In Appendix A we show that when (n ■ V)u = 0 on dB, f |V2u |2 dr > /j, f jb jb y2'i |2 d r > / ||V u ||2 d r, (2.109)
where \x is the least positive eigenvalue of
V20 + = 0, (n -V)0 = 0 r on dB, where 0 is a scalar. Using the result (2.109) in (2.108) we get dE
— < (m — 2|d)E ^ lim E (t) = o if m < 2 id (2.1io)
and so, once again, if the smallest diffusion coefficient is large enough this implies that Vu ^ o and so all spatial patterns tend to zero as t ^ro.
Othmer (1977) has pointed out that the parameter m defined by (2.ioi) and (2.io7) is a measure of the sensitivity of the reaction rates to changes in u since 1/m is the shortest kinetic relaxation time of the mechanism. On the other hand 1/(2|d) is a measure of the longest diffusion time. So the result (2.11o), which is 1/m > 1/(2|d), then implies that if the shortest relaxation time for the kinetics is greater than the longest diffusion time then all spatial patterning will die out as t ^ ro. The mechanism will then be governed solely by kinetics dynamics. Remember that the solution of the latter can include limit cycle oscillations.
Suppose we consider the one-dimensional situation with a typical embryological domain of interest, say, L = O (1 mm). With d = O (1o—6cm2s—1) the result (2.11o) then implies that homogeneity will result if the shortest relaxation time of the kinetics 1/m > L2/(2n2d), that is, a time of O(5oo s).
Consider the general system (2.1o4) rescaled so that the length scale is 1 and the diffusion coefficients are scaled relative to D\ say. Now return to the formulation used earlier, in (2.1o), for instance, in which the scale y appears with the kinetics in the form yf. The effect of this on the condition (2.11o) now produces y m — 2| < o as the stability requirement. We immediately see from this form that there is a critical y , proportional to the domain area, which in one dimension is (length)2, below which no structure can exist. This is of course a similar result to the one we found in Sections 2.3 and 2.4.
We should reiterate that the results here give qualitative bounds and not estimates for the various parameters associated with the model mechanisms. The evaluation of an appropriate m is not easy. In Sections 2.3 and 2.4 we derived specific quantitative relations between the parameters, when the kinetics were of a particular class, to give spatially structured solutions. The general results in this section, however, apply to all types of kinetics, oscillatory or otherwise, as long as the solutions are bounded.
In this chapter we have dealt primarily with reaction or population interaction kinetics which, in the absence of diffusion, do not exhibit oscillatory behaviour in the restricted regions of parameter space which we have considered. We may ask what kind of spatial structure can be obtained when oscillatory kinetics is coupled with diffusion. We saw in Chapter 1 that such a combination could give rise to travelling wavetrains when the domain is infinite. If the domain is finite we could anticipate a kind of regular sloshing around within the domain which is a reflection of the existence of spatially and temporally unstable modes. This can in fact occur but it is not always so. One case to point is the classical Lotka-Volterra system with equal diffusion coefficients for the species. Murray (1975) showed that in a finite domain all spatial heterogeneities must die out (see Exercise 11).
There are now several pattern formation mechanisms, other than reaction-diffusion-chemotaxis systems. One of the best critical and thorough reviews on models for self-organisation in development is by Wittenberg (1993). He describes the models in detail and compares and critically reviews several of the diverse mechanisms includ ing reaction-diffusion-chemotaxis systems, mechanochemical mechanisms and cellular automaton models.
In the next chapter we shall discuss several specific practical biological pattern formation problems. In later chapters we shall describe other mechanisms which can generate spatial patterns. An important system which has been widely studied is the reaction-diffusion-chemotaxis mechanism for generating aggregation patterns in bacteria and also for slime mould amoebae, one model for which we derived in Chapter 11, Volume I, Section 11.4. Using exactly the same kind of analysis we discussed above for diffusion-driven instability we can show how spatial patterns can arise in these model equations and the conditions on the parameters under which this will happen (see Exercise 9). As mentioned above these chemotaxis systems are becoming increasingly important with the upsurge in interest in bacterial patterns and is the reason for including Chapter 5 below. We discuss other quite different applications of cell-chemotaxis mechanisms in Chapter 4 when we consider the effect of growing domains on patterning, such as the complex patterning observed on snakes.
1. Determine the appropriate nondimensionalisation for the reaction kinetics in (2.4) and (2.5) which result in the forms (2.8).
2. An activator-inhibitor reaction diffusion system in dimensionless form is given by u2 2
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