Dynamics of Functional Differential Equations with Applications to Biology and Ecology

泛函微分方程动力学及其在生物学和生态学中的应用

• 批准号: RGPIN-2015-05686
• 负责人: Wang, Lin
• 金额: $ 1.24万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613689
• 关键词: Dynamics; Functional; Differential; Equations; Applications

Many biological and ecological processes involve time lags. For example, a predator needs time to convert its consumption of prey to its biomass; for many infectious diseases, time lags exist due to dispersal of populations and disease state changes. To model these time-lag involving processes, a natural choice is to use functional differential equations (i.e., differential equations subject to delays). In contrast to systems described by ordinary differential equations, the initial state of a functional differential equation is a function specified on an interval and the associated phase space is infinite dimensional. Since the resulting dynamical systems are infinite dimensional, the analysis of functional differential equations is extremely challenging, especially for equations with delay-dependent coefficients, nonlinearity and multiple state variables, which happen to be typical features of equations risen from biology and ecology. In this proposed research, I aim to develop new methodologies and techniques to depict the global dynamic structure of certain nonlinear systems of functional differential equations arising from important applications in biology and ecology. In particular, I plan to formulate biological and ecological models by mainly using functional differential equations and to develop new methodologies, techniques and algorithms to study their short-time (transient) and long-time (asymptotic) behaviors. The focus will be on the effects of time delays involved in the dispersal of populations, in intraguild predation and in induced resistance of plants in responding to insect herbivore's attack on the dynamics. Theoretical and applicable results to be established in this proposed research will increase our understanding of both short-time and long-time qualitative behaviors in biological and ecological models with time delay and nonlinearity, and thereby, provide useful and valuable guidance and suggestions on controlling the spread of infectious diseases, preventing the extinction of endangered species, and maintaining sustainable development of ecosystems. Anticipated newly developed theories, methodologies and techniques in this proposed research will highly enhance the advance of theoretical development in nonlinear functional differential equations and nonlinear dynamical systems. The established theoretical results will greatly improve our ability to deal with real world problems arising from biology and ecology and provide biologists and ecologists powerful tools to solve their specific problems that may be described by functional differential equations. This research will also provide excellent opportunities to train graduate students and postdoctoral researchers in dynamical systems and mathematical biology.

许多生物学和生态过程都涉及时间滞后。例如，捕食者需要时间将其猎物消耗转换为生物量。对于许多传染病，由于种群的分散和疾病状态的变化，时间滞后存在。为了对这些涉及过程的时间段进行建模，一种自然的选择是使用功能微分方程（即，差异方程式可能会延迟）。与普通微分方程描述的系统相反，功能微分方程的初始状态是在间隔上指定的函数，相关的相空间是无限的尺寸。由于所得的动力学系统是无限的维度，因此功能微分方程的分析极具挑战性，尤其是对于具有延迟依赖性系数，非线性和多个状态变量的方程式，这恰好是从生物学和生态学方程式中的典型特征。 在这项拟议的研究中，我旨在开发新的方法和技术，以描述由生物学和生态学中重要应用引起的功能微分方程的某些非线性系统的全球动态结构。特别是，我计划通过主要使用功能微分方程来制定生物学和生态模型，并开发新的方法，技术和算法来研究其短期（瞬态）和长期（渐近）行为。重点将放在种群分散，内部捕食和诱导植物抗药性中涉及的时间延迟的影响上，以应对昆虫草食动物对动态的攻击。在这项拟议的研究中要建立的理论和适用结果将增加我们对以时延迟和非线性的生物和生态模型中短期和长期定性行为的理解，从而为控制控制有用的有用的指导和建议，以控制控制传染病的传播，防止濒危物种的灭绝以及维持生态系统的可持续发展。 在这项拟议的研究中，预期的新开发的理论，方法和技术将在非线性功能微分方程和非线性动力学系统中高度提高理论发展的进步。既定的理论结果将大大提高我们处理生物学和生态学引起的现实世界问题的能力，并为生物学家和生态学家提供强大的工具来解决其特定问题，这些问题可能由功能差分方程描述。这项研究还将为培训动态系统和数学生物学的研究生和博士后研究人员提供绝佳的机会。