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Figure 6.9. Qualitative variation in the dispersion relation a(k2) in (6.52) for the model system (6.50) (and

(6.51)) as the traction parameter t increases. The bifurcation values t£ and tc denote the values of t where b(k2) = 0 and c(k2) = 0 respectively. The wavenumbers of the unstable modes are denoted by the heavy line on the k2-axis.

Figure 6.9. Qualitative variation in the dispersion relation a(k2) in (6.52) for the model system (6.50) (and

(6.51)) as the traction parameter t increases. The bifurcation values t£ and tc denote the values of t where b(k2) = 0 and c(k2) = 0 respectively. The wavenumbers of the unstable modes are denoted by the heavy line on the k2-axis.

It is now clear how to investigate various simpler models derived from the more complicated basic model (6.22)-(6.24). Other examples are left as exercises.

Figures 6.10 and 6.11 indicate the richness of dispersion relation types which exist for the class of mechanical models (6.22)-(6.24). Figure 6.10 shows only some of the dispersion relations which have finite ranges of unstable modes while Figure 6.11 exhibits only some of the possible forms with infinite ranges of unstable modes. A nonlinear analysis in the vicinity of bifurcation to spatial heterogeneity, such as has been done by Maini and Murray (1988), can be used on the mechanisms which have a dispersion relation of the form illustrated in Figure 6.10(a). A nonlinear theory for models with dispersion relations with an infinite range of unstable modes, such as those in Figure 6.11, is, as we noted, still lacking as is that for dispersion relations which exhibit infinite growth modes (Figures 6.10(b),(e)-(g)). Although we anticipate that the pattern will depend more critically on initial conditions than in the finite range of unstable mode situations, this has also not been established.

Mechanical models, as we noted above, are also capable of generating travelling waves: these are indicated by dispersion relations with complex a. Table 6.3 gives examples of models which admit such solutions.

From a biological application point of view two- and three-dimensional patterns are naturally of great interest. With the experience gained from the study of the numerous reaction diffusion chemotaxis and neural models in the book, we expect the simulated

Figure 6.10. Examples of dispersion relations a(k ), obtained from (6.32) with (6.31) for mechanical models based on the mechanism (6.22)-(6.24). The various forms correspond to the specific conditions listed in Table 6.1. Realistic models for those with infinite growth must be treated as singular perturbation problems, with small values for the appropriate parameters in terms which have been omitted so as to make the linear growth finite although large.

Figure 6.10. Examples of dispersion relations a(k ), obtained from (6.32) with (6.31) for mechanical models based on the mechanism (6.22)-(6.24). The various forms correspond to the specific conditions listed in Table 6.1. Realistic models for those with infinite growth must be treated as singular perturbation problems, with small values for the appropriate parameters in terms which have been omitted so as to make the linear growth finite although large.

Table 6.1. Mechanical models, derived from the basic system (6.22)-(6.24) with positive nonzero parameters denoted by •, which have dispersion relations with a finite range of unstable wavenumbers. The corresponding dispersion relation forms are given in Figure 6.10, with k = 0, T1 = T2 = t .

Figure 6.10

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