A matter of maturity: To delay or not to delay? Continuous-time compartmental models of structured populations in the literature 2000–2016

Abstract: Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class-dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous-time compartmental models for age- and stage-structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers. Recommendations for Resource Managers: Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system. Single-population continuous-time models with age- or stage-structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics. Managers should be aware of the maturation delay issue when considering the results of continuous-time models of structured populations. Discrete-time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays.