Functional Differential Equations in Biology and Epidemiology

泛函微分方程的一个显著特点是其所描述的过程的演化速率依赖于过去的历史。在过去的三十年里，人们对偏微分方程的理论和应用的研究非常活跃，与偏微分方程相关的数学模型适用于不同性质的现象，如生物学、生态学、流行病学、工程学和经济学。 这项研究的长期目标包括，深入了解生物和流行病学系统中的动态行为和支配这种行为的原因，特别是时间延迟和系统结构如何影响物种的进化，传染病是否可以传播和持续，或导致传染病的爆发。将主要探讨基于非线性分析、泛函微分方程和动力系统的数学方法，包括某些类型周期/准周期解的存在性、不存在性、稳定性、持久性、分叉。 更真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，虽然对自治和非自治的FDE做了一些工作，但缺乏理论分析，尤其是对稳定性和分叉的分析。在拟议的研究中，我将寻求将稳定性和分支理论推广和发展到非单调自治、非自治、空间和时间涉及的具有时滞的泛函微分方程。并将理论结果应用于具有阶段/年龄结构的生物和流行病系统，甚至疾病传播网络模型中出现的动态行为的定性分析问题，特别强调建立关于时滞、系统结构和动态行为的影响的强联系。我也足够灵活，可以在有前景的研究途径出现时进行探索。 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的定性属性的了解。提案中提出的理论和方法将对非线性动力学领域的发展产生非常大的影响。它们不仅将加强在一大类FDE中进行理论扩展的基础，而且还将为支持生物学和流行病学决策提供实用的观点。这种跨学科的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、优化捕获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.