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Pattern Formation in Dynamical Systems

I use dynamical systems techniques to analyze the mechanisms that lead to pattern formation in differential equation models of physical phenomena. Please see my CV for publications and presentations.

  • I describe my interest in self-organized patterns in a recent feature in DSWeb.

Zigzag Selection from Spatial Inhomogeneity

The Swift-Hohenberg equation was originally derived as a simple model equation for the behavior of phenomena appearing in hydro-thermal convection experiments. It captures the behavior of patterns created by rising and falling fluid in a plate heated from below. By introducing a jump-type inhomogeneity in the linear parameter, we restrict the possible wavenumbers of stripes occuring as solutions to the equation. This restriction leaves the stripes unstable to transverse perturbations, resulting in the zigzags in the image on the right. This is joint work with my advisor.

Our resulting paper appears in Phil. Trans. Roy. Soc. A and an image from one of our simulations was chosen as the cover art for the issue. My ongoing thesis work examines extensions of this work applied to more complicated hexagonal patterns.

Traveling waves of foraging locusts

Locusts gather by the millions to feed on crops, destroying fields of agricultural produce. As juveniles, wingless locusts march together and form a wave of advancing insects. We examine this collective propagation through two models: an agent-based model and a set of partial differential equations. The agent-based model is directly linked to individual behavior, via observations from the biological literature, while the PDE model yields insight into the collective behavior of the aggregate group. This work is done through a collaboration which began at the MRC Program in Agent-Based Modeling in June, 2018.

We are currently writing a preprint on our work.

Traveling Vegetation Bands

In arid grasslands, vegetation patterns emerge as a survival strategy to cope with insufficient water and nutrients. On gradual slopes, ecologists have observed distinct bands of vegestation parallel to contour lines that move very slowly uphill. We studied a simple model equation which exhibited this behavior and much more in the presence of a conservation law -- representing a constant amount of organic matrials, e.g. nitrogen. This is joint work with my advisor and two talented undergraduates from our REU.

Our paper appears in the Journal of Nonlinear Science.

Image credit to Zachary Singer (left) and Google Earth (below).