Amplitude Equations for Time-Dependent Solutions of the McKendrick Equations

The well-known McKendrick equations model the dynamical behavior of age-dependent populations. These equations govern, at time t, the number of individuals of age a in a population, known as the population density, and arise from a conservation law subject to constitutive assumptions for the maternity and mortality rates. In this paper, multiple scale analysis and bifurcation theory are applied to models governed by McKendrick’s equations. A weakly nonlinear analysis has been developed which describes the bifurcation to time-dependent solutions whose amplitudes are governed by a complex Landau-Stuart type equation. This analysis provides information about the population density unlike existing methods which reduce the model to a system of ordinary differential equations, and provide information only on the total population. Moreover, the multiple scale techniques developed for the integro-partial differential equations can be extended to more diverse models. in preparation [email protected] [email protected]; corresponding author, supported by NASA grant NCC-3-494 [email protected] 1