Global Dynamics of Delay Differential Systems Modelling Nonlinear Feedbacks in Spatiotemporally Varying Environments

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The comprehensive Program will develop novel mathematical technologies to investigate implication of feedback delay on long-term dynamical behaviours of nonlinear epidemiological systems, and on computational performance of neural networks for high dimensional data clustering. The mathematical framework is the so-called delay differential equations where the change rate of the system's state variable depends on both current and historical status of the system. These equations are also called functional differential equations as their solutions generate infinite dimensional semiflows on appropriate functional spaces. ******Our overall objectives include: to identify and formulate several classes of delay differential equations arising from important applications in ecology, epidemiology, neural networks and data clustering; to develop frameworks, methodologies and technical tools to describe all possible behaviours of solutions of these equations; and to provide insights into the mechanisms behind observed/predicted dynamical scenarios of the underlying systems.******We will integrate three intertwined thematic areas into a novel, cohesive and interdisciplinary Program with multiple projects on specific fundamental research problems and important applications. In one thematic area, we develop the general theory of the global dynamics of semiflows which are monotone with respect to high rank cones, and its applications to delay equations with non-monotone and/or bi-directionally delayed feedback. This thematic research shall make groundbreaking advance towards identifying a broad class of nonlinear epidemiological and neural systems for which the classical monotone dynamical systems theory cannot be applied, and characterizing the global dynamics of such systems. In another thematic area, we develop technologies that can be effectively used to examine the creation, termination and global continuation of oscillatory patterns in models relevant to vector-borne disease spread. This thematic research will also establish the first generation of models and analytic tools to understand the complexity of co-feeding transmission of tick-borne diseases, contributing to the risk prediction of tick-borne pathogen spread under climatic and environmental changes. This risk prediction tool is much needed to build an effective public health decision support system. The thematic research on data clustering takes a novel dynamical systems approach to construct a clustering machine to address a fundamental challenge to adaptively select similarity thresholds so the change of similarity thresholds is linked to dynamical bifurcation when a new input is on the boundary of the domains of attraction of local attractors. Our innovative approach of clustering and our foundational research of the computational performance of a constructed neural network architecture can potentially lead to a major breakthrough in both theory and applications of high dimensional data clustering.