Stochastic age-structured modelling: dynamics, genetics and estimation
Steinar Engen, Norwegian University of Science and Technology
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-specifc 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.