The Kinship Formula: inferring the numbers of all kin from any structured population projection model

Abstract

Structured population projection models are fundamental to many fields of science. They enable abundance forecasting for populations categorized by various traits such as age (for demography), patch (for spatial ecology), genotype (for genetics), infectious stage (for epidemiology) or capital (economics). The demography of a structured population, determined by the transition rates (e.g., survival, fertility) between its various states, also shapes its relatedness - or kinship - structure. This structure (a probabilistic genealogy) is crucial for understanding how individuals are related to the rest of the population and affects effective population size, inclusive fitness, inbreeding, pedigrees, relatedness, familial structures, etc. Despite its significance, the relationship between demography and kinship remains under-explored. By incorporating the generation number as a trait into the population structure, we derive the Kinship Formula, yielding the expected number of any kin for any structured population. This formula is simple to implement and fast to compute, even for complex models. Most importantly, it promises significant theoretical advances. From the Kinship Formula, one can, for instance, assess the impact of embedded processes (e.g., dispersal, inheritance, growth) on kinship, compute mean population relatedness and the eventual number of kin (including kin already dead or not born yet). The Kinship Formula derived here stems from a one-sex constant environment framework. Its simplicity should allow for extensions to include environmental and demographic stochasticity as well as two-sex models. Data accessibility statementNo new data are used. All data used to illustrate the method are public. The code related to the main text (R and Matlab) can be found in Supplementary Materials (S.M.VIII)