Nonlinear Dynamics of Pattern Forming Invasion Fronts

模式形成入侵前沿的非线性动力学

• 批准号: 1516155
• 负责人: Matt Holzer
• 金额: $ 11.72万
• 依托单位: George Mason University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1516155&HistoricalAwards=false
• 关键词: Nonlinear; Dynamics; Pattern; Forming; Invasion

This project is concerned with the spontaneous formation of periodic structures in complex systems. Examples of such systems include bands of mussels in tidal flats, bacterial aggregation patterns, and hexagonal or stripe patterns in convection problems. Spatial patterns often arise due to the instability of a simpler state. For example, in ecological or chemical applications an equilibrium state may be destabilized due to the introduction of a previously absent species. Initial states lying near this equilibrium state will evolve towards a new profile and one would like to predict what this new state will be. For spatially extended problems, this process involves the formation of moving interfaces, known as invasion fronts, that move at a fixed speed and serve to transition the unstable state to some other configuration. The objective is to uncover the properties of a nonlinear system that lead to selection of a particular invasion front and consequently the patterned state that is created by that front. Transitions from unstable to stable states occur in many applied contexts and this research will help inform scientists to formulate predictions regarding, or even control over, the formation of complex spatiotemporal patterns in those systems. The work has potential application via the role of reaction-diffusion equations in modeling the generation of small scale structures in nanotechnology and materials science. The mathematical focus of the project is the study of pattern formation in systems of reaction-diffusion equations in which periodic structures are created by invasion fronts. Several specific systems of parabolic partial differential equations will be considered with the aim of extracting general principles mediating wavespeed selection. The project areas include the study of competition amongst patterns in reaction-diffusion-advection equations arising in ecological and chemical applications, coarsening modes in biological aggregation models of Keller-Segel type, and the existence of fronts in coupled amplitude equations describing the emergence of and competition between patterned states near a supercritical Turing bifurcation. The project will require a variety of tools including linear analysis, geometric singular perturbation theory, blow-up techniques, and comparison principles. There are several innate challenges in regards to describing pattern-forming fronts in systems of equations that will require extensions of analytical techniques. For example, systems of equations can support invasion fronts wherein different components invade at different speeds for which no single fixed frame can capture the relevant dynamics.

这个项目关注的是复杂系统中周期性结构的自发形成。这种系统的例子包括潮滩上的贻贝带，细菌聚集模式，对流问题中的六边形或条纹模式。空间模式通常是由于一个简单状态的不稳定性而产生的。例如，在生态或化学应用中，平衡状态可能由于引入先前不存在的物种而不稳定。在这个平衡态附近的初始状态将演变成一个新的轮廓，人们想要预测这个新状态是什么。对于空间扩展的问题，这个过程涉及到移动界面的形成，称为入侵前沿，它以固定的速度移动，并将不稳定状态转变为其他一些配置。目标是揭示导致选择特定入侵前沿的非线性系统的特性，以及由此产生的由该前沿产生的模式状态。从不稳定状态到稳定状态的转变发生在许多应用环境中，这项研究将帮助科学家制定预测，甚至控制这些系统中复杂时空模式的形成。通过反应扩散方程在纳米技术和材料科学小尺度结构生成建模中的作用，这项工作具有潜在的应用前景。该项目的数学重点是研究反应扩散方程系统中的模式形成，其中周期结构是由入侵前沿产生的。本文将考虑几种特殊的抛物型偏微分方程组，目的是提取影响波速选择的一般原理。项目领域包括研究生态和化学应用中出现的反应-扩散-平流方程模式之间的竞争，凯勒-塞格尔型生物聚集模型中的粗糙模式，以及描述超临界图灵分岐附近模式状态之间出现和竞争的耦合振幅方程中的锋面的存在。该项目将需要各种工具，包括线性分析、几何奇异摄动理论、爆破技术和比较原理。在描述方程系统中的模式形成前沿方面，有几个固有的挑战，需要扩展分析技术。例如，方程系统可以支持入侵前线，其中不同的组件以不同的速度入侵，没有一个固定的框架可以捕获相关的动态。