Figure 4.

The optimal PDE scaling parameter for movement depends on the mesh resolution but appears to converge. (a) A scatter plot of the numerically optimized value of ${\alpha }_{\mathrm{MM}}$ from equation (2.4) as a function of the mesh resolution, $N_{\mathrm{bin}}$, demonstrates that this scaling parameter is correlated with the mesh resolution, but appears to converge at sufficiently high (i.e. $N_{\mathrm{bin}}\ge 240$) detail. (We omit 95% confidence intervals because these are so narrow that they are difficult to see.) (b) Plotting the $\log L^2$ error, given by equation (2.9), as a function of ${\alpha }_{\mathrm{MM}}$ with $N_{\mathrm{bin}}=240$ for each initial condition suggests that the values in (a) are optimal. As we show in figure 2c, melanophores are typically separated by more than 60 μm in our movement-only ABM simulations—over four times the voxel width in a grid with $N_{\mathrm{bin}}=240$. See §2.4 and electronic supplementary material, S2, for numerical details.