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。