of effective and actual population size) The Siberian Jay Data from Phillip Gienapp, University of Helsinki, recorded , analysed by Bernt-Erik Sæther, NTNU. Environmental stochasticity is also included in this analysis.

32 Lee, A.M. Engen, S. and Sæther, B.E The influence of persistent individual differences and age at maturity on effective population size. Proceedings of the Royal Society, B.

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35 Vindenes,Y., Lee, A.M., Engen, S. and Sæther, B.-E Fixation of slightly beneficial mutations: Effects of life history. Evolution 64:

36 One-dimensional character, stabilizing fluctuating selection For details see Engen et al. 2011, Evolution.

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51 The rigour of its demonstration requires that the terms employed should be used stricktly as defined; the ease of its interpretation may be increased by appropriate conventions of measurement. For example, the ratio p:q should stricktly be evaluated at any instant by enumeration, not necessarily of the census population, but of all individuals having reproductive value, weighted according to the reproductive value of each. For papers by our group on this subject see