Pattern selection by invasion fronts

Emory University, Atlanta GA

Investigators

Abstract

Invasion fronts, or the moving boundaries between unstable and stable states caused by disturbances such as invasive species or a novel disease, play a key role in the self-organized development of coherent structures in many scientific fields, including epidemiology, developmental biology, fluid dynamics, materials science, and more. Historically, these invasion processes have been poorly understood, with clear results available only for special systems of limited interest and are difficult to study computationally. This project involves the development of a broad, model-independent framework for studying complex invasion fronts both theoretically and computationally, which can be used to make efficient predictions of invasion speeds and resulting spatial structures in broad classes of physical systems. The development of computational tools will focus on making these theoretical advances available to be put into practice by a broad range of scientists. Furthering the understanding of the selection of spatial structures in the wake of invasion fronts has particular promise for semiconductor manufacturing technologies which aim to harness this self-organized pattern formation to efficiently create nanomaterials with desirable electrical and optical properties. The project will involve both undergraduate and graduate students in the research, helping to develop the next generation of applied mathematicians. Prior mathematical results on invasion fronts have been limited to models satisfying restrictive monotonicity assumptions, which do not cover invasion fronts which make up many crucial examples in fluid dynamics, materials science, and more. In the first part of the project, the investigator will prove rigorous results for the prediction of invasion speeds and selected wave numbers of patterns in partial differential equations models, under conceptual assumptions on spectral stability of coherent front solutions. In the second part, the investigator will advance the theoretical understanding of when propagating fronts are ``pulled'' along by their tail or ``pushed'' ahead by their interface, and use this understanding to develop efficient algorithms for numerically continuing fronts and predicting selected speeds and wave numbers. In the final part of the project, the investigator will extend theoretical results on front propagation from one spatial dimension to higher dimensions, and investigate related transverse instabilities which play a key role in invasion dynamics in systems modeling bacterial motion, vegetation growth, and cancer dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.