Date of Award

8-1-2014

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Hughes, Harry

Abstract

This dissertation discusses the construction of some stochastic models for population dynamics with a variety of birth and death rate functions. A general model is constructed considering a fundamental growth rate function of the population while allowing random births and deaths in the population. Four stochastic discrete delay models and two non-delay models using the infinitesimal mean and variance given by birth and death rate functions have been produced and analyzed. In these constructions drift terms are in the form of logistic growth or logistic growth with delay. Logistic growth models are well known to biologists and economists. For each model, the existence and uniqueness of the global solution, non-negativeness of the solution is discussed, and for some models, boundedness of the path is also given. Persistence of the population and the boundary behavior have also been discussed through the hitting times. Here, a new method to analyze the hitting times for a specific class of stochastic delay models is presented. This work is related to and also extends the work of Edward Allen, Linda Allen and Bernt Oksendal in population dynamics.

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