Wave propagation study of abstract dynamical systems with applications

本文的主要目的是研究反应扩散系统、积分差分系统、非局部扩散系统等抽象动力系统的波传播动力学。 这些系统可以模拟空间单种群或多种群模型中的运动模式等重要现象，用于研究竞争与协作、生态入侵和疾病感染等问题。虽然描述物种空间演化的模型多种多样，但在我们研究物种空间演化动力学的过程中，普遍采用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以及不变的溶液轮廓。波的确定，特别是波的传播速度的确定，是一项具有挑战性的工作。对于具有单稳定性的单调动力系统，我们将研究系统是具有单一的传播速度还是具有多个传播速度。我们希望深入了解如何以及何时由线性猜想确定传播速度，以及如何以及何时可以出现多个传播速度和叠加波前。在上述任何一种情况下（单一或多个扩展速度），我们将研究线性选择（线性猜想）或非线性选择的扩展速度。 对于一类具有非线性项的单调动力系统，我们将研究如何确定表征两个非线性项竞争结果的波速符号。本文的主要研究内容是：（1）。研究了三种情况下具有单稳态非线性的抽象动力系统传播速度的选择机制：a）动力系统是时间周期的; B）动力系统处于周期栖息地。c）动力系统是时间周期的，并且处于周期栖息地。2）。研究了具有非线性项的抽象动力系统在下列情况下的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、趋化性模型以及生物、物理、化学等科学和工程领域的模型。本文旨在完善以往的一些重要研究成果，解决或回答近期相关研究中存在的问题或疑问。这是一个重要的突破，在深入了解行波的一些复杂的偏微分方程模型。它在理论上发展了一些数学生物学家和物理学家的线性猜想思想。此外，这些方法可以应用于解决生命科学和工程中的问题。

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.