Moving boundary problems in mathematical biology

Overview

Invasion of biological cells or ecological populations involves moving fronts that invade into previously unoccupied regions of space. Such moving fronts are driven by a combination of motility, such as random diffusion, and proliferation, such as logistic growth. Understanding how best to model such invasive fronts is important as moving fronts of cells are associated with wound healing and cancer progression and moving fronts in ecology are associated with the spreading of weeds and invasive species.

Previously both continuum and discrete models related to the Fisher-Kolmogorov equation have been used to model experimental data relating to moving fronts of cells. These kinds of models can successfully capture certain aspects of experiments, such as the formation of moving fronts. However, the Fisher-Kolmogorov model fails to capture other features of experimental observations, such as the formation of a well-defined front of cells with compact support or the ability to model biological invasion and retreat.

An alternative way to model the motion of a moving front of cells is to treat the problem as a moving boundary problem, where the evolution of the moving boundary is part of the solution of the mathematical model. While some theoretical advances have been made in reformulating the Fisher-Kolmogorov model as a moving boundary problem, there are many open questions about how such a moving boundary problem captures the underlying biological processes.

This project will investigate three related aims which seek to explore how such a moving boundary problem can be interpreted biologically.

Research activities

In this project you will:

• formulate partial differential equations of biological invasion, and solve these models using a combination of numerical and analytical techniques
• analyse travelling wave solutions of partial differential equations in the phase plane
• interpret partial differential equation solutions and results from the phase plane in terms of underlying biological phenomena
• extend classical one-dimensional single-population models to deal with more realistic models of multiple interacting populations (e.g. cells and substrate) and to deal with multiple dimensions (e.g. wound closure in 2D/3D).

Depending on your interests, we could work with laboratory data to parameterise the new moving boundary models.

Outcomes

This project will produce new:

• mathematical models of biological invasion
• mathematical insight into travelling wave solutions through approximate methods of analysis (e.g. perturbation techniques, phase plane analysis)
• parameterised mathematical models using, for example, experimental data relating to cell migration and cell proliferation.

If you're enrolled in a Masters of Philosophy, you will produce one publication in the mathematical biology literature. Whereas if you're enrolled in the PhD program, you will produce three or four publications in the mathematical biology literature.

Skills and experience

This project requires experience in solving differential equation models numerically.

Scholarships

You may be eligible to apply for a research scholarship.

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Contact

Contact the supervisor for more information.