CAREER: Pattern formation in singularly perturbed partial differential equations

University Of California-Irvine, Irvine CA

Investigators

Abstract

Spatiotemporal pattern formation appears throughout the natural world; examples include the dynamics of vegetation patterns in dryland ecosystems, phase separation in mixtures, and patterns in chemical reactions. Fundamental theoretical questions concern identifying universal pattern-forming mechanisms and predicting the nature and dynamics of the patterns which appear. This project aims to address these questions through the study of pattern-forming instabilities in the setting of partial differential equations (PDEs) arising in models from biology, ecology, and chemical reactions. In particular, the project focuses on the development of techniques to tackle pattern formation in PDEs which are singularly perturbed; such equations arise naturally in the study of systems where components operate on widely differing length or time scales. Application areas include desertification fronts in dryland ecosystems, as well as wave phenomena in mathematical biology and neuroscience. Included in the project are activities which integrate research and education through the supervision of graduate students as well as outreach programs and summer research opportunities for undergraduate students. This project focuses on three primary research areas, with the goal of providing insight into complex spatiotemporal pattern formation phenomena in singularly perturbed reaction diffusion PDEs. Motivated by the phenomenon of desertification fronts in dryland ecosystems, the first research area concerns pattern-forming instabilities of bistable planar interfaces in multi-component reaction diffusion systems. The second area aims to build a rigorous theory of temporal pulse replication, by which a single traveling pulse (or localized wave) self-replicates, sequentially nucleating additional pulses in a manner resembling a spatiotemporal canard explosion. The third area is concerned with the nature and properties of far-from-onset patterns formed in the wake of pattern-forming invasion processes, whereby a pattern is nucleated when a disturbance invades an unstable steady state. These research areas will grow and advance the theory of far-from-onset pattern-forming mechanisms through the development of geometric singular perturbation methods and spatial dynamical systems tools to be used in the existence, stability, and bifurcation analysis of waves and patterns. Several of the phenomena explored in this project occur in regions where standard singular perturbation methods break down, requiring the development of novel techniques. Graduate students and undergraduate students will participate in various aspects of the project. 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.