Degree Name

Doctor of Philosophy


Department of Mathematics


This thesis is primarily concerned with age dependent population models and in particular with the integral equation initially described by Lotka, A.J. Relationships are developed between this integral equation and various other formulations such as partial differential equations and difference equations. As an example of such relationships the set of differential difference equations representing a multi stage population is reduced to a single integral equation. Many models are systems of autonomous differential equations such as the logistic. For these to be valid population models they must represent the mean behaviour of some associated stochastic model. It is shown that if this stochastic model is to be a birth and death system then the transition probabilities for a single individual must be entirely dependent on the mean and not the actual numbers in the population. In the situation where the transition probabilities are functions of the actual numbers the single autonomous equation for the mean becomes a coupled system of differential equations involving second and higher moments. Techniques are developed that enable the solution of these coupled equations.