Functional Differential Equations in Biology and Epidemiology

泛函微分方程（FDEs）的一个显著特征是，这种方程所描述的过程的演化速率取决于过去的历史。近三十年来，FDE的理论和应用研究非常活跃，FDE相关的数学模型适用于生物学、生态学、流行病学、工程学和经济学等不同性质的现象。******本研究的长期目标包括深入了解fde所能说明的生物和流行病学系统的动力学行为及其原因，特别是时间延迟和系统结构如何影响物种的进化，传染病是否能够传播和持续，或导致传染病爆发。主要探讨基于非线性分析、fde和动力系统的数学技术，包括存在性、不存在性、某类周期/准周期解的稳定性、持续性、分岔。******更现实的模型可能与导致非自治系统的环境变化有关。在文献中，虽然对自主和非自主FDEs做了一些研究，但缺乏理论分析，特别是对稳定性和分岔的分析。在本文的研究中，我将寻求将稳定性和分岔理论扩展和发展到非单调自治、非自治、空间和时间涉及时滞的FDEs。我将应用理论结果来解决在具有阶段/年龄结构的生物和流行病学系统中产生的动态行为的定性分析问题，甚至疾病传播网络模型，特别强调在时间延迟，系统结构和动态行为的影响上建立强有力的联系。我也有足够的灵活性去探索有前途的研究途径。******这项拟议的研究将从数学和生物学/流行病学的角度增加我们对应用FDEs定性性质的认识。本文所提出的理论和方法将对非线性动力学领域的发展产生重大影响。它们不仅将加强理论扩展的基础，而且还将为支持生物学和流行病学决策提供实践观点。跨学科合作及其在渔业中的应用将具有巨大的潜力，为加强海洋种群动态、优化捕捞率和维持生态系统结构提供新的见解。预期的结果可能会产生影响，导致应用动力学领域的进步，并将加强HQP培训和合作，使加拿大受益。

A distinguishing feature of the functional differential equations (FDEs) is that the evolution rate of the process described by such equations depends on the past history. During the past three decades, study on the theory and application of FDEs has been very active, the FDE related mathematical models are applicable to phenomena of different natures, such as, in biology, ecology, epidemiology, engineering and economy.******The long-term goals of the research include, understanding in-depth the dynamical behavior and the reasons that govern such behavior in biological and epidemiological systems that can be illustrated by FDEs, especially how time delay and the system structure affect the evolution of the species, whether the infectious disease can spread and persist, or lead to an infectious disease outbreak. Mathematical techniques based on nonlinear analysis, FDEs and dynamical systems including existence, non-existence, stability of certain type of periodic/quasi-periodic solutions, persistence, bifurcations will be mainly approached.******More realistic models may relate to the variation of environment which result in non-autonomous systems. In the literature, although some work has been done for autonomous and non-autonomous FDEs, there is a lack of theoretical analysis, especially for the stability and bifurcations. In the proposed research I will seek to extend and develop the stability and bifurcation theories to non-monotonic autonomous, non-autonomous, spatial and temporal involved FDEs with delays . And I will apply the theoretical results to tackle problems of qualitative analysis of dynamical behaviors arising in biological and epidemiological systems with stage/age-structures, or even disease transmission network models, with particular emphasis on establishing strong links on the effect of time delay, system construction and the dynamical behavior. I am also flexible enough to explore promising avenues of research as they emerge.******This proposed research will increase our knowledge of the qualitative properties in applied FDEs, from mathematical and biological/epidemiological points of view. The theories and methodologies developed in the proposal will have very high impact on the development in the nonlinear dynamics community. They will not only strengthen the foundation for theoretical expansion in a large class of FDEs, but will also provide a practical view to support biological and epidemiological decision making. The interdisciplinary collaboration and the application to the fishery will have great potential to provide novel insights to strength marine population dynamics, optimize harvesting rates, and maintain ecosystem structure. The anticipated outcomes are likely to have impact by leading to advancements in the field of applied dynamics and will enhance HQP training and collaboration benefiting in Canada.