Mathematical Sciences: Theory and Applications of Functionaland Partial Differential Equations

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This project continues research on theoretical and applied problems involving functional and partial differential equations. Work will be done investigating the stability and Hopf bifurcation problems for a general class of functional differential equations with state-dependent bounded delay. Another line of study concerns singularly perturbed nonlinear differential difference equations where the goal is to show the existence of a pulse-like periodic solutions. Several problems involving the existence of distributional solutions of classes of partial differential equations. Closer to immediate applications, work will be done on the dynamics of population models. It includes a generalization of Volterra's population equation to include spatially distributed population and delay in the birth process. Models for disease transmission of sexually transmitted diseases will also be analyzed. Here the additional element in the model incorporates risk-behavior change. Differential equations form the backbone of mathematical modeling in the biological sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not only to state the relationships, but also to extract qualitative and quantitative meaning from them.