In this example our growing surface is a lattice. This is not particularly realistic as few systems (if any) can be thought of as a growing chessboard, however, it does mean using mathematics is much easier. It is also often the case that lattice-based models will demonstrate the same behaviour as more realistic models. Therefore, a lattice can act as a useful abstraction: it simplifies but does not lose the salient features of the system you are interested in.

The cell population model on our lattice contains two-species. Species A is red, and moves quickly but proliferates slowly. Species B is blue, and moves slowly but proliferates quickly. This model also has volume exclusion, that is, a cell requires an empty site next to it to proliferate. Below are two videos, both of which have the same cell population model on them, but the lattice they are situated on is grown in different ways. Can you see the differences?

In the first video as the surface grows the red species dominates, whereas in the second video the blue species dominates. Why does this happen? Simply put, it is because the two growth mechanisms break down spatial correlations differently. In the growth mechanism in the second video, the larger the system gets, the faster it breaks down spatial correlations. This favours the blue cell population, and ultimately allows it to dominate. Below we have the same videos as before, but we have doubled the initial size of the surface in both cases. Notice how in the second video the blue cells dominate faster than they did previously?

Given the nature of the growth mechanisms above it suggests the following fun idea should be thought about. Imagine you are a little creature (pictured below) who is able to stand on the growing lattices described above. You are also magical, as you are able to see the governing equations of the physics at the place you stand. In the case of the first growth mechanism described (videos 1 and 3), as you move around the lattice you would always see the same equations - they are translationally invariant. As for the second growth mechanism (videos 2 and 4), as you move around the lattice the equations would change - this growth mechanism breaks a symmetry and translational invariance is lost. The work described on this page can be found here, here, and here.

This work was originally motivated by thinking about the formation of mammalian pigmentation patterns during embryonic development, and the second growth mechanism was used in the modelling.