Patterns and rules for sensitivity and elasticity in population projection matrices

Sensitivity and elasticity analysis of population projection matrices (PPMs) are established tools in the analysis of structured populations, allowing comparison of the contributions made by different demographic rates to population growth. In some commonly used structures of PPM, however, there are mathematically inevitable patterns in the relative sensitivity and elasticity of certain demographic rates. We take a simulation approach to investigate these mathematical constraints for a range of PPM models. Our results challenge some previously proposed constraints on sensitivity and elasticity. We also identify constraints beyond those that have already been proven mathematically and promote them as candidates for future mathematical proof. A general theme among these rules is that changes to the demographic rates of older or larger individuals have less impact on population growth than do equivalent changes among younger or smaller individuals. However, the validity of these rules in each case depends on the choice between sensitivity and elasticity, the growth rate of the population, and the PPM structure used. If the structured population conforms perfectly to the assumptions of the PPM used to model it, the rules we describe represent biological reality, allowing us to prioritize management strategies in the absence of detailed demographic data. Conversely, if the model is a poor fit to the population (specifically, if demographic rates within stages are heterogeneous), such analyses could lead to inappropriate management prescriptions. Our results emphasize the importance of choosing a structured population model that fits the demographics of the population.