Dr. John Norbury

Research Interests

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 two books on atmosphere and ocean dynamics have been published by CUP on some of the work studied there, and ideas and progress made since. 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 local and small scale interactions lead to large scale structures, with for instance recognisable weather patterns (in spite of apparently random behaviour in clouds), which recur in spite of many local showers and/or wind gusts. How robust are biological (and other) communities in the face of change ?

The end product of each study is an understanding of the differential equation model, an approximate description of the solutions, and computer methods for more careful calculations to compare with experiments - hopefully leading to better predictions and better understanding !

More particularly, recently we have been looking at

1. how perturbations to the Ginzburg-Landau equations affect the structure of their solutions,
2. how biphasic behaviour arises in models of chemotactic cell invasion,
3. and how a stationary fluid surface behaves in a corner.