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

该综合项目将开发新的数学技术来研究反馈延迟对非线性流行病学系统长期动态行为的影响，以及对高维数据聚类的神经网络计算性能的影响。数学框架是所谓的时滞微分方程，其中系统状态变量的变化率取决于系统的当前和历史状态。这些方程也被称为泛函微分方程，因为它们的解在适当的泛函空间上产生无限维半流。我们的总体目标包括：识别和制定几类延迟微分方程，这些方程在生态学、流行病学、神经网络和数据聚类中有重要应用；开发框架、方法和技术工具来描述这些方程解的所有可能行为；并提供对潜在系统的观测/预测动态情景背后的机制的见解。我们将把三个相互交织的主题领域整合成一个新颖的、有凝聚力的、跨学科的项目，其中包括针对特定基础研究问题和重要应用的多个项目。在一个专题领域，我们发展了关于高阶锥的单调半流整体动力学的一般理论，以及它在具有非单调和/或双向延迟反馈的时滞方程中的应用。这一专题研究将在识别一类广义的非线性流行病学和神经系统方面取得突破性进展，这些系统不能应用经典的单调动力系统理论，并表征这些系统的全局动力学。在另一个专题领域，我们开发可有效用于审查与病媒传播疾病有关的模型中振荡模式的产生、终止和全球延续的技术。该专题研究还将建立第一代模型和分析工具，以了解蜱传疾病共食传播的复杂性，有助于预测气候和环境变化下蜱传病原体传播的风险。建立有效的公共卫生决策支持系统需要这种风险预测工具。数据聚类的专题研究采用一种新颖的动态系统方法构建聚类机，以解决自适应选择相似阈值的基本问题，从而将新输入位于局部吸引子吸引域边界上时相似阈值的变化与动态分岔联系起来。我们的创新聚类方法和我们对构建的神经网络架构的计算性能的基础研究可能会在高维数据聚类的理论和应用方面取得重大突破。

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.