The age-structured population models : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University

Abstract

Mathematical theories of population dynamics have been derived and have been effectively used in the last two hundred years. These theories have appeared both implicitly and explicitly in many important studies of populations: human populations, as well as populations of animals, cells and viruses. The aim of this thesis is to understand how these models have developed with a view to an improved formulation. Nowadays, the structured model can be considered to be of great importance and use. Mathematicians have realised that in real biological and ecological situations, a model should be developed which is at least structured on age especially with higher order animals, for example human and possum populations, so in this thesis we pay much attention to these type of population models. We shall also be discussing the qualitative nature of the solutions to the model: such as the long-term behaviour, steady-age distribution and the stability of the solution in great details. In chapter 1, we begin with the historical background of the unstructured population in which the properties of individuals are ignored and only the total population is considered. The Malthusian and Verhulst model are set as examples. We then proceed in chapter 2 with the simplest McKendrick's age- structured population model. In chapter 3, we shall show how Laplace transform can be used to solved the problem. We have also chosen some arbitrary functions for either one or both the birth and/or death rate, so that we can make deductions from the assumption of these special cases. Chapter 4, discusses the long-term behaviour: steady age distribution (s.a.d.) and the stability of the solution being analysed. We then generalise the linear age- dependent population model in chapter 5 to a non-linear age-dependent model where the limiting effects (overcrowding and limitation of resources) has an effect, on the specific age class only. Chapter 6 discusses the more realistic non-linear model similar to that described in chapter 5 but here the limiting effects have an effect on the whole population. Finally, we realise that since these models need to be tested, we shall, in chapter 7 test our model with possum populations on data collected from the Orongorongo Valley in Wellington. And then make suggestions for future work in Chapter 8.