I was born and educated in Australia, and then took a doctorate in applied mathematics in Cambridge working on the theoretical existence of swirling "doughnuts" (both spherical and toroidal) of fluid known as vortex rings, and on the computation of their shape and energy. I then taught at University College London for seven years, during which I spent 12 months at the Courant Institute, New York University.
Much of life - such as the healing of wounds, forecasting the weather, the flow of money through our economic community, river flooding, the invasion of cancer - can be modelled by differential equations. Since we would all like to predict what will happen next, and why, we need to explain how solutions to differential equations evolve, and what are the interesting relations, or cause and effects, that we might modify to change the future behaviour.
Ian Roulstone of the UK Meteorological Office (research branch JMM at the University of Reading) and I organised a 6 month research workshop at the Newton Institute in Cambridge on large scale atmosphere and ocean dynamics, and we are now sending two books to CUP on some of the work studied there, and ideas and progress made since, to be published in 2001. With several graduate students and research colleagues at other Universities, I have been looking at various models of cancer invasion, at swirling vortex flows, at control chips for use on engines and other feedback systems, and at focusing behaviour in stock markets (where fashionable stocks can get ever more expensive). The first of these differential equation models look at competition between different states (for instance, healthy states vs. diseased, etc.), while some of the other models look at the development of hot spots or clustering behaviour. So we are interested in how these nonlinear balances between competing physical processes are maintained for a long time by the differential equations, and how to reliably compute these persevering balances. The patterned states that survive the many small scale interactions lead to large scale structures for instance recognisable weather pattern, recurs in spite of many local showers and/or wind gusts. How robust are biological communities in the face of change?
The end product of each study is an understanding of the differential equation model, approximate descriptions of the solutions, and computer methods for more careful calculations to compare with experiments - hopefully leading to better predictions and better understanding!
Travelling Shock Waves arising in a Model of Malignant Invasion - SIAM Applied Mathematics Journal. B. P. Marchant, J. Norbury and A. J. Perumpanani
Solution Structure for a Nonautonomous Form of the Keller-Segel Chemotaxis Model, John Norbury and Laurence Mays.
Dynamics of Constrained Differential Delay Equations - Journal of Computational and Applied Mathematics, John Norbury and R. Eddie Wilson.
The Numerical Computation of Axisymmetric, Swirling Vortex Rings, John Norbury and Rhys Williams.
The Location and Stability of Interface Solutions of an Inhomogeneous Parabolic Problem - SIAM Applied Mathematics Journal. John Norbury and Li-Chen Yeh.
Recent DPhil Supervision
Stuart H. Doole, 1993 - Research Assistant of Bristol University, now in Investment Banking, London
"Steady gravity waves on flows with vorticity: bifurcation theory and variational principles"
Stuart and I were interested in the appearance of wave motion on sheared streams of ideal fluid flowing horizontally with gravity acting vertically. We related these waves on streams to their values of pressure head R and flow force S.
The Bifurcation of Steady Gravity Water Waves in (R,S) Parameter Space, Journal of Fluid Mechanics, vol. 302 (1995), pp.287-305
Jan H. van Vuuren, 1995 - Senior Lecturer at Stellenbosch University, Matieland 7602, South Africa
"Permanence and asymptotic stability in diagonally convex reaction - diffusion systems"
Jan and I continue to work on competitive/co-operative species and their survival in changing habitats
Permanence and Asymptotic Stability in Diagonally Convex Reaction -
Diffusion Systems, Proceedings of the Royal Society of Edinburgh, vol. 123A (1998), pp. 147-172. vol. 42 (2000), pp. 195-223
Conditions for Permanence in Well-Known Biological Competition Models,
Anziam J, - Junior Research Fellow, Oxford and the Harvard Medical School, now a Business consultant, Perth, Western Australia
Abbey J. Perumpanani, 1996
"Malignant and morphogenetic waves"
Mathematical Modelling of Capsule Formation and Multinodularity in Benign Tumour Growth, Nonlinearity, vol. 10 (1997), pp. 1599-1614
Laurence J. H. O. Mays, 1997 - Lecturer and research assistant, Lincoln and Mansfield Colleges, Oxford
"Existence of solutions of nonlinear problems using positive square root operators"
Laurence and I are working on the developments of clustering behaviour in ceratin Keller-Segel biological models - this has application in various areas of science and economics
A Note on Solitary and Periodic Waves of a New Kind, Proceedings of the Royal Society of London A, vol. 454 (1998)
Bifurcation of Positive Solutions for a Neumann Boundary Value Problem, Journal of the Australian Mathematical Society: Series B, vol. B42 (2000), pp. 1-17
Brenda Allen, 1997 - Senior scientific civil service, now in Investment Banking, London Richard E. Wilson, 1998 - Lecturer in Department of Engineering Mathematics, University of Bristol
"Non-smooth differential equations"
"Modelling, analysis, and simulation of road traffic networks"
Rhys L. Williams, 1998 - Volunteer service abroad teacher in Africa
"Exact, asymptotic and numerical solutions to certain steady, axisymmetric, ideal fluid flow problems in 3D"
Volunteer service abroad teacher in Africa
Timo I. Taskinen, 1999 - Head of Design Engineering, Nokia, Vekaratie 4, Aanekoski, Finland
"On the steady-state flow of an elastic-plastic material past cones and wedges"
On the Steady Flow of an Incompressible Elastic-Plastic Material Past Thin Cones: Numerical Results, Computers and Geotechnics, vol. 20, no. 2 (1997), pp.143-159
Li-Chin Yeh, 1999 - Lecturer, Department of Applied Mathematics, National Dong Hwa University, Haalien, Taiwan
"Patterned solutions in a reaction diffusion problem with convex nonlinearity"
Leslie and I looked at the large scale shapes and patterns of interaction fronts between competing states when there is variation in the growth and capacity of the states depending on position. These large scale patterns stably evolve from local small scale interactions, and are a general feature of these reaction-small diffusion models The Location and Stability of Interface Solutions of an Inhomogeneous Parabolic Problem, OCIAM Research Report 269, Oct 2000, to appear in SIAM journal of applied math 2001
Ben Marchant, 1999 - Post doctoral research post, Biotechnology and Biological Sciences Research Council, Silsoe Research Institute, Wrest Park, Silsoe, Milton Keynes
"Modelling tumour invasion"
Ben and I are interested in new features of travelling waves of invading cells (that first arose in Perumpanani's post graduate work). We found out that jumps in concentration were possible when chemotaxis/haptotaxis mechanisms are present at the invasion front, and that these blunt invasion profiles could be both stable and of the most physical relevance.
Travelling Shock Waves Arising in a Model of Malignant Invasion, Society for Industrial and Applied Mathematics, vol. 60, no. 2 (2000), pp. 463-476