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

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 classes, 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. Due to its significance, Kinship Demography, the study of the relationship between transition rates and kinship structure, is currently among the most actively growing areas of demography. In this manuscript, 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 and distant kin relationships. 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.