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Algebra Seminar: The Discrete Age-Structured Population Growth Model
1:10 PM, Neill 106W
ABSTRACT: This talk is part one of a two-part series designed to introduce some fundamental techniques used to describe population dynamics as influenced by particular vital population elements. In this talk, we will derive the discrete age-structured population growth model, called the Leslie matrix, and describe some of its properties. This particular model is useful in describing population dynamics among age groups within a population, such as birth, survivorship, mortality, and reproduction, as well as how the population moves about these classes over time - dynamics which non-structured population models, such as Lotka's integral model (which describes the overall birth level) and McKendrick-von Foerster's equation (a continuous PDE model which characterizes the overall impact of aging and mortality, but does not include reproduction), cannot describe. Moreover, these models offer no connection to individual vital rates - a connection provided by structured models. We will see that these matrices are nonnegative, and even primitive in most circumstances, so that they yield well to Perron-Frobenius theory. In addition to deriving the model proper, there will be a discussion regarding how the coefficients appearing in this matrix (which may be unknown) may be computed from known data (i.e. from species life tables, for instance), and an example illustrating its use in practice. Part two of this series will be an analysis of stage-structured population growth models, which are useful for characterizing population dynamics that are driven by vital population elements other than age, such as sex, marital status, and location. Such models are more appropriate for populations in which reproduction and survival are not the most important factors influencing dynamics.