Dynamics of Functional Differential Equations with Applications to Biology and Ecology

许多生物和生态过程涉及时间滞后。例如，捕食者需要时间将猎物的消耗量转化为生物量；对于许多传染病来说，由于种群的分散和疾病状态的变化，存在时间滞后。为了模拟这些涉及时滞的过程，自然的选择是使用泛函微分方程（即，受延迟影响的微分方程）。与常微分方程所描述的系统不同，泛函微分方程的初始状态是在一个区间上指定的函数，其相空间是无限维的。由于所得到的动力系统是无限维的，因此分析泛函微分方程是极具挑战性的，特别是对于具有时滞相关系数、非线性和多状态变量的方程，而这些方程恰好是生物学和生态学中产生的方程的典型特征。***在这项研究中，我的目标是开发新的方法和技术来描述某些非线性泛函微分方程系统的全局动态结构，这些系统在生物学和生态学中有重要应用。特别是，我计划主要使用泛函微分方程来制定生物和生态模型，并开发新的方法，技术和算法来研究它们的短时（瞬态）和长时间（渐近）行为。重点将放在时间延迟对种群扩散的影响，在野外捕食和诱导植物抵抗昆虫食草动物的攻击对动力学的反应。本研究所建立的理论和应用结果将增加我们对具有时滞和非线性的生物和生态模型的短期和长期定性行为的认识，从而为控制传染病的传播、防止濒危物种的灭绝和维持生态系统的可持续发展提供有用和有价值的指导和建议。****本研究中预期的新理论、新方法和新技术将极大地促进非线性泛函微分方程和非线性动力系统的理论发展。已建立的理论结果将大大提高我们处理由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可能由泛函微分方程描述的特定问题。这项研究也将为培养动力系统和数学生物学方面的研究生和博士后提供很好的机会。*** **

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.*** *** **