Age-Structured Population Dynamics

TL;DR

This chapter reviews the mathematical modeling of age-structured populations, covering equations, key concepts, stability analysis, and numerical approximation methods with practical examples.

Contribution

It introduces a comprehensive framework connecting renewal equations, PDEs, and numerical methods for age-structured population models.

Findings

Equivalence of renewal and PDE formulations

Analysis of stability and bifurcation in models

Numerical approximation via pseudospectral methods

Abstract

This chapter reviews some aspects of the theory of age-structured models of populations with finite maximum age. We formulate both the renewal equation for the birth rate and the partial differential equation for the age density, and show their equivalence. Next, we define and discuss central concepts in population dynamics, like the basic reproduction number $R_0$, the Malthusian parameter $r$, and the stable age distribution. We briefly review the sun-star theory that turns the birth term into a bounded additive perturbation, thus allowing to develop stability and bifurcation theory along standard lines. Finally, we review the pseudospectral approximation of the infinite-dimensional age-structured models by means of a finite system of ordinary differential equations, which allows to perform numerical bifurcation analysis with existing software tools. Here, Nicholson's blowfly equation serves as a worked example.