Study level


Master of Philosophy


Vacation research experience scheme


Science and Engineering Faculty

School of Mathematical Sciences

Topic status

We're looking for students to study this topic.


Dr Pascal Buenzli
Division / Faculty
Science and Engineering Faculty
Professor Chris Drovandi
Division / Faculty
Science and Engineering Faculty
Professor Matthew Simpson
Division / Faculty
Science and Engineering Faculty


Random walk models are often used to represent the motion of biological cells. These models are convenient because they allow us to capture randomness and variability. However, these approaches can be computationally demanding for large populations.

One way to overcome the computational limitation of using random walk models is to take a continuum limit description, which can efficiently provide insight into the underlying transport phenomena.

While many continuum limit descriptions for homogeneous random walk models are available, continuum limit descriptions for heterogeneous populations are much harder to obtain.

Research activities

This project will involve:

  • implementing random walk algorithms to capture the movement, proliferation and interactions (e.g. crowing) among agent populations
  • constructing continuum limit approximations using mathematical and computer-aided machine learning techniques
  • obtaining averaged data from repeated simulations
  • comparing this data to numerical solutions of the continuum limit partial differential equations
  • analysing the new partial differential equation models through phase plane analysis and/or perturbation methods.

Depending on your study level, we will parameterise both homogeneous and heterogeneous random walk models using experimental data.


The outcomes of the project include:

  • stochastic random walk algorithms and software
  • partial differential equation models that predict the averaged behaviour of homogeneous and heterogeneous agents through mechanisms, such as
    • migration
    • proliferation
    • death
    • agent-to-agent adhesion
  • analysis of the resulting partial differential equation models
  • approaches to equation learning which will involve penalised regression techniques.

Skills and experience

This project requires you have good programming skills (e.g. MATLAB, Python).


You may be able to apply for a research scholarship in our annual scholarship round.

Annual scholarship round



Contact the supervisor for more information.