Contributions and Aspects of the Work of J.D. Murray Organised By Patrick Nelson

Applications of the work of J.D. Murray to medical problems

Trachette Jackson
University of Michigan, USA
tjacks@umich.edu

In the mid 1990s Prof. J.D. Murray began to focus more attention to the application of mathematical modelling to medical applications. In 1995 he began a collaboration with Dr. E.C. Alvord, Jr. that led to some of the first papers that explored the effects of chemotherapy on diffusive tumours (Gliomas) and predicted the effects of surgical resection on their growth. He has collaborated with noted medical scientists on the dynamics of prostate tumour growth and worked out an acceptable way of determining the disease progression from measurements of prostate specific antigens. He helped pioneer the first models on two-step drug treatment strategies for skin melenomas and was one of the first people to suggest incorporating time delays in models of HIV, an idea that has led to a much better understanding of the disease progression. We will review all of these works and show how each has made a substantial impact in the medical community.

Mathematical models for carnivore territories

Mark Lewis
University of Alberta, Canada
mlewis@math.ualberta.ca

Social carnivores, such as wolves and coyotes, have distinct and well-defined home ranges. During the formation of these home ranges scent marks provide important cues regarding the use of space by familiar and foreign packs. In this talk I will discuss mechanistic rules that can be used to understand the process of territorial pattern formation through interactions with scent marks.

I will consider different model formulations, with and without the den site as an organizing centre for spatial movement. Under realistic assumptions the resulting territorial patterns include spontaneous formation of `buffer zones' between territories which act refuges for prey such as deer. This is supported by detailed radiotracking studies. The model will also be analysed using game theory, where the objective of each pack is to maximize its fitness by increasing intake of prey (deer) and by decreasing interactions with hostile neighboring packs. Predictions will be compared with radio tracking data for wolves and coyotes.

Much of this work was started in collaboration with Prof. J.D. Murray. During my talk, I will trace the effect that these collaborative interactions have had on development of the subject and on researchers involved.

Spatio-temporal pattern formation: Turing pre-pattern systems and
mechanochemical models

Philip Maini
Oxford University, UK
maini@maths.ox.ac.uk

In the early 1980s Prof. J.D. Murray wrote a number of papers on self-organising pattern formation using the Turing reaction-diffusion model and showed how the theory could be applied to pattern formation on butterfly wings and animal coat markings. Then, with Oster and Harris, he wrote a seminal paper on a different approach to such patterns, the now well-known mechanochemical theory for pattern formation. This approach was applied to skeletal limb development and feather germ formation, and served as a precursor for later models in wound healing and cancer. We will review this work and show, in particular with wound healing, how the models agree very closely with recent experimental data.