Our research efforts will center on the use of mathematical techniques to solve and analyze the complex different and differential equations that arise in the modeling of systems in biology, ecology and physiology. The systems to be studied directly relate to significant problems and issues in the biosciences. The fundamental goal is to use the resulting mathematical results to provide a better understanding of the dynamics of these systems. Particular systems of interest include (but, are not limited to): . periodic diseases (discrete models) . the renal concentrating mechanism . biochemical oscillators . reaction-advection diffusion processes . modeling of dieting . interacting population dynamics Exact, approximate and numerical solutions will be obtained and compared with available data/observations to both understand the particular system being studied and to make predictions concerning its dynamical evolution. The mathematical methods to be used include perturbation (both regular and singular) and asymptotic series, harmonic balance procedures, phase-space analysis, the """"""""theory"""""""" of chaotic systems, and numerical integration. Many of these mathematical tools have originated in previous work by the PI, in particular, the use of harmonic balancing for determining periodic solutions to oscillating systems and non-standard finite difference schemes for calculating numerical solutions to differential equations. Secondary, but also important objectives are to expose both undergraduate and graduate students to an area of research in the biosciences for which they are generally not familiar or knowledgeable and to introduce into the science curriculum an introductory course in mathematical biosciences.
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