**Yihong Du is Professor of Mathematics at the ****University of New England**.** He is developing partial differential equation (PDE) theory and techniques to improve the understanding of population phenomena occurring in ecological invasion, disease spreading, krill swarming and elsewhere.**

“I grew up during the Cultural Revolution in China. During this period, universities accepted students according to recommendation – which meant you had to come from the right family. It wasn’t based on academic talent. As my father, a high school teacher, had a grandfather who owned land and a small business before 1949, I would fail the family check. But a big change happened in 1977, when I was 14. The Cultural Revolution ended and the government resumed university entrance examinations. My father thought, wow, that’s good – we should grab this opportunity. So he decided that I should finish high school in one year.

I hadn’t done much work in junior high, so in the first half of that year I tried to pick up all the stuff I had missed. In the second half of the year, I tried to finish everything remaining and be prepared to take the university entrance examination in 1978. That was the plan.

At this time, some levels of government started to organise mathematics competitions for high school students, and I was chosen to take part at county level. And I did very well – I got first place. Next, I went to a maths competition at a higher level, and I also did well.

By coincidence, one of the best provincial universities in China, Shandong University, announced that they wanted to enrol a special class of students based in mathematics. I was invited to sit for a series of exams and, well, they chose me. So I was enrolled at Shandong University to study mathematics in 1978, without the usual entrance examination after all. I was 15 years old.

After my bachelor degree I continued on to do a Masters and then a PhD there, followed by a lecturer position in 1988. But after two years I accepted the chance to visit the UK for a year.

During this period, I realised I wanted to do a postdoc overseas to learn from a top mathematician. I applied to study under Professor Norman Dancer at the University of New England, in Armidale, NSW, and arrived here in 1991.

Coming to Australia made a big impression on me. Norman Dancer is a top-tier mathematician, and I met other first-rate mathematicians who really opened my eyes. I started to work on problems that were much more promising.

My postdoc project was on population dynamics, which uses mathematical ideas and techniques to try to solve problems arising from the real world. My research is mainly theoretical, trying to prove theorems. I sometimes use simulations to help make the abstract result easily communicable to other people.

### The power of mathematics

I don’t create models for any particular species. I’m trying to use simple models to find assumptions which apply to a relatively general group of species that all share something. For example, they might be invasive species, propagating to new territory.

If we want to make good use of mathematical theory, we usually have to take options which are not very real. If you are too realistic, your theories become too complicated. If it’s too complicated, many of the beautiful, deep mathematical theories have no way of being applied.

What you have to do instead is oversimplify, initially at least. If through these processes you then observe some important phenomena, then you try to say, okay, because I made over-simplifications, I will use the powerful mathematics to see if they persist *without* these oversimplifications. It’s only then that I’ll come back to more realistic situations.

Even then, if we try to extend our theory to more general situations, very often we can’t get very far. But sometimes the simulation proves that some kinds of important phenomena exist in the more realistic setting. It’s quite an involved process, but we have made some important discoveries through this powerful mathematical analysis.

Populations change over time, so our models must try to capture this dynamic behaviour. Most of the existing equations, or models, assume that a species moves over and occupies a fixed physical domain or territory. But if you are interested in the introduction of a new invasive species, like cane toads, the change of occupied territory, known as the population range, causes you lots of trouble in mathematics.

If you want to understand the spreading of cane toads, for example, in mathematical theory you must first assume that they occupy the territory that is the entire world – in mathematic language, we call this the entire Euclidean space, which may be two-dimensional, or three-dimensional. This actually helps to simplify the theoretical analysis, but it’s still challenging.

But I found this approach less satisfactory when the boundary of the occupied territory is a very sensitive issue, such as Covid. Cane toads are quite nasty, but your species could be an even nastier virus; and people in the academic area are often very sensitive about defining the boundary of the population range. You could make some assumptions to approximate it, depending on an introduced artificial parameter, but that’s not very satisfactory.

I try to ask if, in some simple situations, we assume the occupied territory of the species changes with time, can we still handle it in mathematics? And fortunately, we found a way to handle at least some simple situations where the occupied territory of the species changes with time. In mathematics, we call this a “free boundary problem”. This allows us to begin to understand the long-term dynamics of the evolving population range of the concerned species, including some nasty viruses causing life-threatening diseases.

It’s very satisfying solving a problem like this, then discovering that other people around the world are following your research.

Previously in the ARC Laureate Fellows series: Jeffrey Walker.