If you are an undergraduate student interested in the MXML, please fill this Undergraduate Student Application by November 23 at 11:59pm.
Uncovering Transient Dynamics in Ecological Systems
Supervisor: Christopher Heggerud
Level: 3000/4000
Prerequisites: At least one course focusing on differential equations such as MATH 3440 or MATH 3460
Programming Skills: Students should be proficient in MATLAB, Python, or R (or similar language capable of producing numerical simulations). Students should also be prepared to write in LaTeX.
Project Description: Mathematical modelling efforts have traditionally focused on long-term steady states dynamics. While this focus has led to many exciting results, such as the development of R_0 and population persistence estimates, shorter-term dynamics, referred to as transient dynamics, have been understudied. For example, in algal blooms, questions like 'how long does the bloom last?', or 'how severe will the bloom be?' cannot easily be answered with traditional modelling tools like stability and bifurcation analysis.
In this project, we will work to uncover which mechanisms and variables drive transient dynamics in ecological systems. Some candidate systems include algal blooms and anaerobic digestion. We will develop new mathematical models for such systems and begin with numerical simulations to understand the general model behavior. Based on the simulations we will extend mathematical toolkits to study transient dynamics. Where possible, our models will be confronted with ecological data so that we can work towards creating predictive tools for transient dynamics. This project has the possibility to be both short term, and extended into future projects including graduate studies and interdisciplinary collaborations.
Expected Outcome: Students should be expected to work towards writing a journal paper and creating an academic poster to disseminate their results.
Additional Information: Differential equations, and coding experience are the key prerequisites. However, students with experience in statistics may be able to progress faster in later stages.
Simulating Non-overlapping Interacting Particle Systems
Supervisor: Jeremy Wu
Level: 3000/4000
Prerequisites: MATH 2090 (Linear Algebra 2), MATH 2160 (Numerical Analysis 1), MATH 3420 (Numerical Analysis 2)
Programming Skills: Basic proficiency in at least one of Python, Maple, or MATLAB is essential. LaTeX proficiency is desirable but not expected (this will be developed throughout the project).
Project Description: This projects aims to visualise a system of interacting particles with a non-overlapping constraint. An ODE system is given to describe the trajectories of particles under this dynamic. Basic programming will be used to solve this system and plot particle trajectories. The non-overlapping system will be compared with its unconstrained counterpart. Time permitting, the particle system will also be compared with numerical approximations of the macroscopic PDE analogue.
Expected Outcome: Novel examples and interesting numerical experiments.
Additional Information: The particle system described in this project focuses on a physical system modelling, for example, crowd motion and tumour growth. Similar particle systems are also extensively studied in other areas such as machine learning. Specific examples of transferable skills developed in this project are 1) theoretical analysis of interacting particle systems and 2) numerical analysis and coding simulation of such systems. These skills are highly desirable in a wide variety of academic and industry contexts beyond the scope of the project.