Delay differential equation models in mathematical biology.

Abstract

In this dissertation, delay differential equation models from mathematical biology are studied, focusing on population ecology. In order to even begin a study of such models, one must be able to determine the linear stability of their steady states, a task made more difficult by their infinite dimensional nature. In Chapter 2, I have developed a method of reducing such questions to the problem of determining the existence or otherwise of positive real roots of a real polynomial. The method of Sturm sequences is then used to make this determination. In particular, I developed general necessary and sufficient conditions for the existence of delay-induced instability in systems of two or three first order delay differential equations. These conditions depend only on the parameters of the system, and can be easily checked, avoiding the necessity of simulations in these cases. With this tool in hand, I begin studying delay differential equations for single species, extending previously obtained results about the existence of periodic solutions, and developing a proof for a previously unproven case. Due to the infinite dimensional nature of these equations, it is quite difficult to prove the existence of periodic solutions. Nonetheless, knowledge of their existence is essential if one is to make decisions about the suitability of such models to biological situations. Furthermore, I explore the effect of delay-dependent parameters in these models, a feature whose use is becoming more common in the mathematical biology literature. Finally, I look at a delayed predator-prey model with delay dependent parameters. Although I was unable to obtain a complete proof for the existence of periodic solutions, significant progress has been made in understanding the nature of this system, and it is hoped that future work will continue to clarify this picture. This model seems to display chaotic behavior for certain parameter regimes, and thus the existence of periodic solutions may be precluded in the most general case.