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

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My research focuses on developing and analyzing mathematical models to study population dynamics. I am interested in how model assumptions influence the predictions, especially the entire range of possible dynamics. Many processes involve time delays that are ignored in models for mathematical tractability. I have been investigating the consequences of ignoring these delays as well as how the delays are modeled, i.e., using different distributions. Many mathematical functions used in models are not mechanistically justified, but rather only their basic form is suggested from data. I have been investigating the implications of using different choices for the mathematical forms (with the same properties) on model predictions. Is one form more likely to predict extinction of species than another? Is one form more likely to predict that a population size oscillates as opposed to approaching a constant population size? Is this influence consistent or model dependent? One goal is to reconcile commonly believed general principles with conflicting experimental observations, and to 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 waste decomposition, and biological remediation. Other potential applications include pest control on the one hand and the prevention of the extinction of endangered species on the other. Recently, I have become interested in modelling: anaerobic digestion related to the production of green energy from animal waste; self-cylcing fermentation used for water purification; exploring the effect of access to light on phytoplankton blooms; and exploring the predictions of various predator-prey and competition models. More generally, I am interested in trying to understand the causes of oscillatory behavior and chaotic dynamics. To explore the dynamics of the models, the qualitative theory of differential equations is used to determine local and if possible global dynamics of the models; persistence theory when appropriate to predict when certain species avoid extinction; bifurcation theory to determine the full spectrum of model behavior and to help identify which key parameters need to be measured and how accurately, in order to improve predictions. Use is made of specialized software to obtain bifurcation diagrams, computer simulations to elucidate complex dynamics, to test conjectures, and to reveal properties that help to develop analytic proofs, and symbolic computation to verify lengthy calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, impulsive, integro, and functional differential equations.