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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668101
• 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.*** *** **

许多生物和生态过程都涉及时间滞后，例如，捕食者需要时间将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而存在时间滞后。过程中，自然的选择是使用函数微分方程（即涉及延迟的微分方程），与常微分方程描述的系统相比，函数微分方程的初始状态是在区间和相关联上指定的函数。相空间是由于所得到的动力系统是无限维的，所以泛函微分方程的分析极具挑战性，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。 .***在这项拟议的研究中，我的目标是开发新的方法和技术来描述生物学和生态学中重要应用产生的某些非线性函数微分方程系统的全局动态结构。主要使用函数式的模型微分方程并开发新的方法、技术和算法来研究它们的短时（瞬态）和长时间（渐近）行为，重点将放在种群分散、群体内捕食和行为中的时间延迟的影响。这项研究中建立的理论和应用结果将增加我们对具有时间延迟的生物和生态模型中的短期和长期定性行为的理解。和非线性，并保持，为控制传染病的传播、防止濒危物种的灭绝和生态系统的可持续发展提供有用和有价值的指导和建议。****本文提出的预期新发展的理论、方法和技术研究将极大地促进非线性泛函微分方程和非线性动力系统理论发展的进步，所建立的理论成果将极大地提高我们处理生物学和生态学所产生的现实世界问题的能力，并为生物学家和生态学家解决他们的具体问题提供有力的工具。可能出现的问题这项研究还将为培训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。*** *** **