Renewal and stability in populations structured by remaining years of life

Abstract

The Lotka-Leslie renewal model is the core of formal demography. This model is structured by chronological age, and it does not account for thanatological age. I derive a specification of the classic renewal equation that is structured by thanatological age rather than by chronological age. I give both continuous and discrete variants of the derived model, and relate these to the Lotka-Leslie renewal model. In stability, the thanatological and chronological renewal models are commensurable, implying identical intrinsic growth rates. I demonstrate approximate symmetry be- tween chronological and thanatological age structure in stability when subject to intrinsic growth rates equal magnitude and opposite sign. Birth-death renewal processes can be expressed as death-birth processes, and vice versa. The thanatological renewal model offers a new perspective on population renewal, and it is valid more generally as an aspect of birth-death processes.