Nonlinear Dynamics of Pattern Forming Invasion Fronts

George Mason University, Fairfax VA

Investigators

Abstract

This project is concerned with the spontaneous formation of periodic structures in complex systems. Examples of such systems include bands of mussels in tidal flats, bacterial aggregation patterns, and hexagonal or stripe patterns in convection problems. Spatial patterns often arise due to the instability of a simpler state. For example, in ecological or chemical applications an equilibrium state may be destabilized due to the introduction of a previously absent species. Initial states lying near this equilibrium state will evolve towards a new profile and one would like to predict what this new state will be. For spatially extended problems, this process involves the formation of moving interfaces, known as invasion fronts, that move at a fixed speed and serve to transition the unstable state to some other configuration. The objective is to uncover the properties of a nonlinear system that lead to selection of a particular invasion front and consequently the patterned state that is created by that front. Transitions from unstable to stable states occur in many applied contexts and this research will help inform scientists to formulate predictions regarding, or even control over, the formation of complex spatiotemporal patterns in those systems. The work has potential application via the role of reaction-diffusion equations in modeling the generation of small scale structures in nanotechnology and materials science. The mathematical focus of the project is the study of pattern formation in systems of reaction-diffusion equations in which periodic structures are created by invasion fronts. Several specific systems of parabolic partial differential equations will be considered with the aim of extracting general principles mediating wavespeed selection. The project areas include the study of competition amongst patterns in reaction-diffusion-advection equations arising in ecological and chemical applications, coarsening modes in biological aggregation models of Keller-Segel type, and the existence of fronts in coupled amplitude equations describing the emergence of and competition between patterned states near a supercritical Turing bifurcation. The project will require a variety of tools including linear analysis, geometric singular perturbation theory, blow-up techniques, and comparison principles. There are several innate challenges in regards to describing pattern-forming fronts in systems of equations that will require extensions of analytical techniques. For example, systems of equations can support invasion fronts wherein different components invade at different speeds for which no single fixed frame can capture the relevant dynamics.