Age-structured models: Short-term transient dynamics

Abstract

Linear age-structured models eventually grow geometrically, and reach a stable age distribution (as in Chapter 3). This chapter describes what happens before “eventually.” That is, it describes the short-term, “transient” dynamics that occur when a population is perturbed, then begins to return to its stable distribution. Transients involve eigenvalues other than the largest (real) one, so the chapter begins by showing how complex eigenvalues can produce population cycles. It then addresses factors that make transients shorter or longer. In some cases, frequent environmental disturbances may prevent populations from ever reaching equilibrium. That scenario can be described by switching from linear models to linearized models varying about an equilibrium. The chapter describes temporal characteristics of that variability (such as time scales and frequencies), which require new tools: Fourier transforms and wavelets. These reveal how age-structured populations are more sensitive to certain environmental frequencies than to others, a phenomenon termed cohort resonance.