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One way of modeling the evolution in time of an age-structured population is to set up the evolution process as a partial differential equation in which time and age are the independent variables. The resulting equation, known as the McKendrick equation, has received attention recently from mathematicians. Some advantages of the PDE are that it can easily be adapted to include more detail in the model, including explicit time-dependence in the coefficients and even some nonlinear effects. The initial-boundary conditions for the McKendrick equation, imposed by the population model, are not the standard side conditions one sees in PDE theory for an evolution equation. In the simplest case, the problem reduces to a well-known model in demography, the Lotka integral equation. In this paper, we explain the solution of the McKendrick model and compare the McKendrick equation with other common models for age-structured populations (the Leslie matrix and the difference equation, as well as the integral equation) in several ways. The approaches differ in their suitability for computation, their ease of generalization, and their adaptability to different demographic objectives and other biological applications. With small intervals of age and time all forms are identical, but if the intervals are finite, differences will appear in the numerical results. The structure of solutions of the partial differential equation contributes to better understanding and computation of population models.
Keyfitz, B. L., & Keyfitz, N. (1997). The McKendrick partial differential equation and its uses in epidemiology and population study. Mathematical and Computer Modelling, 26(6), 1–9. https://doi.org/10.1016/S0895-7177(97)00165-9