Notes on Building Matrix Population Models: Insights From Combining Distinct Matrices for Survival, Reproduction, and Growth

ABSTRACT We propose a unified framework for constructing matrix population models in wildlife and fisheries management that accommodates variations in census timing and both natural and human‐caused mortality. The approach is applicable to age‐, size‐, and stage‐structured populations with a common short breeding season and decomposes the annual process into three components: survival ( S ), reproduction ( B ), and age increment or growth ( G ), each represented by a square matrix. In age‐structured models, reproduction and age increment occur nearly simultaneously, but with reproduction following aging, resulting in matrix formulations such as S ( BG ) or ( BG ) S , depending on whether the census is conducted just before or after parturition. In size‐ or stage‐structured models, where growth and survival occur in parallel, the model becomes ( GS ) B or B ( GS ), depending on the census timing. The framework also accommodates census taken at other times by partitioning survival into intervals from parturition to the census and from the census to the next parturition. Furthermore, harvest data expressed as absolute numbers can be incorporated via a capture vector structured by harvest time. This flexible modeling approach not only integrates diverse biological processes but also simplifies the construction of complex models by eliminating the need for multiple matrix formulations based on census timing, a feature often overlooked in standard textbooks on matrix population models.