# Plankton and tumours captured in mathematical equations

Plants that grow in patterns of stripes on the edge of deserts, plankton that live on the sea bed or oscillate in the water, and melanomas that spread throughout the body. Mathematician Lotte Sewalt discovered the common elements in these three systems. PhD defence 8 September.

### Anthology of models

‘Why is it that nature often appears so structured, even though we don't impose any rules on it?' This was the question that mathematician Lotte Sewalt asked herself before starting her PhD research. In her research she looked for mathematical similarities in three very different biological growth processes. She refers to her dissertation as an anthology of mathematical models.

### Maths as a tool

Most patterns in nature are based on a kind of optimisation process. Sewalt works mainly with scientists - ecologists, geologists and physicists - who study these processes in great depth. ‘They can often explain the factors that are important in this optimisation.' Sewalt then converts these insights into mathematical formulae. 'Maths is a very good tool for understanding these processes. For me, it's a really good application because it gives my maths some direction.'

### How does a system change?

The maths in Sewalt's research consists of differential equations that show, for example, how the characteristics of a system change over time. Sewalt uses systems of differential equations to describe pattern formation.  Many of the formulae needed for her research were worked out using pen and paper, she says. 'I want to understand at analytical maths level what is going on.' Computer simulations supported her analytical work, allowing her to check whether the models agreed with reality.

### Oscillating plankton

Sewalt's research started with the mathematical description of phytoplankton. These organisms are dependent on sunlight that is often strongest just below the surface of the water. At the same time the plankton need nutrients, and these tend to sink to the sea bed. Sewalt's equations show that this can cause the plankton to oscillate up and down in the water. ‘But in shallow water most of the sunlight reaches the bottom. My method shows nicely that a colony of plankton stays motionless on the seabed.'

### Light determines location of plankton

When studying the behaviour of a system, Sewalt is looking particularly for one important parameter that gives rise to the structures, which she can vary. In the case of phytoplankton, for example, this could be a change in the amount of nutrients. The availability of light is another such parameter; the two are related. ‘These are two different ways a system can be triggered: we refer to that as bifurcation.'

### Pattern of stripes in vegetation

Sewalt is also studying patterns in vegetation in semi-arid areas on the edges of deserts. Plants growing there can exhibit some remarkable patterns of stripes, especially if the area has a gentle slope. Sewalt's model has shown that a drastic reduction in the amount of water is the crucial factor in causing this pattern.

### Tumours

And finally she applied her maths to tumours. 'You can describe many melanomas as having a stable condition, as in a mole that stays the same size, for example. Then something happens at cell level which allows the melanoma cells to invade healthy cells. That could be something like the concentration of collagen in the skin.'

### Melanomas become active

This pattern formation seems different from the other two examples, Sewalt explains, but in mathematical terms the switch of a melanoma from stable to spreading follows a definite pattern. ‘That's what interests me. In these different applications you are dealing with very different laws, but if you describe them in mathematical terms and leave out all the details, they are very similar. That's why it is important to gain in-depth mathematical insights. In maths it often happens that a set of equations proves to be applicable to very different systems than those they were intended for.'