Fourier modes and dispersion relations
(It's not you: this entry makes little sense. One day it might.)
The natural mathematical framework for the analysis of Turing's spot-formation theory comes from the idea of Fourier modes. In order to make sense of the pictures in the archive it's necessary to understand a little of this theory, at least graphically.
Fourier modes and dispersion relations
Figure 6 A dispersion relation.
Consider a pattern like that of Figure 5: this can be thought of as a wave rising and falling around the ring four times. This number of times that the wave pattern fits around the ring is called the wavenumber, and we call it k. We can imagine other patterns with wavenumbers of none, one, two, three, and so on. Now suppose we set up an island with an initial pattern of each of these wavenumbers, and allow the reaction-diffusion dynamics to act. Over time, each of these patterns will either grow or decline, and we can measure this rate of growth and plot it against the wavenumber (Figure 6). This graph is called the dispersion relation. Clearly, it is those patterns which grow fastest in time which will be most visible. Thus the highest region of the curve (or more precisely the highest point corresponding to an integer wavenumber) gives the wavenumber, and thus the characteristic length scale, of the pattern which the reaction-diffusion mechanism will generate.
'Cooking' the Turing instability
Figure 7 Cooking the Turing instability
This simple description of the Turing instability explains where a characteristic length scale emerges, but things are a little more complicated. For example, given that all waves of the maximal wavenumber will be growing at an equal rate, and that perturbations are equally likely to occur at all places round the ring, the model can't predict the phase of the pattern: that is it might say there will be four troughs around the ring but it can't say where they will begin. Another complexity is that the simple model predicts that patterns will go on growing forever: in order to prevent this we have to change the conditions after a while. These and other questions made it necessary for Turing to introduce the notion of 'cooking'. As shown in Figure 7, this is simply increasing the dispersion relation, so that the system moves from having no unstable mode, to having one and then possibly more.
In two dimensions, what happens to the dispersion relation? For a one dimensional lattice, we saw that patterns on rings could be described by wavenumbers. What happens when we widen the beach: that is when we consider two dimensional lattices on cylinders? In particular, Turing considered very wide beaches, that is infinitely tall cylinders. To describe a pattern on a cylinder, we still have a wavenumber describing how many times the pattern repeats around the cylinder; but we must also have a number describing how frequently it repeats vertically: unlike the around-wavenumber, this isn't constrained to be an integer, although if the pattern is a lattice it will be. Thus the analogue in two-dimensions of the dispersion-relation is the dispersion plot.
There are examples of these two-dimensional Fourier pictures in the archive:
