Figure 2.

The PDE for cell migration does not accurately describe the EA ABM result when its scaling parameter, ${\alpha }_{\mathrm{MM}}$, is set to unity. (a) To compute our EA ABM result, we solve equation (2.2) using an initial condition of 400 melanophores placed uniformly at random in a 1 × 1 mm² square, group cell positions in a 240 × 240 histogram, and average such data over 10⁴ ABM realizations. (b) We compute the corresponding PDE solution by simulating equation (2.4) with ${\alpha }_{\mathrm{MM}}=1$ from a uniform density of 400 cells mm⁻² in the same square region. The ABM and PDE solutions use the same potential (given by equation (2.3) with parameters in table 1). We overlay an example ABM realization for comparison; the results demonstrate that the support of the PDE is larger than that of the ABM by about 200–250 μm. Because melanophore–melanophore distances have been measured to be roughly 50 μm in vivo [78] and stripes are only about 7–12 cells wide [16], this is a large difference. (c) The distribution of nearest-neighbour distances across 100 ABM realizations demonstrates that cell–cell separation ranges from roughly 60 to 100 μm. Based on visual inspection of the graphs, nearest-neighbour distances appear inversely proportional to the EA cell density. In (a–c), we show results at t = 150 days.