The smaller, the better: Collapsing projection matrices while preserving population dynamics

Matrix projection models have long served as powerful tools for tracking populations and their structure over time. One of the main appeals of projection matrices is their simplicity, which arises from grouping individuals into classes that are assumed to share identical vital rates (e.g., mortality, birth rates). In natural populations, however, there are many traits that influence vital rates, resulting in a trade-off between accurately capturing this heterogeneity and minimizing model complexity. Here, we prove two theorems about when and how projection matrices for populations structured by both age and phenotypic classes can be collapsed to smaller and more easily evaluated matrices that generate, or allow recovery of, the original abundance dynamics after at most one maximum lifespan. We refer to such collapsed matrices as "dynamically sufficient" matrices. The proof identifies two kinds of redundant variables, one arising from phenotype-space directions that are not produced or are lost as cohorts age ("survival collapsing") and one from directions that do not make independent contributions to future reproduction ("reproductive collapsing"), which can each be removed from projection matrices while maintaining dynamical sufficiency. We follow this formal proof with a biological example that illustrates how the two theorems can be applied to simplify projection matrices in ecological and evolutionary analyses. Finally, we warn that empirical studies averaging over hidden heterogeneity without collapsing using these theorems will typically produce inaccurate estimates of vital rates, and we quantify the proportional error in the predicted population growth rate. Our collapsing theorems can be used in future studies to incorporate individual heterogeneity within age classes while minimizing the numerical and empirical costs of added complexity.