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

该综合计划将开发新的数学技术，以研究反馈延迟对非线性流行病学系统的长期动力学行为的影响，以及对高维数据聚类神经网络的计算性能的影响。 数学框架是所谓的延迟微分方程，其中系统的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穷维半流。 我们的总体目标包括：确定和制定生态学，流行病学，神经网络和数据聚类中的重要应用所产生的几类延迟微分方程;开发框架，方法和技术工具，以描述这些方程的所有可能的解决方案的行为;并提供对基础系统的观测/预测动态场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，有凝聚力和跨学科的计划，其中包括多个关于特定基础研究问题和重要应用的项目。 在一个专题领域，我们开发的一般理论的整体动力学的半流是单调的高秩锥，其应用程序的延迟方程与非单调和/或双向延迟反馈。这一专题研究将取得突破性的进展，确定了广泛的一类非线性流行病学和神经系统，经典的单调动力系统理论不能应用，并表征这种系统的全局动力学。在另一个主题领域，我们开发的技术，可以有效地用于检查相关的媒介传播疾病传播模型中的振荡模式的创建，终止和全球延续。本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，有助于气候和环境变化下蜱传病原体传播的风险预测。建立一个有效的公共卫生决策支持系统需要这种风险预测工具。数据聚类的主题研究采用了一种新的动态系统方法来构建聚类机，以解决自适应地选择相似性阈值的基本挑战，因此当新输入位于局部吸引子的吸引域的边界上时，相似性阈值的变化与动态分叉相关联。我们创新的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导致高维数据聚类理论和应用的重大突破。

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.