Linear and nonlinear integral equation population models.
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis is concerned with population models by using integral equations. These equations are formulated by using concepts of the continuous time delay (bounded and unbounded), concepts of population history, concepts of birth of new offsprings and concepts of survival of each individuals. The thesis consists of four parts. The first part deals with formulation of the integral equations for population dynamics. This part starts with the introduction of the integral equation models for population dynamics. Then, the rest of the first part will discuss the issues of time delay, formulation of the birth rate, the issues of population history and the formulation of survival rate. The second part will cover the topics about the linear equations. This part will deal with the assumptions of the linear equations, the relationships with Volterra integral equations, the reducibility of the integral equations (unbounded delay) to systems of ordinary differential equations and the asymptotic stability of zero solutions of the linear integral equations. The third part concerns only global stability of zero solutions of the special forms of the integral equations by using Lyapunov functionals. Finally, the fourth part is devoted to the analysis of the nonlinear integral equations I formulated in the beginning of this thesis. This part is concerned with the assumptions of the equations, the steady states and local stability of the steady states. This part will also consider the analysis of an example, its steady states, the local stability of the steady states, the construction of the periodic solutions and the numerical solutions.