dynamical systems

pattern formation


I study self-organized patterns arising in models of natural phenomena, especially from the life sciences. Mathematical techniques from dynamical systems provide a framework to understand the formation of stripes, spots, and waves.

This page describes some past projects that have published works associated with them.

A band of locusts marches left to right through pasture.

Adapted from https://www.abc.net.au/radionational/programs/scienceshow/9-nymphal-band.jpg/5615018

A solution to the PDE model.

Foraging Locusts

Food-based rules lead to collective motion

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. I 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 through a rigorous analysis of traveling wave solutions. I am working on this project through a collaboration of six researchers across the US. We met and began working together at the MRC Program in Agent-Based Modeling in June, 2018. We have been grateful to receive support for follow up collaboration meetings from the AMS in August 2018 and from the Institute for Advanced Study in the summer of 2019.

  • AJ Bernoff, M Culshaw-Maurer, R Everett, ME Hohn, WC Strickland, J Weinburd. Agent-based and continuous models of hopper bands for the Australian plague locust: How resource consumption mediates pulse formation and geometry. PLOS Comp Biol (2020). (Preprint)

Summary of our agent-based model:

A simulation of our agent-based model. Try slowing down the play-back speed to watch individual locusts (maroon and blue dots).

Fundamental Pattern Selection

Local change in space creates a global transition

Why is one pattern favored over another? I examine selection that results from an environment that suppresses patterns in half the domain. Introducing this spatial inhomogeneity restricts the period of the emergent patterns. My thesis work proves and quantifies this restriction in two cases.

From stripes to zigzags

In a fundamental partial differential equation model for spatially periodic patterns, the Swift-Hohenberg equation, this selection process becomes explicitly observable. Moving this environmental inhomogeneity forces patterns to form that are unstable to transverse perturbations. As this instability takes hold straight stripes start to wiggle, evolving into zigzags.

We derive an explicit formula for the selected period by using spatial dynamics and a normal form transformation. An image from one of our simulations was chosen as the cover art for the issue. This work is joint with my PhD advisor Arnd Scheel.

Deforming Hexagons

More complicated hexagonal patterns arise in a modified Swift-Hohenberg equation by breaking one of its symmetries. In the video (left), watch as the hexagons stretch to a maximal horizontal period before adding each new row. In my thesis work, I develop a framework for calculating that maximal period by tracking terms in normal form transformations across the spatial inhomogeneity.

For more, see my thesis. I am currently adapting this material for publication with my PhD advisor Arnd Scheel.

Traveling Vegetation Bands

Satellite image over the Haud region in the Horn of Africa.

Retrieved 2018 from Google Earth.

In arid grasslands, vegetation patterns emerge as a survival strategy that allows plants 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 more complex dynamics in the presence of a conservation law -- representing a constant amount of organic nutrients, e.g. nitrogen. This is joint work with my advisor and two talented undergraduates from our REU.

Image credit to Zachary Singer.