Formulation and Analysis of Nonlinear Mathematical Models with Applications to Population Dynamics

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I develop and analyze mathematical models to predict how the average abundance of populations changes and if the abundance is expected to oscillate or remain relatively constant over time. In particular, I study how certain minor differences in model formulations influence the predictions of the entire range of dynamics exhibited by models. Many processes involve delays that are often ignored for the sake of mathematical tractability. I investigate how to formulate models to include delays properly and the consequences of ignoring delays. I study the impact on the dynamics of including delays in different ways by comparing the dynamics of models without delays with the analogous discrete models or continuous models with fixed or distributed delays with different distributions.  Also, many functions in models are not mechanistically justified, but rather their basic form is suggested from data, I have been comparing the differences in the range of dynamics possible for models formulated with different choices of  mathematical forms that have the same basic properties. Is one form more likely to predict extinction of species or  more likely to predict that a population size equilibrates or oscillates indefinitely. More generally, I am especially interested in trying to understand the causes of oscillatory behavior and chaotic dynamics. One goal is to reconcile what are commonly believed general principles with conflicting experimental observations, and hence suggest new or modified principles. Another goal is to develop measurable criteria that would enable scientists to predict which combination of microorganisms would be most effective and safest for use in such processes as water purification, biological remediation, biological waste decomposition, green energy production from animal waste using anaerobic digestion by microbes, and prevention of harmful phytoplankton blooms. Other potential applications include pest control on the one hand and prevention of extinction of endangered species on the other. The qualitative theory of differential equations is used to determine local and when possible global dynamics of the models. Students learn to use persistence theory to predict under what circumstances certain species avoid extinction. Bifurcation theory helps to determine the full spectrum of behavior for all appropriate parameter ranges and initial data and helps to identify key parameters that need to be measured to improve predictions. If there are significant time delays involved in any interactions, integro- and functional differential equations are used. Students develop computational skills to test conjectures and reveal properties of the models useful in developing analytic proofs. Symbolic computation is used for complicated calculations. The analysis often leads to interesting abstract mathematical problems in dynamical systems, including difference equations, ordinary, impulsive, integro- and functional differential equations.