New Insights into Population Dynamics from the Continuous McKendrick Model

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.