Delay differential equation models in mathematical biology.

Delay Differential Equation Models in Mathematical Biology by Jonathan Erwin Forde Chair: Patrick W. Nelson 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 devel- oped 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 solu-

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