Avery, Montie2022-09-132022-09-132022-06https://hdl.handle.net/11299/241612University of Minnesota Ph.D. dissertation. June 2022. Major: Mathematics. Advisor: Arnd Scheel. 1 computer file (PDF); 208 pages.This thesis focuses on understanding the spatiotemporal dynamics of instabilities in largephysical systems. The onset of instability, either through a change in system parameters or the introduction of an external agent, plays a central role in mediating state transitions and structure formation in many physical systems. Examples common to our every day experience include viral epidemics and invasive species in ecology. Dynamics in the wake of instability are often governed by an invasion process, in which localized perturbations to an unstable background state grow, spread, and select a new state in the wake of invasion. A fundamental question is to predict the speed of propagation and the selected state in the wake. The mathematical study of invasion processes began in the 1920s with the Fisher-KPP equation,a model for the spread of advantageous genes in biological populations. This started a long avenue of research into related equations, predicting propagation speeds by constructing appropriate super- and sub- solutions and controlling propagation using comparison principles. On the other hand, there is a substantial body of experimental and theoretical work in the physics literature, dating back to the plasma physics literature in the 1950s, providing universal predictions for invasion speeds based only on certain spectral stability properties. This universal guiding principle is known as the marginal stability conjecture. In this thesis, we give the first proof of the marginal stability conjecture in a model independentframework, in the case where the invasion process selects a spatially constant state in its wake. We expect the framework we develop to remain useful in predicting invasion speeds in pattern-forming systems. In the process of our proof, we develop new mathematical techniques for establishing stability estimates in the presence of essential spectrum, which we expect to have broader use in diffusive stability problems.enDiffusive stabilityFront propagationNonlinear wavesTraveling wavesThesis or Dissertation