Dynamics of Functional Differential Equations with Applications to Biology and Ecology

许多生物学和生态学过程都涉及时滞。例如，捕食者需要时间将其对猎物的消耗转化为生物量;对于许多传染病，由于种群的分散和疾病状态的变化，存在时滞。为了对这些涉及时滞的过程进行建模，自然的选择是使用泛函微分方程（即，微分方程的延迟）。与常微分方程描述的系统相反，泛函微分方程的初始状态是在区间上指定的函数，并且相空间是无限维的。由于所得到的动力系统是无限维的，泛函微分方程的分析是非常具有挑战性的，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，这些恰好是生物学和生态学方程的典型特征。在这项研究中，我的目标是开发新的方法和技术来描绘全球动态结构的某些非线性系统的功能微分方程在生物学和生态学的重要应用。 特别是，我计划制定生物和生态模型，主要使用泛函微分方程，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）的行为。重点将是时间延迟的影响，在种群的扩散，在intraguild捕食和诱导的植物在响应昆虫草食动物的攻击的动态。本研究所建立的理论和应用结果将加深我们对具有时滞和非线性的生物和生态模型中的短时和长时定性行为的理解，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。本研究所提出的新理论、新方法和新技术将极大地推动非线性泛函微分方程和非线性动力系统的理论发展。所建立的理论结果将大大提高我们处理生物学和生态学中真实的问题的能力，并为生物学家和生态学家解决他们可能用泛函微分方程描述的具体问题提供强有力的工具。这项研究还将为培养动力系统和数学生物学方面的研究生和博士后研究人员提供极好的机会。*** **

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.*** *** **