Delay differential equations: theory and applications of periodicity, multistability and global dynamics

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A delay differential equation (DDE) describes the evolution of a system, for which the change rate of the state variable depends on the system's current and historical status. The initial state must be specified on an (initial) interval and an appropriate phase space must be infinite dimensional. This infinite dimensional functional analytic framework was developed in the last century, and the qualitative study of DDEs has been among major sources of inspiration for advance in nonlinear analysis and infinite dimensional dynamical systems. The objective of this proposal is to describe the global dynamics (how local structures such as equilibria, periodic/quasi-periodic solutions are connected together to form the global attractor) for several classes of nonlinear systems of DDEs arising from important applications including spatial dynamics of migratory birds and avian influenza; transmission dynamics of vector-borne diseases such as West Nile virus and Lyme disease; and neural networks for pattern storage and recognition. This project is the core component of a more comprehensive research program to understand long-term behaviors of biological or epidemiological systems with highly heterogenous individual/community structures. This core component focuses on developing general mathematical theories, methodologies and techniques to explore the implication of structure heterogeneity including environment/climate change for the spatiotemporal patterns of population dynamics. The heterogeneity to be addressed involve both temporal and spatial variations, including seasonality, long range dispersal vs local spatial diffusion, physiological structures and maturation stages, and state-dependent delays in signal processing and feedbacks. This project is an important step towards the long-term objective to understand the joint effect of time delay and spatial diffusion on dynamics of nonlinear systems, and its implications for a wide range of issues in biology, epidemiology, neuroscience and data mining. This project will provide ample opportunities for thesis topics and postdoctoral research, and NSERC support will be critical to form a well structured project team to fully capitalize a network of complementary expertise.