Modeling and Control of Pattern-Forming Dynamical Systems

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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.