A branching process approach to lifetime reproductive success of structured populations: Variance–covariance and distribution

Abstract Lifetime reproductive success (LRS) is a key metric in ecology and evolution. It measures the number of offspring produced by an individual during its lifetime. In epidemiology, it corresponds to the number of secondary cases generated by an infected individual. For structured populations, it is crucial to understand how the distribution of LRS is shaped by survival, reproduction and other processes embedded in the projection model (such as dispersal or trait inheritance). Previous approaches have used diverse tools, such as Markov chains with rewards, to tackle this question, but generally consider only the total number of offspring produced, ignoring the distribution of their types (newborn states). Here, we use the framework of branching processes to derive formulas for the variance–covariance matrices and probability‐generating functions (allowing us to obtain the joint distribution) of LRS structured by the type of the parent and the type of the offspring. Furthermore, this framework leads to a simple algorithm providing a numerical approximation of the distribution of total LRS that does not require explicit expression of probability‐generating functions. We illustrate the power of the branching process approach, further, by studying the asymptotic behaviour of LRS, that is, the probability of producing many offspring. Finally, our general approach is applicable to any structured population model, and we provide R and Matlab code to facilitate implementation.