Hello! Are you wondering, "What is applied math research, anyway?" This page describes recent student projects related to my long-term program to understand collective behavior in locusts. Check these out to get a sense of what doing research with me might be like, and shoot me an email if you have any questions.
Unfortunately, I do not anticipate taking on students this upcoming summer. If you're looking for research projects similar to the ones described below, I'd be happy to chat with you about future opportunities or other faculty who are engaged in similar work and might be looking for students.
If you are a Harvey Mudd student, keep an eye on the Research Opportunities Program website. New projects should appear on November 22nd.
Locusts form swarms with distinctive geometries that appear to aid in foraging. Fig A shows locusts moving perpendicular to the line of advancing insects through a lush agricultural field. In contrast, Fig C shows locusts moving parallel to the collective stream towards an isolated patch of vegetation. No leader directs the swarm to aggregate or move in these ways, instead both collective behaviors can be attributed to the interaction of rules that dictate an individual locust's attraction to food and social attraction/repulsion from other locusts. Understanding that interaction may eventually help identify efficient strategies for controlling locust outbreaks.
Most of the recent projects below involve data science in an effort to advance our empirical understanding of locust behavior within a swarm.
Motion Tracking Individual Locusts in a Swarm
The form, speed, and density of these swarms is in large part due to social interactions, such as a locust orienting itself to move in the same direction as its neighbors. In order to deduce these interactions, we extracted numerical trajectories from raw video footage of locusts marching in the field. The video was provided by our collaborator Jerome Buhl (University of Adelaide), an eminent locust biologist. We conducted exploratory analysis of these trajectories, which continued in additional projects below.
On the left is a visual representation of our tracking process. Top: Locusts (dark dots) walk and hop in a naturally occurring swarm. Bottom: Processed video shows locusts (bright spots) detected by the algorithm (purple circles) and linked into trajectories (rainbow lines).
This project was part of the HMC Data Science REU, supported by NSF Grant DMS - 1757952.
Classifying Motion State with Support Vector Machines
Starting with the trajectories we collected from video, we investigated the movement behavior of individuals within the swarm. We developed a support vector machine (SVM) to classify motion into three states: stationary, walking, and hopping. The SVM used motion characteristics of the trajectories such as mean speed averaged over the near future and near past and standard deviation of speed over the same time window. We trained the SVM with data from diligently collected manual classification. Through cross-validation we estimated that the SVM was between 85-90% accurate. Approximately half the data was classified as hopping and the other half was split almost evenly between stationary and walking. This provided the first quantitative insight into how locusts move in a natural swarm!
The figure on the right shows an iteration of our SVM that shows clear clustering of the data. Figure credit to our collaborator Michael Culshaw-Maurer.
Deducing Insect Interactions from Field Data
We set out to infer how individuals interacted with their nearby neighbors. Our aim was to deduce biologically realistic rules for these interactions that could inform the creation of a future agent-based model that can produce swarms comparable to those observed in nature.
On the right, we plotted two-dimensional histograms of the position of nearby neighbors relative to a focal locust (red arrow) that is stopped (top) and crawling (bottom). Yellow indicates higher counts of neighbors while blue indicates lower counts. Note that crawling locusts have fewer neighbors in front of them. This suggested to us that when a crawling locust sees a neighbor directly in front, it either stops moving or turns to avoid a collision.
Figure credit goes to Jacob Landsberg (Haverford College '21) who conducted the bulk of this work as part of his senior thesis!
The Sociobiology of Foraging Strategies
Perhaps the most striking characteristic of locust behavior is that they manifest an epigenetic phase change where they transition from solitary individuals (when resources are plentiful) to gregarious foragers (when resources are sparse). It is believed that this behavioral transition evolved independently in multiple locust species on at least three continents.
Our hypothesis is that this behavioral transition increases foraging efficiency for the swarm and survival potential for the species. We formulated this as a question in game theory and compared predictions using an extension of the agent-based model developed by Hannah Larson (HMC 2020) shown in the Overview Fig F. Dominant ecological theories of foraging describe the optimal search strategy of an individual only. Whereas game theory provides routes for optimizing strategies for groups. Our goal was to explain mathematically why locust behavior is bimodal.