New Insights into Population Dynamics from the Continuous McKendrick Model

Abstract

This article presents a comprehensive study of the continuous McKendrick model, which serves as a foundational framework in population dynamics and epidemiology. The model is formulated through partial differential equations that describe the temporal evolution of the age distribution of a population using continuously defined birth and death rates. In this work, we provide rigorous derivations of the renewal equation, establish the appropriate boundary conditions, and perform a detailed analysis of the survival functions. The central result demonstrates that the population approaches extinction if and only if the net reproduction number $R_{n}$ is strictly less than unity. We present two independent proofs: one based on Laplace transform techniques and Tauberian theorems, and another employing a reformulation as a system of ordinary differential equations with eigenvalue analysis. Additionally, we establish the connection between the deterministic framework and stochastic process formulations, showing that the McKendrick equation emerges as the fluid limit of an individual-based stochastic model.

Figures (4)

• Figure 1: Approximation quality for Example 1. Comparison of the exact function $\beta(a) = 1/(1+a)$ (solid curve) with its two-term Gauss--Laguerre exponential approximation \ref{['eq:example1_approx']} (dashed curve). Left panel: On the interval $[0, 0.3]$, the approximation is visually indistinguishable from the exact function, with relative error below $0.1\%$. Right panel: On the extended interval $[0, 1.0]$, the approximation remains accurate with maximum relative error approximately $3\%$ near $a = 1$. The exponential approximation slightly overestimates the true function for intermediate ages, a consequence of matching the first two moments of the Laguerre weight function.
• Figure 2: Polynomial approximation for Example 2. Comparison of the exact birth law $\beta(a) = e^{-\mu_1 a}/(1 - a/2)$ (solid curve) with its third-order polynomial approximation \ref{['eq:example2_approx']} (dashed curve) for $\mu_1 = 1$. Left panel: Excellent agreement on $[0, 0.3]$ with relative error below $0.5\%$. Right panel: On $[0, 1.9]$, the polynomial approximation captures the qualitative behavior but underestimates the singularity as $a \to 2$. The approximation is valid for biological applications where ages remain bounded away from the singularity.
• Figure 3: Convergence of polynomial approximations for Example 3. Comparison of the exact Bessel function birth law \ref{['eq:bessel_birth']} (solid blue curve) with polynomial approximations of order $N = 2$ (red dashed) and $N = 5$ (green dotted) for $\mu_1 = 1$. The figure demonstrates rapid convergence: the $N = 2$ approximation is accurate for $a \lesssim 3$, while $N = 5$ extends accuracy to $a \lesssim 7$. The factorial-squared denominators in \ref{['eq:bessel_expansion']} ensure rapid convergence of the series, making low-order polynomial approximations effective for practical applications.
• Figure 4: Comparison of deterministic and stochastic population dynamics.Left panel: Total population $P(t)$ over time. The blue solid curve shows the numerical solution $P_{\mathrm{numeric}}(t)$ obtained from finite-difference discretization of the McKendrick PDE. The red dashed curve represents the analytical approximation $P_{\mathrm{approx}}(t)$ from \ref{['eq:P_approx']}. Both curves demonstrate exponential decay to zero, confirming extinction when $R_n < 1$. The excellent agreement (relative error $< 1\%$) validates the analytical solution. Right panel: The renewal function $B(t)$ showing the birth rate over time. The oscillatory decay pattern reflects the complex conjugate eigenvalues $s_2, s_3$ in the solution \ref{['eq:B_explicit']}, with the envelope decaying exponentially due to the dominant real eigenvalue structure.

Theorems & Definitions (14)

• Lemma 2.1: Solution representation via characteristics
• proof
• Theorem 2.2: Extinction criterion
• proof : Proof of Theorem \ref{['main']}
• Lemma 2.3: Monotonicity of $R_n$
• proof
• Lemma 2.4: Eigenvalue criterion
• proof
• Theorem 2.5: Extinction criterion for exponential sums
• Definition 1: Exact numerical solution