I am in the last year of my PhD at the University of Minnesota where I work with Arnd Scheel in pattern formation and dynamical systems. This fall, I am incredibly grateful for the support of an NSF Postdoctoral Fellowship which allows me to head to Harvey Mudd College. There, I will collaborate with Andrew Bernoff on models of swarming and collective behavior in locusts. I am extremely excited to research, mentor, and teach in an undergraduate-focused environment.
Figure 1: Cloud streets over Vancouver Island, Feb 13, 2015. From CTV News.
Imagine stripes in the sky that form as long rows of cumulus clouds. These cloud streets result from just-right atmospheric conditions. These conditions include a vertical temperature difference great enough to trigger convective airflow. In other words, there are updrafts of hot air and downdrafts of cool air (rather than a uniform heat gradient with diffusive transfer). Up/down drafts align to form convection rolls, each contributing hot, wet air in long thin stripes of cloud.
What if that temperature difference occurs only over the surface of a large lake and ends abruptly at the shoreline? The stripes should form over the water only, and not over the land. Should they have the same characteristics as the stripes that form on a day where the temperature difference is constant over both the water and the land? Perhaps they will be narrower? Perhaps wider? Or perhaps they will wiggle, creating zigzags in the sky? Answering these questions on a conceptual level provides insight into a range of patterned phenomena.
In my thesis work [1, 2], I study a conceptual PDE model known as the Swift-Hohenberg equation (SHE) . I examine the SHE with a spatial jump in the parameter that controls the pattern-forming instability. Stripes are suppressed in half the domain and allowed in the other half. Using spatial dynamics, center manifold reduction, and normal form theory , I construct heteroclinic orbits representing spatial transitions from stripes to the non-patterned state. The primary difficulty arises from the parameter jump. I bridge the jump with an explicit calculation that leads to a relation dictating the width of stripes as a function of the size of the jump. Notably, this is a significant restriction on the width as compared to the constant-parameter case. I will be presenting on work that extends this approach to hexagonal patterns at the upcoming Snowbird meeting.
The wider pattern formation community often uses SHE as a first model in the study of self-organized patterns. Mathematically speaking, I simply mean that the patterns are not baked into the PDE (there is no periodic forcing, for instance). Instead, one may regard SHE as a gradient flow and patterns arise as a minimizer of the associated energy functional. Physically, no one flies an airplane up and down the lake to get the stripes started; the stripe configuration is simply the best way to transfer heat between the atmospheric layers. Self-organization provides a clue towards understanding the ubiquity of common patterns in nature; all that’s required is a basic set of mechanisms and the system self-organizes the rest.
Another example of self-organization emerges in my interest in swarming. Insects, fish, and humans each behave according to individual rules. But once aggregated in large swarms, schools, or crowds they exhibit behavior that appears to follow collective rules. Because it can be difficult to empirically measure individual interactions amidst a large aggregation, a common approach is to infer individual information by quantifying and analyzing collective behavior, see  for instance. When aggregations are large enough and interactions are simple enough, the collective behavior may be described by a continuum model for density, in the spirit of . My particular interest is in phenomena observed in a biblical bug.
Figure 2. Coherent structures in locust hopperbands: locusts march parallel to the collective stream over bare ground. From wikipedia.
Figure 3. Coherent structures in locust hopperbands: locusts march perpendicular to the collective front through pasture. Modified from ABC.
Locust nymphs (wingless juveniles) aggregate in hopperbands to forage. These hopperbands exhibit a variety of distinctive shapes. In Figure 2, locusts move parallel to the columnar stream towards an isolated patch of vegetation. In Figure 3, locusts move perpendicular to the line of advancing insects through an agricultural field. These two collective behaviors can be attributed to the interaction of rules that dictate an individual locusts attraction to food and attraction/repulsion from other locusts. For instance, it appears the planar front in Figure 3 is at odds with insect-insect attraction, but is stabilized by strong attraction to food. My approach is to use continuum models to classify collective behavior in parallel with investigations of agent-based models that directly encode individual behavior.
The continuum model takes the form of a PDE or integro-differential equation. Typical results on existence, bifurcations, and stability of traveling wave solutions with various profiles determine profile-selection principles that can be observed in the agent-based model. Careful comparison with empirical data, for instance , links these to the individual level and sparks conversations with collaborators in the biological sciences. I aim to contribute results that inform the function and evolutionary origin of individual behaviors. I hope that this work will eventually inspire efficient methods for locust control to mitigate the loss of agricultural crops.
This research agenda grew out of the Mathematical Research Communities program, administered by the AMS with support from the NSF. To my peers, I cannot recommend this program highly enough! It combines a workshop, summer school, and research introduction with additional funding for travel to the JMM and follow-up collaboration. I participated in a week on Agent-Based Modeling in the Biological and Social Sciences, but they offer three new themes each year. I added peers and mentors to my network. I developed new interests that inspired a successful grant proposal. Finally, I joined a research team scattered across the country with diverse interests in PDE, stochastic processes, math bio, and parasitic insect behavior. This summer we are looking forward to reuniting for two more weeks of collaboration, graciously hosted by the Institute for Advanced Study.
It’s important to remember that nothing we do is truly independent. I owe a deep gratitude for the support of my advisor, mentors, peers, and the dynamical systems community. I hope to see you at the next conference!
A. SCHEEL AND J. WEINBURD, Wavenumber selection via spatial parameter jump, Phil Trans Roy Soc A, https://doi.org/10.1098/rsta.2017.0191 (2018).
J. WEINBURD, Planar patterns deformed by spatial inhomogeneity, PhD Thesis, in prep, (2019).
J. SWIFT AND P. C. HOHENBERG, Hydrodynamic fluctuations at the convective instability, Phys Rev A, 15 (1977), pp. 319–328.
M. HARAGUS AND G. IOOSS, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Springer (2011).
R. LUKEMAN, Y. X. LI, AND L. EDELSTEIN-KESHET, Inferring individual rules from collective behavior, Proc Natl Acad Sci, 107 (2010), pp. 12576–12580.
Y. L. CHUANG, M. R. D’ORSOGNA, D. MARTHALER, A. L. BERTOZZI, AND L. S. CHAYES, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D, 232 (2007), pp. 33–47.
J. BUHL, G. SWORD, F. CLISSOLD, AND S. SIMPSON, Group structure in locust migratory bands, Behav Ecol Sociobiol, 65 (2011), pp. 265–273.