Mentoring through Predator-Pray: Polymath Junior Realizes Hopes

What is the Polymath Jr. Program?

Fortuna est quae fit cum praeparatio in occasionem incidit.
Luck is what happens when preparation meets opportunity. —Seneca

The Polymath Jr. program provides summer opportunities for group undergraduate research in mathematics, including applied mathematics and modeling, via remote mentoring online. The experiences of the program have provided many opportunities for the students and mentors involved. The purpose of this note is to describe and advertise the program, and encourage people to apply to be mentors and student researchers. For another article by many of the professors involved in the program from the start, see Adarichva et al. [2021].

The two of us find that our experiences are a great realization of the quotation above. We gave a talk at the Joint Mathematics Meetings in January 2026. Author El Khateeb was then a graduate student on the job market. Brian Hollenbeck, Mathematics Chair at Emporia State University, heard our presentation (and El Khateeb’s remark that he was seeking employment); fast forward a few months, and the two of them are now colleagues, as El Khateeb has begun a tenure-track position there!

This story illustrates the driving force behind the Polymath Jr. program: we create opportunities. Not everyone will seize them, but those who do can have life-changing outcomes, ranging from acceptance to strong graduate programs to having their prayers answered and getting a good job.

We describe the program’s history and how it works. We discuss issues and end with a brief summary of a project that the two of us led, a predator-prey problem. For more details, about how to participate as a student or help to run a project as a graduate mentor or professor, contact author Miller at This email address is being protected from spambots. You need JavaScript enabled to view it. or see the program’s webpage

https://geometrynyc.wixsite.com/polymathreu

Note: The information below was created with the assistance of AI.

Mathematics Level

This article is best suited for undergraduate mathematics, especially students interested in research, applied mathematics, mathematical modeling, and mentoring. The article describes the Polymath Jr. program, a remote undergraduate research program, and includes a representative predator-prey modeling project. The mathematical content ranges from early undergraduate to advanced undergraduate, depending on how deeply the predator-prey model is studied. The mentoring and program-design portions are accessible to a broad audience, while the research example involves more advanced topics such as differential equations, dynamical systems, Jacobian matrices, eigenvalues, local stability, phase portraits, bifurcations, and computational modeling.

Application Areas

The article connects mathematics to several applied and professional areas:

• Undergraduate research and mentoring
• Mathematical biology
• Ecology and predator-prey systems
• Applied mathematics
• Dynamical systems
• Differential equations
• Computational modeling
• Data science
• Professional development in mathematics
• Remote and collaborative learning

The featured project studies ecological interactions among Burmese pythons, alligators, and shared prey in the Florida Everglades, using mathematical and computational tools to analyze coexistence, stability, hunting pressure, and environmental sensitivity.

Prerequisites

For the general article, students need only an interest in mathematics research and some familiarity with undergraduate mathematics. For participation in Polymath Jr., the article notes that applicants are expected to have taken at least one proof-based course. For the predator-prey project, helpful prerequisites include:

• Calculus
• Proof-writing experience
• Introductory differential equations
• Systems of ordinary differential equations
• Linear algebra
• Eigenvalues and eigenvectors
• Basic mathematical modeling
• Introductory programming or computational tools such as MATLAB, Maple, or Mathematica

Some topics can be introduced during the project, making the article useful for students who are still developing their advanced mathematical background.

Subject Matter

The article is partly a description of the Polymath Jr. Summer Research Program and partly a case study in mentoring undergraduate research. It explains how remote group research can create opportunities for many students, especially when traditional in-person research experiences are limited. The article emphasizes mentorship, collaboration, accessibility, professional growth, conference presentation, mathematical communication, and the development of future mentors.

The mathematical case study focuses on a predator-prey research project. Students were introduced to differential equations, phase portraits, local stability, Jacobian matrices, and bifurcation ideas before helping build and analyze a model of species interaction in the Everglades. Figures on page 10 show students presenting their research at undergraduate and SIAM meetings, reinforcing the article’s emphasis on communication and professional development.

Correlation to Mathematics Standards

This article aligns strongly with undergraduate mathematical modeling and research standards rather than traditional high school content standards. It supports MAA CUPM recommendations by emphasizing mathematical modeling, communication, collaboration, computation, proof-based reasoning, interdisciplinary applications, and undergraduate research experiences.

It also aligns well with SIAM/COMAP modeling practices, including identifying a real-world problem, making assumptions, building a model, using computational tools, interpreting results, revising work through feedback, and communicating conclusions. The predator-prey project is especially relevant to modeling standards because it connects mathematical structure to ecological interpretation.

For Common Core High School Mathematics, the alignment is more limited but still present in modeling, quantitative reasoning, and interpreting relationships. For AP Calculus or AP Statistics, the article may be useful as an enrichment reading, but the main mathematical standards connection is stronger at the undergraduate level.

Overall Classification

This article is best classified as an undergraduate applied mathematics and mathematical research mentoring article. Its strongest educational uses are in courses or programs focused on mathematical modeling, undergraduate research preparation, differential equations, mathematical biology, data science, and professional development in mathematics. It is especially valuable for showing how students can move from classroom mathematics into authentic research, collaboration, conference presentations, and applied problem solving.