PDE Models and Methods in Mathematical Biology

Lecturers

Associate Professor Peter Kim, University of Sydney
Dr Justin Tzou, Macquarie University

Synopsis

Mathematics can be useful for studying biological systems. In this course, we will show how to utilise partial differential equations (PDEs) to model and analyse a range of biological systems. PDE models capture a wide range of biological phenomena, including spatial and age-structured interactions. Particular topics will include age/maturity-structured models, diffusion and reaction-diffusion models for predator-prey dynamics, chemotaxis, pattern formation, and mean first-passage time models for applications such as intracellular transport. We will also discuss a recently developing area of mathematical modelling, that of bridging agent (or individual)-based models and differential equations.

Course Overview

  • Age-structured models
  • Diffusion
  • PDE approach for mean first passage time problems in 1-D and 2-D
  • Chemotaxis
  • Fisher’s equation
  • Travelling waves/fronts
  • Reaction-diffusion systems
  • Turing stability analysis
  • Weakly nonlinear analysis for small amplitude patterns
  • Asymptotic methods for large amplitude patterns in 1-D and 2-D
  • Connecting PDEs to agent-based models

Prerequisites

Introduction to PDEs (e.g., a 2nd-year course on PDEs) which cover topics on classification of PDEs, separation of variables, heat equation, wave equation, Laplace equation and its solution via separation of variables.

Ability to write a basic MATLAB (or equivalent) program is an advantage. Otherwise, one can learn during the course.

Assessment

  • Mid-school assignment: 40%
  • Final examination: 60%

Resources and Pre-reading

J.D. Murray, Mathematical Biology, any edition
J.P. Keener, J. Sneyd, Mathematical Physiology
S. Redner, A guide to first-passage processes.

Models in Mathematical Biology

 Biography

Associate Professor Peter Kim, University of Sydney

I work in mathematical biology, a rapidly growing interdisciplinary field that requires a synthesis of information from a variety of perspectives, creative modelling, and interaction with researchers from a range of areas.

Specific areas that I work on include mathematical immunology, cancer dynamics, virus dynamics, and human evolution. To address these problems, I apply ordinary, delay, and partial differential equations and agent/individual-based models.

More information is available at https://sydney.edu.au/science/people/peter.kim.php

Models in Mathematical Biology

Biography

Dr Justin Tzou, Macquarie University

Justin is a lecturer in applied mathematics at Macquarie University. Prior to that, he was a PIMS CRG postdoctoral fellow at the University of British Columbia, and an AARMS postdoctoral fellow at Dalhousie University. He obtained his PhD at Northwestern University under the supervision of Bernard Matkowsky.

His research centers on the development and application of asymptotic methods for studying the stability and dynamics of localised pattern formation, as well as narrow capture problems in mean first-passage time theory.

More information is available at http://web.science.mq.edu.au/~jtzou/

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