Wave propagation study of abstract dynamical systems with applications

抽象动力系统的波传播研究及其应用

• 批准号: RGPIN-2022-03842
• 负责人: Ou, Chunhua
• 金额: $ 1.53万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759086
• 关键词: Wave; propagation; study; abstract; dynamical

The primary goal of this proposal is to investigate wave propagation dynamics of abstract dynamical systems arising from reaction diffusion systems, integrodifference systems, nonlocal dispersal systems.  These systems  can model important phenomena of moving patterns in spatial single or multiple species models occurred in the study of competition and collaboration, ecological invasion and disease infection. Although there are various models describing spatial evolution of species, a universal  approach  in our studies of the dynamics is to consider the solution semiflow of the modeling system. This semiflow is also called as the dynamical system of the given model. Traveling wave patterns are important biological phenomena observed in these models, with a movement of a fixed speed as well as an unchanged solution profile. Determinacy of the waves, especially the spreading speed, is a challenging job. For a monotone dynamical system with monostability, we will study whether the system has a single spreading speed or there are multiple spreading speeds. We want to deeply understand how and when the spreading speed is determined by a linear conjecture, and how and when multiple spreading speeds and stacked wavefronts can appear. In either of above cases (single or multiple spreading speeds) we will study linear selection (linear conjecture) or nonlinear selection of the spreading speeds.  For a monotone dynamical system with bistable nonlinearity, we will study how to determine the wave speed sign which indicates the outcome of competition between two bistable states. The main scope of my research is: 1). Study the selection mechanism of the spreading speed(s) of abstract dynamical systems with monostable nonlinearity in three cases: a) The dynamical system is time periodic. b) The dynamical system is in a periodic habitat. c) The dynamical system is time periodic and in a periodic habitat. 2). Study the uniqueness and sign of the traveling speed of the abstract dynamical system with bistable nonlinearity in the following cases: a) Continuous or discrete in time or space (lattice). b) Particularly in time periodic case. c) In periodic habitat and/or time periodic case. 3). Further extend our idea and investigation to nonmonotone systems and stochastic dynamical systems. We will apply our theory to infectious disease models, chemotaxis models as well as models in biological, physical, chemical and other science and engineering fields. We intend to improve research results in some important past contributions, solve or answer open problems or conjectures in recent related investigations. This is designed to lead to  an important breakthrough in the deep understanding of traveling waves to some complex partial differential equation models. Theoretically it develops the idea of linear conjecture of some mathematical biologists and physicists. Moreover, these methods can be applied to solve problems in the life sciences and engineering.

该方案的主要目的是研究由反应扩散系统、积分差分系统、非局部扩散系统引起的抽象动力系统的波传播动力学，这些系统可以模拟在竞争与合作、生态入侵和疾病感染研究中出现的空间单种群或多种群模型中的运动模式的重要现象。虽然有各种各样的模型来描述物种的空间演化，但在我们对动力学的研究中，一个普遍的方法是考虑模型系统的解半流。这个半流也被称为给定模型的动力系统。行波模式是在这些模型中观察到的重要生物现象，具有固定速度的运动和不变的溶液轮廓。海浪的确定性，特别是传播速度，是一项具有挑战性的工作。对于具有单稳态的单调动力系统，我们将研究该系统是具有单一扩展速度还是存在多个扩展速度。我们想深入了解线性猜想如何以及何时决定扩散速度，以及多重扩散速度和叠加波前何时如何出现。在上述任何一种情况下(单个或多个扩展速度)，我们将研究扩展速度的线性选择(线性猜想)或非线性选择。对于具有双稳非线性的单调动力系统，我们将研究如何确定表示两个双稳之间竞争结果的波速符号。我的主要研究范围是：1)。研究了单稳非线性抽象动力系统在三种情况下的传播速度(S)的选择机制：a)动力系统是时间周期的。B)动力系统处于周期性生境中。C)动力系统是时间周期的，处于周期生境中。2)。研究了具有双稳态非线性的抽象动力系统在下列情况下运动速度的唯一性和符号：a)在时间或空间(格)上连续或离散。B)特别是在时间周期的情况下。C)在周期性生境和/或时间周期性的情况下。3)。进一步将我们的思想和研究推广到非单调系统和随机动力系统。我们将把我们的理论应用于传染病模型、趋化模型以及生物、物理、化学等科学和工程领域的模型。我们打算改进过去一些重要贡献的研究结果，解决或回答最近相关研究中的公开问题或猜想。这旨在导致对行波的深入理解到一些复杂的偏微分方程模型的重要突破。从理论上讲，它发展了一些数学生物学家和物理学家的线性猜想的想法。此外，这些方法还可以应用于解决生命科学和工程中的问题。