Age-Structured Population Dynamics

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.

Figures (6)

• Figure 1: Left: correspondence between the initial condition of the RE and the PDE, where the age-density $\psi$ at time $t=0$ is related to the initial birth rate history $\phi$ via \ref{['psi-phi']}. Right: integration along characteristics maps $b(t)$ to $n(t,a)$ via \ref{['n-solution']}.
• Figure 2: Solution of the Euler--Lotka equation \ref{['CE']} for $\lambda \in \mathbb{R}$.
• Figure 3: Illustration of the polynomial interpolation process described by \ref{['pM']}.
• Figure 4: Stability boundaries of the positive equilibrium of \ref{['Nicholson-scaled']}, for different values of $\mu$ and $a_{max}$: the equilibrium exists for $R_0>1$ (i.e., $R_0^\infty > 1+\frac{\beta}{\mu}\mathrm{e}^{-\mu a_{max}}$) and is stable below the curves. The stability boundary for $a_{max}=\infty$ is computed analytically.
• Figure 5: Left: bifurcation diagram of \ref{['Nicholson-scaled']}, with $a_{max}=\mu=5$ and varying $R_0^\infty$, showing a stable branch of periodic orbits (minimum and maximum values plotted) starting from Hopf (H), which becomes unstable through a period doubling (PD) bifurcation. Right: stable periodic orbits corresponding to $R_0^\infty=15$ (before PD) and $R_0^\infty=20$ (after PD). The periods are normalized to 1, and the estimated period $T$ is shown in the legend.

Theorems & Definitions (11)

• Theorem 3
• Remark 7
• Definition 17
• Theorem 18
• Corollary 19
• Lemma 21: Lemma III.2.3
• Theorem 22: Theorem III.2.4
• Remark 23
• Theorem 24: Section VII.5
• Lemma 25