Table A.2.

Model parameters for two populations in one dimension.

	Description
u(x, t) v(x, t)	Population densities at position x and time t.
Ku(u, v) Kv(u, v)	Non-local adhesion terms.
Su, Sv, C	Self-population adhesion strength coefficient of population u, self-population adhesion strength coefficient of population v and cross-population adhesion strength coefficient respectively. Each taken to be a non-negative constant.
guu(u, v) guv(u, v) gvv(u, v) gvu(u, v)	These functions define the dependence of the adhesive force on cell density. To represent attractive forces we require that g(u, v) are non-negative. We consider two simple forms, $$1.g_{\mathit{uu}}=g_{vu}=u,g_{\mathit{vv}}=g_{uv}=v$$ $$\begin{array}{cc}\hfill 2.& g_{\mathit{uu}}=g_{\mathit{vu}}=\{\begin{array}{cc}u(1-u-v)\hfill & 0<(u+v)\le 1\hfill \\ 0\hfill & (u+v)>1\hfill \end{array}\phantom{\}},\hfill \\ \hfill & g_{\mathit{vv}}=g_{\mathit{uv}}=\{\begin{array}{cc}v(1-u-v)\hfill & 0<(u+v)\le 1\hfill \\ 0\hfill & (u+v)>1\hfill \end{array}\phantom{\}}.\hfill \end{array}$$ We find that the linear form (1) permits aggregating behaviour but that the density limiting form (2) is required to show cell sorting.
ωuu(x0) ωuv(x0) ωvv(x0)	These functions define the dependence of the adhesive forces on the position of the cells. The direction of the force will depend on the relative positions of the cells. To model an attractive force we require ω(x0) is an odd function and ω(x0) ≥ 0 for 0 < x0 < 1. The magnitude of the force may depend on the distance between cells but for simplicity we assume this is not the case. We assume that ωuu,uv,vv(x0) all take the same form, $$\omega \left(x_0\right)=\{\begin{array}{c}-1-1<x_0<0\hfill \\ 10<x_0<1\hfill \end{array}\phantom{\}}$$