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Numerical solutions were obtained for two sets of parameters: one from the upper stripes region, and one from the spots region. Simulations were performed on square domains just large enough to hold one full period of the pattern. This required the choice lx — ly —

So, we expected to obtain one stripe for the stripe parameters and one spot for the spot parameters. These were found, and typical results for the cell density are shown in Figure 5.14. Each simulation has two plots which indicate the initial conditions and the steady state pattern.

Note that since the boundary conditions are periodic, the maximum of the spot or stripe pattern can occur anywhere in the domain. Also, for the stripe pattern, the orientation of the stripe is not determined from the analysis, since we are working with a square domain. So, for one set of random initial conditions the stripe will appear vertical, whereas for another set it will appear horizontal.

Figure 5.14. Simulation results from the simple model parameter domain where spots were predicted in Figure 5.13. Cell density is plotted as an image at t = 0 and at steady state. Cell density profiles are shown to demonstrate that the pattern has indeed reached a steady state, and the predicted cell density profile is also shown for comparison. The perturbation was s = 0.1. The times given are in nondimensional units. White indicates high cell density, black the opposite. The parameter values are: (a) du = 0.25, a = 1.50, ß = 0.1, S = 15.0, p = 0.01, w = 1.0, u0 = u* = 15.0, v0 = v* = 1.50, k* = 3.0 and lx = ly = l = 3.6276 (domain which can sustain one 2n oscillation), a spot.

Figure 5.14. Simulation results from the simple model parameter domain where spots were predicted in Figure 5.13. Cell density is plotted as an image at t = 0 and at steady state. Cell density profiles are shown to demonstrate that the pattern has indeed reached a steady state, and the predicted cell density profile is also shown for comparison. The perturbation was s = 0.1. The times given are in nondimensional units. White indicates high cell density, black the opposite. The parameter values are: (a) du = 0.25, a = 1.50, ß = 0.1, S = 15.0, p = 0.01, w = 1.0, u0 = u* = 15.0, v0 = v* = 1.50, k* = 3.0 and lx = ly = l = 3.6276 (domain which can sustain one 2n oscillation), a spot.

The Full Model

We consider now the more biologically accurate model d U dt dv ~dt

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