This example demonstrates the effect nonuniform surface growth can have on a simple model of a two-species cell population. The cell population contains yellow and blue cells, which are exactly the same apart from their colour. These cells can only proliferate, and new cells are the same colour as their parent cell. Finally, these cells are situated on a finite disk, and to proliferate they require enough space to fit in (typically referred to as volume exclusion).

We now grow the disk the cell population is situated on in three ways. In the first video growth is nonuniform and occurs more towards the centre of the circle, notice how the contour lines move further apart near the centre. In the second video growth is uniform. In the third video growth is again nonuniform, but this time occurs more towards the perimeter of the circle. It is straightforward to write these three types of surface growth as a function with a single parameter. For example, the radial growth function used here is f(r,ρ,t) = (r/R(t))^ρ, where R(t) is the radius of the circle and r is the distance from the centre. This means ρ = 1 is uniform stretch, ρ < 1 is growth biased towards the centre, and ρ > 1 is growth biased towards the perimeter.

Now we place our cell population model on these differently growing disks and see what happens. Can you understand why the three outcomes differ?

In the first video we get a 'spot'. This is because one of the cells, in this instance a blue one, manages to colonise the middle of the circle. From here it has a positional advantage, due to more new space being added here, and so begins to take over the disk. This occurs if ρ < 1. In the middle video we get a pattern that appears 'critical'. This indicates the existence of a phase transition at ρ = 1. For any physicist reading the decay of interfaces at ρ = 1 follows a power law, however, it is not the same as the voter model. In fact, it appears to have a novel exponent. In the third video we get 'stripes' because more space is being added at the perimeter, which occurs if ρ > 1. The reason for this, albeit for the limiting case ρ → ∞, was originally explained here.

This work is currently being prepared for submission.