Modeling and Control of Pattern-Forming Dynamical Systems

研究者研究形成时空格局的生物和化学系统。其中一个研究项目侧重于蝗虫群的建模和分析。蝗虫可以在单独（躲避邻居）或群居（寻找邻居）阶段活动。为了了解这些因素如何影响蝗虫的相变动态、种群的空间格局和群居群体的形成，研究者构建了一个包含社会行为、环境变量和随机运动的PDE模型。第二个模型描述了迁移的群居群体的滚动运动。高维ODE模型结合了社会互动、重力、风和地面形成的边界。这些模型的研究工具包括动力系统、统计力学、偏微分方程分析和数值模拟。另一个项目是研究化学反应扩散系统中图灵模式的控制。受最近通过使用外部光场控制化学图灵模式的实验的启发，研究者开发了一个通过周期性时空强迫来操纵反应扩散模式的框架。这项研究是建立在基于对称的范式分析、反应扩散型偏微分方程的摄动分析和数值模拟的基础上的，所有这些都揭示了模式稳定性是如何受到施加强迫的影响的。时空格局的形成在无数的自然系统中是自发发生的。关键的挑战包括这些系统的数学模型的开发，模型的分析，以及控制模式的方法的开发。该调查员对蝗灾的研究主要针对前两个挑战。蝗虫是一种重要的模式形成生物系统，因为它们的破坏性群体具有严重的环境和经济影响。一些结果是普遍的，足以说明其他社会生物聚集，如鸟群和鱼群。化学系统中的图灵模式研究解决了第三个挑战。研究结果解释了实验观察结果，并提出了可以在实验室测试的控制策略。此外，一些结果适用于其他图灵模式，这些模式出现在诸如生物形态发生、种群动力学、神经科学、生态学和材料科学等不同的背景中。研究学生的培训和指导以及使用新颖的教学技术发展数学和科学教学都在研究者的计划中发挥着关键作用。

The investigator studies biological and chemical systems that form spatiotemporal patterns. One research project focuses on the modeling and analysis of locust swarms. Locusts can operate in either a solitary (neighbor-avoiding) or gregarious (neighbor-seeking) phase. The investigator constructs a PDE model incorporating social aspects of behavior as well as environmental variables and random motion in order to understand how each of these factors affects the dynamics of phase change, the spatial patterns of the locust population, and the nucleation of gregarious groups. A second model describes the rolling motion of migrating gregarious groups. The high-dimensional ODE model incorporates social interactions, gravity, wind, and the boundary formed by the ground. The models are studied with tools from dynamical systems, statistical mechanics, analysis of PDE, and numerical simulation. Another project examines the control of Turing patterns in chemical reaction-diffusion systems. Inspired by recent experiments which controlled chemical Turing patterns by using an external light field, the investigator develops a framework for manipulating reaction-diffusion patterns via periodic spatiotemporal forcing. The study is built around a symmetry-based analysis of normal forms, a perturbation analysis of reaction-diffusion type PDE, and numerical simulations, all of which reveal how pattern stability is affected by the applied forcing.Spatiotemporal pattern formation occurs spontaneously in myriad natural systems. Key challenges include the development of mathematical models of these systems, the analysis of the models, and the development of methods for controlling the patterns. The investigator's research on locust plagues primarily addresses the first two challenges. Locusts are an important pattern-forming biological systems because their destructive swarms have severe environmental and economic impacts. Some results are general enough to shed light on other social biological aggregations such as bird flocks and fish schools. The research on Turing patterns in chemical systems addresses the third challenge. The results explain experimental observations and suggest control strategies that can be tested in the lab. Furthermore, some of the results apply to other Turing patterns, which occur in such diverse contexts as biological morphogenesis, population dynamics, neuroscience, ecology, and materials science. The training and mentoring of research students and the development of math and science instruction using novel pedagogical techniques all play a key role in the investigator's program.