k unstable modes

Figure 6.32. (a) Typical composite characteristic polynomial for a tissue interaction model mechanism with corresponding dispersion relations in (b). Note how an increase in the interaction parameters, S and T, introduces more complex spatial patterns by exciting other l(k) which have linearly unstable modes.

k unstable modes

Figure 6.32. (a) Typical composite characteristic polynomial for a tissue interaction model mechanism with corresponding dispersion relations in (b). Note how an increase in the interaction parameters, S and T, introduces more complex spatial patterns by exciting other l(k) which have linearly unstable modes.

The dominant modes associated with each (k) are denoted by k and the solution of the linearized system is dominated by where mg (0), mD (0) are the uniform steady states of the coupled system. From this we see that the solution is a superposition of the several dominant unstable modes in Figure 6.32 and since growth of those with wavenumbers k1 and k3 is larger than that for k2 these suggest that the final patern is a superposition of two patterns with wavelengths 2n/k1 and 2n/k3. Numerical simulation of the full nonlinear system studied by Nagorcka et al. (1987), namely, a reaction diffusion system coupled to a mechanical system, showed that this is the case when the pattern formation system is near the bifurcation from homogeneity to structure. The specific mechanical system they used in the dermis is a version of the model (6.22)-(6.24), namely, where the traction depends on the morphogen V from the basal layer of the epidermis and is given by

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