When will human beings outstrip the carrying capacity of the Earth, or numbers of a pest, such as rats, blow out to plague proportions in an agricultural region?

The mathematical description of such thresholds has vexed the best minds around the world for 100 years. If mathematicians could predict when systems will be pushed over the edge, action could be taken before irreversible damage is done.

Fundamental research by a mathematician at the Australian National University promises to formalise such problems, clearing the way for more accurate predictions about the physical world.

Florica Cirstea is studying a class of mathematical relationships known as second order elliptic partial differential equations, which has applications in fields ranging from population dynamics, through fluid dynamics to astrophysics. The ANU is at the forefront of the field. Cirstea is focusing on "singularities", or "boundary blow-up", where conditions change abruptly, leading to catastrophic results such as a population explosion.

"The study of singularities is also relevant to many applications beyond mathematical biology, in electrochemical machining, for example," she says, citing as an example the minimisation of current leakage between the two conductors of a coaxial cable.

"The mathematical theory of boundary blow-up has been actively investigated by researchers since 1916," she adds. But big questions remain, especially in complex systems.

She is working on ways to determine which systems have singularities, what causes the singularities and whether more than one factor will push a system to a point of no return.

Cirstea, whose research takes her to the institutions of her collaborators in France and Italy, says she was attracted to mathematics because "it lies at the foundation of everything that surrounds us".

"It is the language that engineers, physicists, astronomers, biologists and other scientists use to build their theories that enable us to understand and explore the world around us."