Pattern Formation in the Drylands: Self Organization in Semi-Arid Ecosystems
, Northwestern University
- Much of our understanding of spontaneous pattern formation in spatially extended systems was developed in the “wetlands" of fluid mechanics. That setting allowed well-controlled table-top laboratory experiments; it came with fundamental equations governing the system; it benefitted from a back-and-forth between theory and experiment. These investigations identified robust mechanisms for spontaneous pattern formation, and inspired the development of equivariant bifurcation theory. Recently, these pattern formation perspectives have been applied to modeling the vegetation in dryland ecosystems, where satellite images have revealed strikingly regular spatial patterns on large scales. Ecologists have proposed that characteristics of vegetation pattern formation in these water-limited ecosystems may serve as an early warning sign of impending desertification. We use the framework of equivariant bifurcation theory to investigate the mathematical robustness of this approach to probing an ecosystem’s robustness. Additionally, we identify new applied pattern formation research directions in this far-from-pristine setting, where there are no fundamental equations and no controlled laboratory experiments.
University of North Carolina at Chapel Hill Mathematics