Figure 10: Melanophore movement and birth models with our Box initial condition. Results in (a)–(h) and (j)–
(l) are for Nbir = 150 positions/day. We (a)–(d) compute the EA ABM result using 1000 simulations, and (e)–(h)
generate the PDE solution of Eqn. (8) with c+ = 7.0686 cells and the values of αMM and γ that we estimated
in §3.1 and §3.2, respectively. (i) The time evolution of the PDE cell mass agrees well with the mean number
of cells for the ABM under different Nbir values. (j) Depending on the time scales of migration and birth, the
approximate PDE radius of support overtakes or trails the corresponding EA ABM result. We compute the
radius of support for each ABM realisation by finding the most distant cell from the origin at each time step;
we then average these values across our simulations. In the PDE case, we find the furthest voxel with non-zero
density from the origin based on the L∞ distance, after setting the density to zero if it is below single-digit
precision of 10^(−7). (k)–(l) We show the difference between the PDE and EA ABM solutions from (a)–(h) at two
sample times. We overlay cell positions form one ABM simulation to illustrate how the continuous and discrete
solutions are related. In (i) and (j), shaded regions denote plus or minus one standard deviation of the EA ABM
solution.
(l) are for Nbir = 150 positions/day. We (a)–(d) compute the EA ABM result using 1000 simulations, and (e)–(h)
generate the PDE solution of Eqn. (8) with c+ = 7.0686 cells and the values of αMM and γ that we estimated
in §3.1 and §3.2, respectively. (i) The time evolution of the PDE cell mass agrees well with the mean number
of cells for the ABM under different Nbir values. (j) Depending on the time scales of migration and birth, the
approximate PDE radius of support overtakes or trails the corresponding EA ABM result. We compute the
radius of support for each ABM realisation by finding the most distant cell from the origin at each time step;
we then average these values across our simulations. In the PDE case, we find the furthest voxel with non-zero
density from the origin based on the L∞ distance, after setting the density to zero if it is below single-digit
precision of 10^(−7). (k)–(l) We show the difference between the PDE and EA ABM solutions from (a)–(h) at two
sample times. We overlay cell positions form one ABM simulation to illustrate how the continuous and discrete
solutions are related. In (i) and (j), shaded regions denote plus or minus one standard deviation of the EA ABM
solution.